Transformations of Measuring Scales
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Measures of Relative Location - Standard Score
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Measures of Relative Location - Percentiles in Categories
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Percentiles in a Discrete Frequencies Table
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Linear Transformations
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The Probability Function
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Expectation, Variance, and Standard Deviation
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- Introduction
- Example 1
- Exercise 1
- Exercise 2
- Exercise 3 - Part a
- Exercise 3 - Part b
- Exercise 4 - Part a
- Exercise 4 - Part b
- Exercise 5
- Exercise 6
- Exercise 7
- Binomial Probability Density Function
- Example 8 - Part a
- Example 8 - Parts b-c
- Exercise 8 - Parts a-c
- Exercise 8 - Parts d-f
- Exercise 9
- Exercise 10 - Parts a-b
- Exercise 10 - Parts c-d
- Exercise 11
- Exercise 12
- Exercise 13 - Parts a-b
- Exercise 13 - Part c
- Exercise 14
- Exercise 15
- Uniform Probability Density Function
- Exercise 16
- Exercise 17
- Exercise 18
- Exercise 19
- Exercise 20
- Exercise 21

General Probabilities without Integrals
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General Probabilities with Integrals
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- Tutorial 1
- Tutorial 2
- Exercise 1 - Parts a-b
- Exercise 1 - Parts c-e
- Exercise 2 - Parts a-b
- Exercise 2 - Part c
- Exercise 3 - Parts a-b
- Exercise 3 - Parts c-d
- Exercise 4 - Parts a-b
- Exercise 4 - Parts c-e
- Exercise 5
- Exercise 6 - Parts a-b
- Exercise 6 - Part c
- Exercise 7 - Part a
- Exercise 7 - Parts b-c
- Exercise 8

Normal Probability
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- Tutorial
- Example - Part a
- Example - Parts b-c
- Example - Part d
- Exercise 1 - Part a
- Exercise 1 - Parts b-c
- Exercise 1 - Parts d-e
- Exercise 2 - Part a
- Exercise 2 - Parts b-c
- Exercise 2 - Part d
- Exercise 3 - Part a
- Exercise 3 - Parts b-c
- Exercise 3 - Parts d-e
- Exercise 4 - Part a
- Exercise 4 - Parts b-c
- Exercise 5 - Parts a-b
- Exercise 5 - Parts c-e
- Exercise 6 - Parts a-b
- Exercise 6 - Part c
- Exercise 7 - Part a-b
- Exercise 7 - Parts c-d
- Exercise 7 - Part e
- Exercise 8
- Exercise 9 - Parts a-b
- Exercise 9 - Parts c-d
- Exercise 10 - Part a
- Exercise 10 - Parts b-c
- Exercise 11 - Part a
- Exercise 11 - Part b
- Exercise 11 - Parts c-d

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[{"Name":"Transformations of Measuring Scales","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"7m 51s","ChapterTopicVideoID":25433,"CourseChapterTopicPlaylistID":64519,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"In this chapter, we\u0027ll be talking about transformations of measuring scales."},{"Start":"00:04.320 ","End":"00:09.270","Text":"Now, a transformation is an operation that changes the values of a variable."},{"Start":"00:09.270 ","End":"00:14.235","Text":"For example, I have a variable and I collect data on that variable,"},{"Start":"00:14.235 ","End":"00:16.095","Text":"and for some reason,"},{"Start":"00:16.095 ","End":"00:19.560","Text":"I\u0027d like to change the values of the data that I\u0027ve collected."},{"Start":"00:19.560 ","End":"00:21.945","Text":"For example, if I\u0027ve collected,"},{"Start":"00:21.945 ","End":"00:28.379","Text":"let\u0027s say weights in kilograms and now I want to change the data into grams."},{"Start":"00:28.379 ","End":"00:31.110","Text":"Or if I have numerical data and I want to"},{"Start":"00:31.110 ","End":"00:34.260","Text":"take the square roots of the values for some reason."},{"Start":"00:34.260 ","End":"00:37.060","Text":"So these are valid transformations."},{"Start":"00:37.060 ","End":"00:40.765","Text":"Now, the following, right and we\u0027ll be discussing that,"},{"Start":"00:40.765 ","End":"00:45.440","Text":"are valid transformations that may be performed for each measuring scale."},{"Start":"00:45.440 ","End":"00:48.020","Text":"Now let\u0027s take a look at the first measuring scale."},{"Start":"00:48.020 ","End":"00:49.475","Text":"That\u0027s a nominal scale."},{"Start":"00:49.475 ","End":"00:52.670","Text":"Now, a valid transformation for"},{"Start":"00:52.670 ","End":"00:56.615","Text":"a nominal scale is a transformation that preserves identity."},{"Start":"00:56.615 ","End":"00:58.295","Text":"Now, what does that mean?"},{"Start":"00:58.295 ","End":"01:02.930","Text":"Well, let\u0027s take an example of bus routes."},{"Start":"01:02.930 ","End":"01:08.165","Text":"Now, let\u0027s say I have 3 bus routes."},{"Start":"01:08.165 ","End":"01:11.690","Text":"Number 4, number 5, and number 25."},{"Start":"01:11.690 ","End":"01:16.760","Text":"Now, what I want to do is I want to transform these values right here."},{"Start":"01:16.760 ","End":"01:22.730","Text":"As we see, this is a nominal variable. Why is that?"},{"Start":"01:22.730 ","End":"01:26.300","Text":"Well, because 4 is just the identity of"},{"Start":"01:26.300 ","End":"01:31.850","Text":"the bus route or the bus that I\u0027m taking in order to get from one place to another."},{"Start":"01:31.850 ","End":"01:33.440","Text":"5 is another bus route,"},{"Start":"01:33.440 ","End":"01:35.210","Text":"25 is another bus route."},{"Start":"01:35.210 ","End":"01:38.382","Text":"There\u0027s no meaning of more or less here,"},{"Start":"01:38.382 ","End":"01:41.420","Text":"there\u0027s no quantitative context,"},{"Start":"01:41.420 ","End":"01:46.655","Text":"5 is not larger than 4 with respect to bus routes."},{"Start":"01:46.655 ","End":"01:49.850","Text":"This then is a nominal variable."},{"Start":"01:49.850 ","End":"01:52.909","Text":"Now what I want to do is I want to take the values"},{"Start":"01:52.909 ","End":"01:56.345","Text":"here and I want to transform them into other values."},{"Start":"01:56.345 ","End":"01:59.750","Text":"Let\u0027s say I wanted to take the number 1 and I want to"},{"Start":"01:59.750 ","End":"02:04.230","Text":"add that 1 to the left of these numbers."},{"Start":"02:04.230 ","End":"02:07.230","Text":"So 4 now becomes 14,"},{"Start":"02:07.230 ","End":"02:12.715","Text":"5 becomes 15, and 25 becomes 125."},{"Start":"02:12.715 ","End":"02:17.015","Text":"Well, did I preserve the identity of the bus routes?"},{"Start":"02:17.015 ","End":"02:18.440","Text":"Well, of course I did."},{"Start":"02:18.440 ","End":"02:21.650","Text":"Because if I get on bus number 14, well,"},{"Start":"02:21.650 ","End":"02:26.820","Text":"that\u0027s the same thing as getting on bus number 4,"},{"Start":"02:26.820 ","End":"02:31.175","Text":"where 4 was the old name of the bus."},{"Start":"02:31.175 ","End":"02:33.075","Text":"If I got on bus number 15,"},{"Start":"02:33.075 ","End":"02:35.600","Text":"where previously I would have gotten on bus number 5."},{"Start":"02:35.600 ","End":"02:38.075","Text":"It\u0027s the same route, just different name."},{"Start":"02:38.075 ","End":"02:41.685","Text":"Right. And likewise for 125,"},{"Start":"02:41.685 ","End":"02:44.210","Text":"if I got on bus number 125,"},{"Start":"02:44.210 ","End":"02:48.349","Text":"it would be the same if I would\u0027ve gotten on bus number 25."},{"Start":"02:48.349 ","End":"02:51.005","Text":"All that\u0027s changed is the name."},{"Start":"02:51.005 ","End":"02:53.270","Text":"The identity remained the same,"},{"Start":"02:53.270 ","End":"02:55.100","Text":"but the name has been changed."},{"Start":"02:55.100 ","End":"02:59.594","Text":"So, therefore, transformations that preserve identity"},{"Start":"02:59.594 ","End":"03:05.850","Text":"is a valid transformation for nominal measuring scale."},{"Start":"03:05.850 ","End":"03:09.490","Text":"Let\u0027s take a look at the ordinal measuring scale."},{"Start":"03:09.490 ","End":"03:12.400","Text":"What valid transformations can we do here?"},{"Start":"03:12.400 ","End":"03:16.825","Text":"Well, valid transformations are transformations that preserve order."},{"Start":"03:16.825 ","End":"03:19.690","Text":"Now, in the previous measuring scale,"},{"Start":"03:19.690 ","End":"03:21.100","Text":"the nominal measuring scale,"},{"Start":"03:21.100 ","End":"03:26.140","Text":"what we were interested in is transformations that preserved identity only."},{"Start":"03:26.140 ","End":"03:30.955","Text":"Well here we\u0027re adding to that and we want to preserve order as well."},{"Start":"03:30.955 ","End":"03:32.650","Text":"Well, let\u0027s take an example."},{"Start":"03:32.650 ","End":"03:41.635","Text":"Let\u0027s say I\u0027m a psychologist and I want to measure the motivation of employees at work."},{"Start":"03:41.635 ","End":"03:43.210","Text":"So I create a scale,"},{"Start":"03:43.210 ","End":"03:46.235","Text":"1, 2, and 3,"},{"Start":"03:46.235 ","End":"03:51.625","Text":"where 3 is person with a high level of motivation,"},{"Start":"03:51.625 ","End":"03:54.290","Text":"2 with a medium level of motivation,"},{"Start":"03:54.290 ","End":"03:56.720","Text":"and 1 with a low level of motivation."},{"Start":"03:56.720 ","End":"04:00.439","Text":"We can see that this is an ordinal variable."},{"Start":"04:00.439 ","End":"04:03.590","Text":"We have here nominal characteristics."},{"Start":"04:03.590 ","End":"04:08.175","Text":"1 is the name for the low level of motivation,"},{"Start":"04:08.175 ","End":"04:11.300","Text":"2 for the medium, and 3 for the high level of motivation."},{"Start":"04:11.300 ","End":"04:15.980","Text":"Now, apart from that, we have order."},{"Start":"04:15.980 ","End":"04:17.660","Text":"An order is defined here."},{"Start":"04:17.660 ","End":"04:20.900","Text":"Somebody with a value of 3 is a more"},{"Start":"04:20.900 ","End":"04:25.425","Text":"motivated and somebody with a value of 2 and likewise between 2 and 1."},{"Start":"04:25.425 ","End":"04:30.200","Text":"We have here order between the various values."},{"Start":"04:30.200 ","End":"04:37.745","Text":"Now, what I want to do is I want to transform these values into other values."},{"Start":"04:37.745 ","End":"04:39.200","Text":"For example, 0,"},{"Start":"04:39.200 ","End":"04:41.220","Text":"5, and 10."},{"Start":"04:41.220 ","End":"04:44.250","Text":"1 would be transformed into the value 0,"},{"Start":"04:44.250 ","End":"04:46.700","Text":"2 will be transformed into the value 5,"},{"Start":"04:46.700 ","End":"04:50.375","Text":"and 3 would be transformed into the value of 10."},{"Start":"04:50.375 ","End":"04:53.750","Text":"Now, again, what do I have here?"},{"Start":"04:53.750 ","End":"04:55.805","Text":"Did I preserve the order?"},{"Start":"04:55.805 ","End":"04:57.800","Text":"Well, of course, I did right."},{"Start":"04:57.800 ","End":"04:59.420","Text":"10 is greater than 5,"},{"Start":"04:59.420 ","End":"05:01.175","Text":"and 5 is greater than 0."},{"Start":"05:01.175 ","End":"05:03.425","Text":"If a person gets 10,"},{"Start":"05:03.425 ","End":"05:07.490","Text":"he\u0027s still identified as somebody having"},{"Start":"05:07.490 ","End":"05:12.920","Text":"a high level of motivation which is greater than somebody receiving a value of 5,"},{"Start":"05:12.920 ","End":"05:15.725","Text":"which has only a medium level of motivation."},{"Start":"05:15.725 ","End":"05:21.725","Text":"So this is a valid transformation for the ordinal measuring scale."},{"Start":"05:21.725 ","End":"05:24.950","Text":"Next is the interval measuring scale."},{"Start":"05:24.950 ","End":"05:30.080","Text":"Now here, valid transformation would be a positive linear transformation."},{"Start":"05:30.080 ","End":"05:31.415","Text":"What do I mean by that?"},{"Start":"05:31.415 ","End":"05:35.465","Text":"Well, a linear transformation that equals to y,"},{"Start":"05:35.465 ","End":"05:38.201","Text":"equaling to a x plus b,"},{"Start":"05:38.201 ","End":"05:40.190","Text":"that\u0027s a linear transformation."},{"Start":"05:40.190 ","End":"05:41.660","Text":"We take the old variable,"},{"Start":"05:41.660 ","End":"05:43.670","Text":"we multiply it by a constant,"},{"Start":"05:43.670 ","End":"05:45.440","Text":"add another constant to it,"},{"Start":"05:45.440 ","End":"05:50.045","Text":"and we get a new or a transformed value here."},{"Start":"05:50.045 ","End":"05:52.610","Text":"Now what\u0027s a positive linear transformation?"},{"Start":"05:52.610 ","End":"05:54.095","Text":"Well, we take this template,"},{"Start":"05:54.095 ","End":"05:59.180","Text":"but a here the x multiplier has to be greater than 0."},{"Start":"05:59.180 ","End":"06:01.295","Text":"If a is greater than 0,"},{"Start":"06:01.295 ","End":"06:05.150","Text":"then we\u0027re talking about a positive linear transformation, b on the other hand,"},{"Start":"06:05.150 ","End":"06:07.475","Text":"can be any number whatsoever,"},{"Start":"06:07.475 ","End":"06:09.860","Text":"but as long as a is positive,"},{"Start":"06:09.860 ","End":"06:12.635","Text":"then we have a positive linear transformation."},{"Start":"06:12.635 ","End":"06:14.540","Text":"Let\u0027s just take an example here."},{"Start":"06:14.540 ","End":"06:16.850","Text":"Let\u0027s say I have temperature in Fahrenheit,"},{"Start":"06:16.850 ","End":"06:22.505","Text":"and I want to transform that into temperatures in centigrade. How do I do that?"},{"Start":"06:22.505 ","End":"06:31.685","Text":"Well, the equation is this centigrade equals 5 divided by 9 times Fahrenheit minus 32."},{"Start":"06:31.685 ","End":"06:39.170","Text":"We have here a equals to 5 divided by 9 and b equals minus 32."},{"Start":"06:39.170 ","End":"06:40.880","Text":"So there we go."},{"Start":"06:40.880 ","End":"06:45.335","Text":"This then is the positive linear transformation and that\u0027s"},{"Start":"06:45.335 ","End":"06:51.545","Text":"a valid transformation for the interval measuring scale."},{"Start":"06:51.545 ","End":"06:55.670","Text":"Lastly, we have the racial measuring scale."},{"Start":"06:55.670 ","End":"06:57.980","Text":"Now what\u0027s a valid transformation here?"},{"Start":"06:57.980 ","End":"07:02.705","Text":"Well, a transformation would be multiplying or dividing by a positive number."},{"Start":"07:02.705 ","End":"07:04.730","Text":"So if I have x,"},{"Start":"07:04.730 ","End":"07:08.570","Text":"my old value, and I want to transform that into y."},{"Start":"07:08.570 ","End":"07:15.290","Text":"Well, the only way it can do this is to take y and that would be equal to a times x,"},{"Start":"07:15.290 ","End":"07:17.510","Text":"where a is a positive number,"},{"Start":"07:17.510 ","End":"07:18.965","Text":"a would be greater than 0."},{"Start":"07:18.965 ","End":"07:20.930","Text":"Now, let\u0027s just take an example here."},{"Start":"07:20.930 ","End":"07:24.830","Text":"Let\u0027s say x is weight in kilograms,"},{"Start":"07:24.830 ","End":"07:27.905","Text":"and I want to transform that into y,"},{"Start":"07:27.905 ","End":"07:30.170","Text":"which is in grams."},{"Start":"07:30.170 ","End":"07:32.000","Text":"Now, how do I do that?"},{"Start":"07:32.000 ","End":"07:37.590","Text":"Well then that\u0027ll be y would be equal to 1000 times x,"},{"Start":"07:37.590 ","End":"07:39.865","Text":"where a equals 1000."},{"Start":"07:39.865 ","End":"07:44.870","Text":"Now, this then would be a valid transformation when"},{"Start":"07:44.870 ","End":"07:51.150","Text":"dealing with a variable in the ratio measuring scale."}],"ID":26260},{"Watched":false,"Name":"Exercise 1","Duration":"4m 48s","ChapterTopicVideoID":12883,"CourseChapterTopicPlaylistID":64519,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12883.jpeg","UploadDate":"2018-07-30T21:14:53.6470000","DurationForVideoObject":"PT4M48S","Description":null,"MetaTitle":"Exercise 1 - Transformations of Measuring Scales: Practice Makes Perfect | Proprep","MetaDescription":"Studied the topic name and want to practice? Here are some exercises on Transformations of Measuring Scales practice questions for you to maximize your understanding.","Canonical":"https://www.proprep.uk/general-modules/all/statistics/data-distributions-and-random-variables/transformations-of-measuring-scales/vid13358","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"In this question, we\u0027re asked to state for which measuring"},{"Start":"00:02.730 ","End":"00:05.580","Text":"scales the following transformations are permitted."},{"Start":"00:05.580 ","End":"00:08.160","Text":"The first 1 is multiplying by 0."},{"Start":"00:08.160 ","End":"00:12.404","Text":"We want to know whether this is a valid transformation for the nominal,"},{"Start":"00:12.404 ","End":"00:13.920","Text":"ordinal, interval,"},{"Start":"00:13.920 ","End":"00:16.275","Text":"and ratio measuring scale."},{"Start":"00:16.275 ","End":"00:18.930","Text":"Let\u0027s take our example of the bus routes."},{"Start":"00:18.930 ","End":"00:22.020","Text":"We had bus route numbers 4 and bus route number 5."},{"Start":"00:22.020 ","End":"00:25.720","Text":"Now, if we multiply these values by 0,"},{"Start":"00:25.730 ","End":"00:29.370","Text":"then 4 times 0 equals 0,"},{"Start":"00:29.370 ","End":"00:31.170","Text":"5 times 0 equals 0."},{"Start":"00:31.170 ","End":"00:35.990","Text":"Now, did we retain any form of identity of the bus routes?"},{"Start":"00:35.990 ","End":"00:38.390","Text":"Well, we didn\u0027t because we\u0027ve created"},{"Start":"00:38.390 ","End":"00:43.070","Text":"a transformation where all of the bus routes are now 0."},{"Start":"00:43.070 ","End":"00:45.860","Text":"There\u0027s no differentiation between the bus routes."},{"Start":"00:45.860 ","End":"00:53.105","Text":"That means that this is not a relevant transformation for the nominal measuring scale."},{"Start":"00:53.105 ","End":"00:58.520","Text":"As such, then it wouldn\u0027t be a valid transformation for the ordinal,"},{"Start":"00:58.520 ","End":"01:01.615","Text":"interval, and ratio measuring scale."},{"Start":"01:01.615 ","End":"01:05.750","Text":"Now, let\u0027s take a look at this transformation multiplying by 2."},{"Start":"01:05.750 ","End":"01:10.295","Text":"We have our variable x and we want to multiply it by 2."},{"Start":"01:10.295 ","End":"01:16.250","Text":"Now, this is a valid transformation for the ratio measuring scale."},{"Start":"01:16.250 ","End":"01:22.399","Text":"As such, it\u0027s also valid for the interval,"},{"Start":"01:22.399 ","End":"01:26.615","Text":"ordinal, and nominal measuring scale as well."},{"Start":"01:26.615 ","End":"01:29.540","Text":"What about multiplying by minus 1?"},{"Start":"01:29.540 ","End":"01:35.065","Text":"Well, let\u0027s take a look at our example of the motivation scale where we had 1, 2,"},{"Start":"01:35.065 ","End":"01:38.990","Text":"and 3, where 1 was the low motivation level,"},{"Start":"01:38.990 ","End":"01:41.885","Text":"2 was medium, and 3 was high motivation level."},{"Start":"01:41.885 ","End":"01:47.510","Text":"If we multiply these values by minus 1, then what will happen?"},{"Start":"01:47.510 ","End":"01:48.920","Text":"This becomes minus 1,"},{"Start":"01:48.920 ","End":"01:50.315","Text":"this becomes minus 2,"},{"Start":"01:50.315 ","End":"01:52.130","Text":"and this becomes minus 3."},{"Start":"01:52.130 ","End":"01:57.630","Text":"Before the transformation, we had a semblance of order here."},{"Start":"01:57.630 ","End":"02:03.980","Text":"We had an increasing level of motivation from 1-3."},{"Start":"02:03.980 ","End":"02:07.330","Text":"That means that 3 was the highest motivation."},{"Start":"02:07.330 ","End":"02:09.340","Text":"By multiplying by minus 1,"},{"Start":"02:09.340 ","End":"02:15.095","Text":"the highest motivation now became the lowest value."},{"Start":"02:15.095 ","End":"02:18.965","Text":"That means that we\u0027ve just switched the order around."},{"Start":"02:18.965 ","End":"02:22.400","Text":"That means that this transformation did"},{"Start":"02:22.400 ","End":"02:28.435","Text":"not retain the order of the original values and as such,"},{"Start":"02:28.435 ","End":"02:34.790","Text":"this transformation is not a valid transformation for the ordinal measuring scale."},{"Start":"02:34.790 ","End":"02:38.870","Text":"That means that it\u0027s not a valid transformation for"},{"Start":"02:38.870 ","End":"02:43.175","Text":"the interval or the ratio measuring scale as well,"},{"Start":"02:43.175 ","End":"02:47.240","Text":"but it is valid for the nominal measuring scale."},{"Start":"02:47.240 ","End":"02:49.910","Text":"What about adding 3?"},{"Start":"02:49.910 ","End":"02:54.885","Text":"If we have x, our original variable,"},{"Start":"02:54.885 ","End":"02:56.925","Text":"and we add 3 to it,"},{"Start":"02:56.925 ","End":"02:58.470","Text":"if we add 3,"},{"Start":"02:58.470 ","End":"03:00.915","Text":"then what happens now?"},{"Start":"03:00.915 ","End":"03:03.515","Text":"Let\u0027s take a look at a linear transformation,"},{"Start":"03:03.515 ","End":"03:06.380","Text":"y equals a x plus b."},{"Start":"03:06.380 ","End":"03:07.955","Text":"In our case here,"},{"Start":"03:07.955 ","End":"03:09.545","Text":"a would be equal to 1,"},{"Start":"03:09.545 ","End":"03:11.240","Text":"that\u0027s 1 times x,"},{"Start":"03:11.240 ","End":"03:14.320","Text":"and b would be equal to 3."},{"Start":"03:14.320 ","End":"03:21.125","Text":"This is obviously a valid transformation for the interval measuring scale."},{"Start":"03:21.125 ","End":"03:22.490","Text":"What about ratio?"},{"Start":"03:22.490 ","End":"03:25.600","Text":"Is it a valid transformation for the ratio measuring scale?"},{"Start":"03:25.600 ","End":"03:30.020","Text":"It\u0027s not because the transformation for a ratio measuring"},{"Start":"03:30.020 ","End":"03:37.820","Text":"scale needs for x to be only multiplied by positive integer."},{"Start":"03:37.820 ","End":"03:39.740","Text":"We can\u0027t do anything else to it."},{"Start":"03:39.740 ","End":"03:42.485","Text":"We can\u0027t add another constant to it."},{"Start":"03:42.485 ","End":"03:45.020","Text":"It\u0027s not a valid transformation here."},{"Start":"03:45.020 ","End":"03:49.265","Text":"But since it\u0027s a valid transformation for the interval measuring scale,"},{"Start":"03:49.265 ","End":"03:53.615","Text":"it\u0027s also a valid transformation for the ordinal and nominal."},{"Start":"03:53.615 ","End":"03:55.715","Text":"What about subtracting 3?"},{"Start":"03:55.715 ","End":"03:59.270","Text":"Again, y equals ax plus b."},{"Start":"03:59.270 ","End":"04:02.780","Text":"That\u0027s our linear transformation here, that\u0027s the template."},{"Start":"04:02.780 ","End":"04:04.310","Text":"But in our case now,"},{"Start":"04:04.310 ","End":"04:10.125","Text":"a would be equal to 1 and b would be equal to minus 3 as opposed to plus 3 here."},{"Start":"04:10.125 ","End":"04:13.775","Text":"It really doesn\u0027t matter what the value of b is."},{"Start":"04:13.775 ","End":"04:18.230","Text":"What matters is what we multiply our x with,"},{"Start":"04:18.230 ","End":"04:21.950","Text":"whether we multiply it by a positive integer or a negative integer."},{"Start":"04:21.950 ","End":"04:25.117","Text":"Now, in both cases and specifically here,"},{"Start":"04:25.117 ","End":"04:28.040","Text":"we multiply x by positive integer."},{"Start":"04:28.040 ","End":"04:36.965","Text":"The answer to this transformation would be the same as the transformation of adding 3."},{"Start":"04:36.965 ","End":"04:41.000","Text":"It\u0027s not a valid transformation for the ratio measuring scale,"},{"Start":"04:41.000 ","End":"04:43.940","Text":"but it is a valid transformation for interval,"},{"Start":"04:43.940 ","End":"04:48.180","Text":"ordinal, and nominal measuring scales."}],"ID":13358},{"Watched":false,"Name":"Exercise 2","Duration":"2m 8s","ChapterTopicVideoID":12884,"CourseChapterTopicPlaylistID":64519,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.755","Text":"This is a multiple choice question where we\u0027re asked,"},{"Start":"00:02.755 ","End":"00:05.160","Text":"which transformation preserves the measuring scale"},{"Start":"00:05.160 ","End":"00:07.650","Text":"of the variable \"the temperature in the Oval Office?"},{"Start":"00:07.650 ","End":"00:10.005","Text":"Is it a transformation that preserves order,"},{"Start":"00:10.005 ","End":"00:12.075","Text":"a positive linear transformation,"},{"Start":"00:12.075 ","End":"00:14.220","Text":"a transformation that preserves ratio,"},{"Start":"00:14.220 ","End":"00:17.400","Text":"or maybe both B and C are correct?"},{"Start":"00:17.400 ","End":"00:20.985","Text":"Well, first of all, let\u0027s look at the variable temperature."},{"Start":"00:20.985 ","End":"00:25.405","Text":"From discussions here in this chapter and the previous chapters,"},{"Start":"00:25.405 ","End":"00:30.425","Text":"we saw that temperature belongs to the interval measuring scale."},{"Start":"00:30.425 ","End":"00:33.650","Text":"Now, why not the ratio measuring scale?"},{"Start":"00:33.650 ","End":"00:38.240","Text":"Well, it just doesn\u0027t pass the 0 test. What does that mean?"},{"Start":"00:38.240 ","End":"00:40.505","Text":"In the ratio measuring scale,"},{"Start":"00:40.505 ","End":"00:43.920","Text":"a 0 means that there\u0027s an absence of something."},{"Start":"00:44.570 ","End":"00:47.480","Text":"If we measure temperature at 0,"},{"Start":"00:47.480 ","End":"00:50.705","Text":"it would depend on the scale that we\u0027re using."},{"Start":"00:50.705 ","End":"00:52.280","Text":"If we\u0027re using the centigrade scale,"},{"Start":"00:52.280 ","End":"00:56.645","Text":"a 0 measure of temperature means that water freezes,"},{"Start":"00:56.645 ","End":"01:01.435","Text":"which is totally different if we measure 0 on the Fahrenheit scale."},{"Start":"01:01.435 ","End":"01:07.227","Text":"Now that we know that temperature is an interval variable,"},{"Start":"01:07.227 ","End":"01:13.670","Text":"well, let\u0027s just review the valid transformations for the measuring scales."},{"Start":"01:13.670 ","End":"01:17.315","Text":"Here we have the measuring scales, nominal, ordinal,"},{"Start":"01:17.315 ","End":"01:21.080","Text":"interval and ratio, and these are the valid transformation."},{"Start":"01:21.080 ","End":"01:24.125","Text":"For the interval measuring scale,"},{"Start":"01:24.125 ","End":"01:27.470","Text":"a valid transformation is a positive linear transformations."},{"Start":"01:27.470 ","End":"01:30.590","Text":"That means answer B is correct."},{"Start":"01:30.590 ","End":"01:35.270","Text":"Now, what about transformation that preserves the ratio?"},{"Start":"01:35.270 ","End":"01:38.667","Text":"We know the transformations that preserve ratio"},{"Start":"01:38.667 ","End":"01:43.010","Text":"also preserve lower-level measuring scales."},{"Start":"01:43.010 ","End":"01:47.720","Text":"For example, the transformations that preserve the ratio,"},{"Start":"01:47.720 ","End":"01:49.220","Text":"well, they preserve interval,"},{"Start":"01:49.220 ","End":"01:51.020","Text":"ordinal, and nominal as well."},{"Start":"01:51.020 ","End":"01:57.480","Text":"A transformation that preserves ratio also preserves interval,"},{"Start":"01:57.480 ","End":"02:01.910","Text":"that means that both b and c are correct answers."},{"Start":"02:01.910 ","End":"02:04.280","Text":"We have to choose d here,"},{"Start":"02:04.280 ","End":"02:07.980","Text":"which means that answers b and c are correct."}],"ID":13359},{"Watched":false,"Name":"Exercise 3","Duration":"2m 9s","ChapterTopicVideoID":12885,"CourseChapterTopicPlaylistID":64519,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"This is a multiple choice question where we\u0027re asked,"},{"Start":"00:02.820 ","End":"00:06.780","Text":"in which measuring scale is subtracting a constant from each number permitted?"},{"Start":"00:06.780 ","End":"00:11.850","Text":"So we\u0027re looking at a transformation where y equals to"},{"Start":"00:11.850 ","End":"00:19.360","Text":"x minus c. We have our original data and we\u0027re subtracting a constant from each number."},{"Start":"00:19.360 ","End":"00:24.830","Text":"Now, let\u0027s just review what are the valid transformations for each measuring scale here."},{"Start":"00:24.830 ","End":"00:27.710","Text":"Now, let\u0027s take a look at the ratio measuring scale."},{"Start":"00:27.710 ","End":"00:33.080","Text":"Now here, a valid transformation would be multiplying or dividing by a positive number."},{"Start":"00:33.080 ","End":"00:35.315","Text":"This isn\u0027t what we have here."},{"Start":"00:35.315 ","End":"00:40.610","Text":"This transformation is not valid for the ratio measuring scale,"},{"Start":"00:40.610 ","End":"00:44.195","Text":"so that means that answer D is wrong."},{"Start":"00:44.195 ","End":"00:48.485","Text":"Now, let\u0027s take a look at the interval measuring scale here."},{"Start":"00:48.485 ","End":"00:53.765","Text":"Now, a valid transformation is a positive linear transformation. What does that mean?"},{"Start":"00:53.765 ","End":"00:57.695","Text":"That means that we have a linear transformation,"},{"Start":"00:57.695 ","End":"01:01.940","Text":"which is y equaling ax plus b."},{"Start":"01:01.940 ","End":"01:04.580","Text":"That\u0027s a linear transformation on x."},{"Start":"01:04.580 ","End":"01:06.800","Text":"Now, what\u0027s a positive linear transformation?"},{"Start":"01:06.800 ","End":"01:09.245","Text":"That means that a, the x multiplier,"},{"Start":"01:09.245 ","End":"01:11.290","Text":"has to be positive."},{"Start":"01:11.290 ","End":"01:14.525","Text":"What do we have here in our transformation?"},{"Start":"01:14.525 ","End":"01:23.115","Text":"Well, we can write this as 1 times x minus c. Here we have our x multiplier,"},{"Start":"01:23.115 ","End":"01:26.130","Text":"that\u0027ll be a and that equals to 1,"},{"Start":"01:26.130 ","End":"01:28.335","Text":"and that\u0027s greater than 0."},{"Start":"01:28.335 ","End":"01:30.305","Text":"1 is a positive number."},{"Start":"01:30.305 ","End":"01:32.930","Text":"Now, what about c here?"},{"Start":"01:32.930 ","End":"01:34.595","Text":"Well, in our template,"},{"Start":"01:34.595 ","End":"01:37.850","Text":"b can be either positive or negative."},{"Start":"01:37.850 ","End":"01:43.445","Text":"It really doesn\u0027t matter with regards to positive linear transformation."},{"Start":"01:43.445 ","End":"01:47.210","Text":"So here we have a positive linear transformation,"},{"Start":"01:47.210 ","End":"01:51.575","Text":"y equals to x minus c. That\u0027s a positive linear transformation."},{"Start":"01:51.575 ","End":"01:58.220","Text":"Now, if we have a valid transformation for the interval"},{"Start":"01:58.220 ","End":"02:01.670","Text":"measuring scale then this transformation is also valid"},{"Start":"02:01.670 ","End":"02:05.420","Text":"for the ordinal and nominal measuring scales as well."},{"Start":"02:05.420 ","End":"02:10.050","Text":"That means that answer C here is the correct answer."}],"ID":13360},{"Watched":false,"Name":"Exercise 4","Duration":"2m 17s","ChapterTopicVideoID":12886,"CourseChapterTopicPlaylistID":64519,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"This is a multiple-choice question where we\u0027re asked,"},{"Start":"00:02.820 ","End":"00:06.210","Text":"which transformation preserves the scale of bus numbers?"},{"Start":"00:06.210 ","End":"00:10.140","Text":"Well, to which measuring scales do bus numbers belong?"},{"Start":"00:10.140 ","End":"00:13.335","Text":"Well, let\u0027s take a look at some examples of bus numbers."},{"Start":"00:13.335 ","End":"00:15.870","Text":"Bus number 4, bus number 5,"},{"Start":"00:15.870 ","End":"00:19.460","Text":"bus number 23, 152,"},{"Start":"00:19.460 ","End":"00:20.855","Text":"and so on and so forth."},{"Start":"00:20.855 ","End":"00:26.160","Text":"We see here then that these numbers have no quantitative meanings."},{"Start":"00:26.160 ","End":"00:31.760","Text":"There\u0027s no meaning of more or less here, they\u0027re just identifiers."},{"Start":"00:31.760 ","End":"00:37.295","Text":"As such, they belong to the nominal measuring scale."},{"Start":"00:37.295 ","End":"00:43.809","Text":"Now, what have we learned about transformations of measuring scales?"},{"Start":"00:43.809 ","End":"00:48.830","Text":"We know that if we have a higher-order measuring scale,"},{"Start":"00:48.830 ","End":"00:53.000","Text":"the valid transformation for that higher-order scale,"},{"Start":"00:53.000 ","End":"00:56.420","Text":"that\u0027s also valid for lower-level scales as well."},{"Start":"00:56.420 ","End":"01:01.460","Text":"For example, if we have a valid transformation for the ratio measuring scale,"},{"Start":"01:01.460 ","End":"01:04.805","Text":"that would also be a valid transformation for the interval,"},{"Start":"01:04.805 ","End":"01:07.520","Text":"ordinal and nominal measurement scales."},{"Start":"01:07.520 ","End":"01:15.710","Text":"Now here bus numbers belong to the lowest level measuring scale, the nominal scale."},{"Start":"01:15.710 ","End":"01:19.045","Text":"That means that all valid transformation of"},{"Start":"01:19.045 ","End":"01:24.335","Text":"higher-order measuring scales are valid for the nominal measuring scale."},{"Start":"01:24.335 ","End":"01:28.620","Text":"Now let\u0027s take a look at these answers right here."},{"Start":"01:28.620 ","End":"01:32.300","Text":"We\u0027re asked, which transformation preserves the scale of bus numbers?"},{"Start":"01:32.300 ","End":"01:34.760","Text":"Well, a transformation that preserves order."},{"Start":"01:34.760 ","End":"01:38.300","Text":"A transformation that preserves order is a valid transformation for"},{"Start":"01:38.300 ","End":"01:42.960","Text":"the ordinal measuring scale."},{"Start":"01:42.960 ","End":"01:46.670","Text":"That\u0027s also a valid transformation for the nominal."},{"Start":"01:46.670 ","End":"01:48.880","Text":"That\u0027s a correct answer."},{"Start":"01:48.880 ","End":"01:52.700","Text":"The next transformation, a positive linear transformation."},{"Start":"01:52.700 ","End":"01:55.370","Text":"That\u0027s a valid transformation for the interval."},{"Start":"01:55.370 ","End":"01:58.400","Text":"That\u0027s also a valid transformation for the nominal."},{"Start":"01:58.400 ","End":"02:00.335","Text":"That\u0027s also a correct answer."},{"Start":"02:00.335 ","End":"02:03.670","Text":"A transformation that preserves ratio,"},{"Start":"02:03.670 ","End":"02:05.895","Text":"that\u0027s right here,"},{"Start":"02:05.895 ","End":"02:09.285","Text":"then that\u0027s also a valid transformation for the nominal."},{"Start":"02:09.285 ","End":"02:11.250","Text":"That\u0027s also a correct answer."},{"Start":"02:11.250 ","End":"02:13.580","Text":"That means that all the answers are correct,"},{"Start":"02:13.580 ","End":"02:17.700","Text":"and therefore D is the correct answer."}],"ID":13361}],"Thumbnail":null,"ID":64519},{"Name":"Measures of Relative Location - Standard Score","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"1m 30s","ChapterTopicVideoID":12887,"CourseChapterTopicPlaylistID":64520,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.699","Text":"This chapter I\u0027ll be talking about descriptive statistics,"},{"Start":"00:02.699 ","End":"00:05.190","Text":"specifically relative measures of location."},{"Start":"00:05.190 ","End":"00:07.425","Text":"I\u0027ll be talking about the standard score here."},{"Start":"00:07.425 ","End":"00:10.590","Text":"Now the goal is to measure the position of"},{"Start":"00:10.590 ","End":"00:14.970","Text":"an observation relative to other observations in the probability distribution."},{"Start":"00:14.970 ","End":"00:17.850","Text":"For example, let\u0027s say I\u0027m a student in a class."},{"Start":"00:17.850 ","End":"00:24.470","Text":"I want to know whether I\u0027m taller or shorter relative to the average height in the class,"},{"Start":"00:24.470 ","End":"00:31.610","Text":"or do I have better marks on average than the average marks in my class."},{"Start":"00:31.610 ","End":"00:36.350","Text":"Now, the formula for the standard score of an observation is this."},{"Start":"00:36.350 ","End":"00:40.760","Text":"We denote the standard scores Z and that equals to X,"},{"Start":"00:40.760 ","End":"00:42.939","Text":"the observation, minus X bar,"},{"Start":"00:42.939 ","End":"00:46.950","Text":"that\u0027s the average of the distribution divided by S,"},{"Start":"00:46.950 ","End":"00:49.590","Text":"that\u0027s the standard deviation of the distribution."},{"Start":"00:49.590 ","End":"00:51.900","Text":"We see that X bar and S,"},{"Start":"00:51.900 ","End":"00:55.805","Text":"they\u0027re measures that describe the distribution,"},{"Start":"00:55.805 ","End":"00:58.910","Text":"but Z, that\u0027s personalized,"},{"Start":"00:58.910 ","End":"01:01.595","Text":"that describes the actual observation."},{"Start":"01:01.595 ","End":"01:03.500","Text":"Now, the standard score shows"},{"Start":"01:03.500 ","End":"01:09.295","Text":"how many standard deviations the observation is above or below the average."},{"Start":"01:09.295 ","End":"01:12.170","Text":"If I have a positive standard score,"},{"Start":"01:12.170 ","End":"01:15.110","Text":"that means that the observation is above the average."},{"Start":"01:15.110 ","End":"01:17.195","Text":"If I have a negative standard score,"},{"Start":"01:17.195 ","End":"01:20.600","Text":"that means that the observation is below the average."},{"Start":"01:20.600 ","End":"01:23.435","Text":"If I have a 0 standard score,"},{"Start":"01:23.435 ","End":"01:27.235","Text":"that means that the observation is equal to the average."},{"Start":"01:27.235 ","End":"01:31.380","Text":"Let\u0027s take a look at an example right now."}],"ID":13362},{"Watched":false,"Name":"Tutorial 2","Duration":"3m 52s","ChapterTopicVideoID":12888,"CourseChapterTopicPlaylistID":64520,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.610","Text":"In this example, we have a certain workplace."},{"Start":"00:02.610 ","End":"00:07.575","Text":"The average salary there is $8,000 with a standard deviation of $2,000."},{"Start":"00:07.575 ","End":"00:10.230","Text":"The average education of the employees is"},{"Start":"00:10.230 ","End":"00:13.875","Text":"14 years with a standard deviation of 1.5 years."},{"Start":"00:13.875 ","End":"00:15.990","Text":"Now, we\u0027re given an employee,"},{"Start":"00:15.990 ","End":"00:20.230","Text":"let\u0027s call him Dave, and he earns $11,000."},{"Start":"00:20.230 ","End":"00:22.414","Text":"He has 16 years of education."},{"Start":"00:22.414 ","End":"00:27.520","Text":"Now, we\u0027re asked, is Dave relatively more educated than this level of earnings?"},{"Start":"00:27.520 ","End":"00:31.985","Text":"Let\u0027s first write down the data."},{"Start":"00:31.985 ","End":"00:34.130","Text":"Let\u0027s define x,"},{"Start":"00:34.130 ","End":"00:38.795","Text":"will then be the salary and the salary,"},{"Start":"00:38.795 ","End":"00:41.150","Text":"that will be in thousands of dollars."},{"Start":"00:41.150 ","End":"00:47.800","Text":"We\u0027ll define y as the education in years."},{"Start":"00:47.800 ","End":"00:51.740","Text":"What\u0027s the average salary here?"},{"Start":"00:51.740 ","End":"00:57.275","Text":"Well, that\u0027s $8,000 and the standard deviation,"},{"Start":"00:57.275 ","End":"00:59.045","Text":"that equals to 2."},{"Start":"00:59.045 ","End":"01:00.620","Text":"That\u0027s 8 and 2, right?"},{"Start":"01:00.620 ","End":"01:02.615","Text":"We\u0027re in thousands of dollars here."},{"Start":"01:02.615 ","End":"01:05.380","Text":"What about levels of education,"},{"Start":"01:05.380 ","End":"01:08.185","Text":"what\u0027s the average education?"},{"Start":"01:08.185 ","End":"01:09.780","Text":"That\u0027s 14 years,"},{"Start":"01:09.780 ","End":"01:14.925","Text":"and the standard deviation of the education that\u0027s 1.5 years."},{"Start":"01:14.925 ","End":"01:19.820","Text":"Now, how can we compare salary and education?"},{"Start":"01:19.820 ","End":"01:24.870","Text":"We have to bring these 2 variables to a common denominator."},{"Start":"01:24.870 ","End":"01:27.830","Text":"The way we do that is by calculating"},{"Start":"01:27.830 ","End":"01:33.365","Text":"the standard score of the salary and the standard score of the education,"},{"Start":"01:33.365 ","End":"01:35.780","Text":"and that\u0027s a common denominator."},{"Start":"01:35.780 ","End":"01:40.100","Text":"Now, how do we calculate the standard score? Well, that\u0027s z."},{"Start":"01:40.100 ","End":"01:42.020","Text":"That equals to x,"},{"Start":"01:42.020 ","End":"01:48.800","Text":"the observation minus the average divided by the standard deviation."},{"Start":"01:48.800 ","End":"01:53.975","Text":"Now, what\u0027s our observation here?"},{"Start":"01:53.975 ","End":"01:56.870","Text":"Well, that\u0027s what Dave has right here."},{"Start":"01:56.870 ","End":"01:59.255","Text":"He earns $11,000,"},{"Start":"01:59.255 ","End":"02:04.605","Text":"so x would be $11,000 and y,"},{"Start":"02:04.605 ","End":"02:07.590","Text":"his level of education is 16 years."},{"Start":"02:07.590 ","End":"02:13.230","Text":"Let\u0027s just calculate the standard score for x and from y,"},{"Start":"02:13.230 ","End":"02:17.254","Text":"and only then can we compare salary and education."},{"Start":"02:17.254 ","End":"02:22.580","Text":"The standard score for the salary,"},{"Start":"02:22.580 ","End":"02:23.655","Text":"that\u0027ll be 11,"},{"Start":"02:23.655 ","End":"02:25.290","Text":"that\u0027s what Dave earns,"},{"Start":"02:25.290 ","End":"02:26.910","Text":"minus the average, well,"},{"Start":"02:26.910 ","End":"02:28.465","Text":"the average here is 8,"},{"Start":"02:28.465 ","End":"02:30.860","Text":"divided by the standard deviation,"},{"Start":"02:30.860 ","End":"02:34.120","Text":"that\u0027s 2 and that equals to 11 minus 8,"},{"Start":"02:34.120 ","End":"02:37.385","Text":"that\u0027s 3, 3 divided by 2, that\u0027s 1.5."},{"Start":"02:37.385 ","End":"02:43.430","Text":"Now, what about the standard score for the education?"},{"Start":"02:43.430 ","End":"02:46.460","Text":"Well, Dave has 16 years of education."},{"Start":"02:46.460 ","End":"02:48.140","Text":"We have to take away the average,"},{"Start":"02:48.140 ","End":"02:54.475","Text":"that\u0027s 14 divided by the standard deviation, that\u0027s 1.5."},{"Start":"02:54.475 ","End":"03:01.215","Text":"Now, 16 minus 14 divided by 1.5 that\u0027s 1 and1/3."},{"Start":"03:01.215 ","End":"03:05.280","Text":"What do these numbers mean?"},{"Start":"03:05.280 ","End":"03:08.640","Text":"Well, with the salary, 1.5,"},{"Start":"03:08.640 ","End":"03:14.670","Text":"that means that Dave earns more than the average. By how much?"},{"Start":"03:14.670 ","End":"03:17.820","Text":"By 1.5 standard deviations."},{"Start":"03:17.820 ","End":"03:19.860","Text":"What about his education?"},{"Start":"03:19.860 ","End":"03:27.315","Text":"Well, Dave has more education than the average employee by how much?"},{"Start":"03:27.315 ","End":"03:30.435","Text":"By 1 and 1/3 standard deviations."},{"Start":"03:30.435 ","End":"03:40.770","Text":"We see here that relative to the education, Dave earns more."},{"Start":"03:40.770 ","End":"03:42.650","Text":"When answering the question,"},{"Start":"03:42.650 ","End":"03:46.940","Text":"is Dave relatively more educated than his levels of earnings?"},{"Start":"03:46.940 ","End":"03:53.610","Text":"No, Dave earns more relative to his levels of education."}],"ID":13363},{"Watched":false,"Name":"Exercise 1 - Parts a-b","Duration":"4m 28s","ChapterTopicVideoID":12889,"CourseChapterTopicPlaylistID":64520,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.680","Text":"In this question, we have students from"},{"Start":"00:01.680 ","End":"00:04.815","Text":"the eighth grade who took exams in English and in math."},{"Start":"00:04.815 ","End":"00:07.035","Text":"Now, the following results were obtained."},{"Start":"00:07.035 ","End":"00:08.985","Text":"Here we have the table."},{"Start":"00:08.985 ","End":"00:15.570","Text":"The average grade in English for the whole class was 74 with a standard deviation of 12."},{"Start":"00:15.570 ","End":"00:21.284","Text":"For math, the average grade was 80 with a standard deviation of 16."},{"Start":"00:21.284 ","End":"00:26.595","Text":"Now, Mike, a student in the class who received 68 in English and 70 in math."},{"Start":"00:26.595 ","End":"00:31.635","Text":"We\u0027re asked in which subject is Mike better relative to the rest of the class?"},{"Start":"00:31.635 ","End":"00:37.890","Text":"Now, we can\u0027t just compare 68 and 70."},{"Start":"00:37.890 ","End":"00:44.630","Text":"We have to compare these marks relative to what\u0027s going on here. How do we do that?"},{"Start":"00:44.630 ","End":"00:50.505","Text":"Well, we calculate the standard score for English and for math."},{"Start":"00:50.505 ","End":"00:52.050","Text":"How do we calculate that?"},{"Start":"00:52.050 ","End":"00:53.545","Text":"With a standard score, that\u0027s Z,"},{"Start":"00:53.545 ","End":"01:00.815","Text":"that equals to the observation minus the average divided by the standard deviation."},{"Start":"01:00.815 ","End":"01:05.550","Text":"Let\u0027s calculate the standard score for English."},{"Start":"01:05.550 ","End":"01:08.850","Text":"Z, that\u0027s for English,"},{"Start":"01:08.850 ","End":"01:09.990","Text":"that equals to what?"},{"Start":"01:09.990 ","End":"01:11.910","Text":"Mike received 68 in English."},{"Start":"01:11.910 ","End":"01:14.130","Text":"That\u0027ll be 68 minus,"},{"Start":"01:14.130 ","End":"01:17.700","Text":"what\u0027s the average of the class? That\u0027s 74."},{"Start":"01:17.700 ","End":"01:20.460","Text":"Divided by the standard deviation."},{"Start":"01:20.460 ","End":"01:22.530","Text":"The standard deviation is 12."},{"Start":"01:22.530 ","End":"01:27.405","Text":"That equals to minus 0.5."},{"Start":"01:27.405 ","End":"01:31.895","Text":"What about the standard score for math?"},{"Start":"01:31.895 ","End":"01:34.155","Text":"Well, what did Mike get?"},{"Start":"01:34.155 ","End":"01:42.180","Text":"Mike got 70 minus the average of the class,"},{"Start":"01:42.180 ","End":"01:47.040","Text":"well that\u0027s 80, divided by the standard deviation, that\u0027s 16."},{"Start":"01:47.040 ","End":"01:52.550","Text":"That equals to minus 0.625."},{"Start":"01:52.550 ","End":"01:58.725","Text":"Now, we see here that in both subjects, English and math,"},{"Start":"01:58.725 ","End":"02:03.920","Text":"Mike received a grade which was lower than the average,"},{"Start":"02:03.920 ","End":"02:06.560","Text":"first of all, and by how much?"},{"Start":"02:06.560 ","End":"02:11.210","Text":"Well, in English, he received a grade that was lower than"},{"Start":"02:11.210 ","End":"02:16.235","Text":"the average by 0.5 standard deviations and in math,"},{"Start":"02:16.235 ","End":"02:20.525","Text":"by 0.625 standard deviations."},{"Start":"02:20.525 ","End":"02:23.165","Text":"But in any case,"},{"Start":"02:23.165 ","End":"02:25.370","Text":"although he received lower grades than"},{"Start":"02:25.370 ","End":"02:29.285","Text":"the average relative to the rest of the class in both subject,"},{"Start":"02:29.285 ","End":"02:36.770","Text":"here this standard score is greater than this standard score."},{"Start":"02:36.770 ","End":"02:43.400","Text":"So he\u0027s much better in English relative to the class than he is in math."},{"Start":"02:43.400 ","End":"02:44.885","Text":"In this section we\u0027re asked,"},{"Start":"02:44.885 ","End":"02:50.050","Text":"what mark did Mike need in math to be the equivalent to his mark in English?"},{"Start":"02:50.050 ","End":"02:54.855","Text":"First of all, we\u0027re looking for a mark in math, let\u0027s call that x."},{"Start":"02:54.855 ","End":"02:57.440","Text":"We want this new mark,"},{"Start":"02:57.440 ","End":"02:59.060","Text":"the mark that we\u0027re looking for,"},{"Start":"02:59.060 ","End":"03:01.220","Text":"to be equivalent to his mark in English."},{"Start":"03:01.220 ","End":"03:03.670","Text":"Well, his mark in English is 68."},{"Start":"03:03.670 ","End":"03:06.820","Text":"When we standardize this mark,"},{"Start":"03:06.820 ","End":"03:09.470","Text":"when we calculated the standard score in English,"},{"Start":"03:09.470 ","End":"03:19.040","Text":"we found out that 68 was minus 0.5 standard deviations away from the class average of 74."},{"Start":"03:19.040 ","End":"03:24.245","Text":"Now this is exactly what we want for his mark in math."},{"Start":"03:24.245 ","End":"03:28.685","Text":"We want this new mark to be exactly"},{"Start":"03:28.685 ","End":"03:33.695","Text":"minus 0.5 standard deviations away from the class average of 80."},{"Start":"03:33.695 ","End":"03:36.560","Text":"Let\u0027s calculate the standard score here."},{"Start":"03:36.560 ","End":"03:39.090","Text":"That will be x, the mark that we\u0027re looking for,"},{"Start":"03:39.090 ","End":"03:48.480","Text":"minus 80 divided by 16. x minus the average divided by the standard deviations of math."},{"Start":"03:48.480 ","End":"03:52.265","Text":"Now, that has to be equal to minus 0.5."},{"Start":"03:52.265 ","End":"03:53.615","Text":"Now again, why is that?"},{"Start":"03:53.615 ","End":"04:00.915","Text":"Because we want this mark to be equivalent to his mark in English."},{"Start":"04:00.915 ","End":"04:02.630","Text":"What does that equal?"},{"Start":"04:02.630 ","End":"04:05.315","Text":"Well, let\u0027s multiply both sides by 16."},{"Start":"04:05.315 ","End":"04:10.815","Text":"That means that x minus 80 will now become minus 8."},{"Start":"04:10.815 ","End":"04:14.774","Text":"That means that x then equals to 72."},{"Start":"04:14.774 ","End":"04:21.080","Text":"A mark of 72 in math is equivalent to a mark of 68 in"},{"Start":"04:21.080 ","End":"04:28.900","Text":"English with respect or relative to the class averages."}],"ID":13364},{"Watched":false,"Name":"Exercise 2","Duration":"3m 33s","ChapterTopicVideoID":12890,"CourseChapterTopicPlaylistID":64520,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"In this question, we have a car battery factory and we"},{"Start":"00:02.910 ","End":"00:06.680","Text":"counted the number of employees per day and measured their daily output,"},{"Start":"00:06.680 ","End":"00:07.815","Text":"the number of batteries,"},{"Start":"00:07.815 ","End":"00:10.515","Text":"in 100s for 40 continuous days."},{"Start":"00:10.515 ","End":"00:14.205","Text":"Now, the following table shows the information collected about the 2 variables."},{"Start":"00:14.205 ","End":"00:16.635","Text":"Here we have output, that\u0027ll be our x,"},{"Start":"00:16.635 ","End":"00:20.145","Text":"and here that\u0027ll be the number of employees, that\u0027s y."},{"Start":"00:20.145 ","End":"00:22.470","Text":"In the 40 continuous days,"},{"Start":"00:22.470 ","End":"00:24.810","Text":"we had an average of 48 or"},{"Start":"00:24.810 ","End":"00:30.090","Text":"4,800 car batteries that were produced with a standard deviation of 10."},{"Start":"00:30.090 ","End":"00:32.820","Text":"Here, for the number of employees,"},{"Start":"00:32.820 ","End":"00:38.930","Text":"we had an average 15 employees per day with a standard deviation of 2."},{"Start":"00:38.930 ","End":"00:41.735","Text":"Now, on 1 of the 40 days,"},{"Start":"00:41.735 ","End":"00:44.855","Text":"there were 13 employees that were working"},{"Start":"00:44.855 ","End":"00:48.410","Text":"and they produced 5,000 or 50 hundred batteries."},{"Start":"00:48.410 ","End":"00:49.586","Text":"Let\u0027s just write that down."},{"Start":"00:49.586 ","End":"00:55.790","Text":"Here we have 13 and here we have 50 and we\u0027re asked the following."},{"Start":"00:55.790 ","End":"00:58.905","Text":"We\u0027re asked, what was more exceptional about that day"},{"Start":"00:58.905 ","End":"01:02.600","Text":"relative to other dates on which the output and the number of employees were examined?"},{"Start":"01:02.600 ","End":"01:05.945","Text":"We\u0027re looking at these numbers right here,"},{"Start":"01:05.945 ","End":"01:08.990","Text":"50 for the output and 13 for the employees."},{"Start":"01:08.990 ","End":"01:10.354","Text":"What\u0027s more exceptional?"},{"Start":"01:10.354 ","End":"01:11.675","Text":"That\u0027s what we\u0027re asked."},{"Start":"01:11.675 ","End":"01:15.530","Text":"Well, in order to answer this question,"},{"Start":"01:15.530 ","End":"01:19.805","Text":"we need to find the standard score because the standard score gives me"},{"Start":"01:19.805 ","End":"01:26.780","Text":"the position of these values relative to the average for each 1 of the variables here."},{"Start":"01:26.780 ","End":"01:30.500","Text":"The other thing that we have to take into account is that we\u0027re"},{"Start":"01:30.500 ","End":"01:34.235","Text":"asked about the exceptions which is more exceptional."},{"Start":"01:34.235 ","End":"01:36.140","Text":"Now, in order to do that,"},{"Start":"01:36.140 ","End":"01:40.610","Text":"we have to take a look at the standard score,"},{"Start":"01:40.610 ","End":"01:42.590","Text":"but not the actual values."},{"Start":"01:42.590 ","End":"01:46.460","Text":"We can\u0027t take a look at the sign of the standard score,"},{"Start":"01:46.460 ","End":"01:49.795","Text":"but we\u0027re going to have to take a look at the magnitude."},{"Start":"01:49.795 ","End":"01:54.815","Text":"The magnitude is calculated by taking the absolute value of the standard score."},{"Start":"01:54.815 ","End":"02:00.530","Text":"First of all, let\u0027s just calculate the standard score for each 1 of the variables here."},{"Start":"02:00.530 ","End":"02:04.380","Text":"For output, that\u0027s z sub x,"},{"Start":"02:04.380 ","End":"02:06.000","Text":"that\u0027s the standard score of the output,"},{"Start":"02:06.000 ","End":"02:08.985","Text":"that will be 50 minus 48."},{"Start":"02:08.985 ","End":"02:13.055","Text":"That\u0027s 50 minus the average divided by the standard deviation, that\u0027s 10."},{"Start":"02:13.055 ","End":"02:16.445","Text":"That equals to 2 divided by 10, that\u0027s 0.2."},{"Start":"02:16.445 ","End":"02:19.145","Text":"Now, the standard score for y,"},{"Start":"02:19.145 ","End":"02:20.270","Text":"the number of employees,"},{"Start":"02:20.270 ","End":"02:23.544","Text":"that will be 13 minus the average,"},{"Start":"02:23.544 ","End":"02:25.835","Text":"that\u0027s 15, divided by 2."},{"Start":"02:25.835 ","End":"02:28.190","Text":"Well that\u0027s minus 2 divided by 2,"},{"Start":"02:28.190 ","End":"02:30.020","Text":"that equals to minus 1."},{"Start":"02:30.020 ","End":"02:32.850","Text":"Here for the output,"},{"Start":"02:32.850 ","End":"02:37.910","Text":"we see that the number of batteries that were"},{"Start":"02:37.910 ","End":"02:44.145","Text":"produced was 0.2 standard deviations above the average."},{"Start":"02:44.145 ","End":"02:46.370","Text":"With respect to the number of employees,"},{"Start":"02:46.370 ","End":"02:50.450","Text":"they did that with a number of employees"},{"Start":"02:50.450 ","End":"02:55.375","Text":"that was minus 1 standard deviation below the average."},{"Start":"02:55.375 ","End":"02:57.285","Text":"But again, as we said,"},{"Start":"02:57.285 ","End":"02:59.430","Text":"we don\u0027t look at the sign,"},{"Start":"02:59.430 ","End":"03:00.855","Text":"we look at the magnitude."},{"Start":"03:00.855 ","End":"03:06.335","Text":"We\u0027re looking at the absolute value of the standard score for x,"},{"Start":"03:06.335 ","End":"03:10.910","Text":"for the output, and the absolute value of the standard score for y."},{"Start":"03:10.910 ","End":"03:13.415","Text":"Now here, that equals to 0.2,"},{"Start":"03:13.415 ","End":"03:15.805","Text":"here that equals to 1."},{"Start":"03:15.805 ","End":"03:20.100","Text":"1 is obviously greater than 0.2,"},{"Start":"03:20.100 ","End":"03:23.210","Text":"so that means that what was more exceptional,"},{"Start":"03:23.210 ","End":"03:25.805","Text":"well, that was the number of employees."},{"Start":"03:25.805 ","End":"03:28.850","Text":"Let\u0027s look at the answers here."},{"Start":"03:28.850 ","End":"03:31.695","Text":"Well, that\u0027s answer B."},{"Start":"03:31.695 ","End":"03:34.450","Text":"Answer B is the correct answer."}],"ID":13365},{"Watched":false,"Name":"Exercise 3 - Parts a-b","Duration":"6m ","ChapterTopicVideoID":12891,"CourseChapterTopicPlaylistID":64520,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"In this question, we\u0027re given that the average height of people"},{"Start":"00:02.820 ","End":"00:05.820","Text":"that are drafted to the army is a 175 centimeters,"},{"Start":"00:05.820 ","End":"00:08.490","Text":"with a standard deviation of 10 centimeters."},{"Start":"00:08.490 ","End":"00:13.590","Text":"The average weight is 66 kilograms with a standard deviation of 8 kilograms."},{"Start":"00:13.590 ","End":"00:20.600","Text":"Now, Steve, he\u0027s a recruit and he is 180 centimeters tall and weighs 59 kilograms."},{"Start":"00:20.600 ","End":"00:23.230","Text":"We\u0027re asked, in what does Steve deviate"},{"Start":"00:23.230 ","End":"00:26.825","Text":"more in comparison with the other recruits; height or weight?"},{"Start":"00:26.825 ","End":"00:28.160","Text":"Well, first of all,"},{"Start":"00:28.160 ","End":"00:32.180","Text":"let\u0027s define our variables, let\u0027s define x."},{"Start":"00:32.180 ","End":"00:38.780","Text":"Well, that will be the height in centimeters."},{"Start":"00:38.780 ","End":"00:40.605","Text":"The average height, well,"},{"Start":"00:40.605 ","End":"00:43.310","Text":"that equals to 175 centimeters,"},{"Start":"00:43.310 ","End":"00:46.775","Text":"that\u0027s right here, and the standard deviation,"},{"Start":"00:46.775 ","End":"00:48.790","Text":"that equals to 10 centimeters."},{"Start":"00:48.790 ","End":"00:50.375","Text":"That\u0027s right here. Now,"},{"Start":"00:50.375 ","End":"00:58.010","Text":"y would be defined as the weight in kilograms."},{"Start":"00:58.010 ","End":"01:00.812","Text":"Now, the average weight, well,"},{"Start":"01:00.812 ","End":"01:06.485","Text":"that equals to 66 kilograms and the standard deviation of the weight,"},{"Start":"01:06.485 ","End":"01:09.815","Text":"that equals to 8 kilograms that\u0027s right here."},{"Start":"01:09.815 ","End":"01:17.169","Text":"Now, we have Steve and he is 180 centimeters tall,"},{"Start":"01:17.169 ","End":"01:20.100","Text":"so x, that would be 180 centimeters tall,"},{"Start":"01:20.100 ","End":"01:23.300","Text":"and y, that\u0027s how much he weighs."},{"Start":"01:23.300 ","End":"01:25.696","Text":"He weighs 59 kilograms."},{"Start":"01:25.696 ","End":"01:28.100","Text":"That\u0027s this thing and this thing right here."},{"Start":"01:28.100 ","End":"01:30.650","Text":"We\u0027re asked, in what does Steve deviate"},{"Start":"01:30.650 ","End":"01:34.090","Text":"more in comparison with the other recruits; height or weight?"},{"Start":"01:34.090 ","End":"01:39.230","Text":"We\u0027re going to have to calculate the standard score for x and for y,"},{"Start":"01:39.230 ","End":"01:40.520","Text":"the height and weight,"},{"Start":"01:40.520 ","End":"01:44.000","Text":"in order to see what\u0027s the relative position of"},{"Start":"01:44.000 ","End":"01:47.930","Text":"Steve\u0027s height and weight with respect to the other recruits."},{"Start":"01:47.930 ","End":"01:50.495","Text":"That\u0027s 1 thing. But the other thing,"},{"Start":"01:50.495 ","End":"01:53.640","Text":"is that we\u0027re asked about the deviation,"},{"Start":"01:53.640 ","End":"01:58.301","Text":"how much does he deviate more in comparison with the other recruits?"},{"Start":"01:58.301 ","End":"02:01.175","Text":"When we calculate the standard score,"},{"Start":"02:01.175 ","End":"02:06.080","Text":"we have to look at the absolute value of the standard score"},{"Start":"02:06.080 ","End":"02:11.530","Text":"in order to see what the magnitude of the deviation is."},{"Start":"02:11.530 ","End":"02:14.420","Text":"First of all, let\u0027s calculate the standard score."},{"Start":"02:14.420 ","End":"02:16.670","Text":"The standard score z,"},{"Start":"02:16.670 ","End":"02:23.640","Text":"that equals to an observation minus its average divided by the standard deviation."},{"Start":"02:23.690 ","End":"02:28.095","Text":"The standard score of x,"},{"Start":"02:28.095 ","End":"02:30.570","Text":"the height, that equals to 180,"},{"Start":"02:30.570 ","End":"02:33.570","Text":"that\u0027s how tall he is,"},{"Start":"02:33.570 ","End":"02:39.380","Text":"minus 175, that\u0027s the average of all recruits divided by 10."},{"Start":"02:39.380 ","End":"02:41.735","Text":"That\u0027s the standard deviation of the height."},{"Start":"02:41.735 ","End":"02:45.755","Text":"Now, that equals to 5 divided by 10, that\u0027s 0.5."},{"Start":"02:45.755 ","End":"02:49.725","Text":"Now, what about the standard score of y the weight?"},{"Start":"02:49.725 ","End":"02:52.995","Text":"Well, Steve weighs 59 kilograms."},{"Start":"02:52.995 ","End":"02:54.732","Text":"We have to take away the average,"},{"Start":"02:54.732 ","End":"02:58.240","Text":"that\u0027s 66 divided by 8."},{"Start":"02:58.240 ","End":"02:59.510","Text":"That\u0027s a standard deviation."},{"Start":"02:59.510 ","End":"03:03.815","Text":"That equals to minus 0.875."},{"Start":"03:03.815 ","End":"03:13.220","Text":"We say that Steve is 0.5 standard deviations above the average with"},{"Start":"03:13.220 ","End":"03:17.615","Text":"respect to height and he\u0027s below the average weight"},{"Start":"03:17.615 ","End":"03:23.225","Text":"by a factor of minus 0.875 standard deviations."},{"Start":"03:23.225 ","End":"03:26.725","Text":"But again, we want to see the magnitude here."},{"Start":"03:26.725 ","End":"03:29.070","Text":"We want to see which is greater."},{"Start":"03:29.070 ","End":"03:35.385","Text":"We have to take the absolute value of the standard score."},{"Start":"03:35.385 ","End":"03:39.875","Text":"Here, the absolute value of the standard score for the height,"},{"Start":"03:39.875 ","End":"03:44.775","Text":"that equals to 0.5 and the absolute value of the weight,"},{"Start":"03:44.775 ","End":"03:47.820","Text":"that equals to 0.875."},{"Start":"03:47.820 ","End":"03:51.410","Text":"We see here that the absolute value of the standard score of"},{"Start":"03:51.410 ","End":"03:57.025","Text":"weight is greater than the absolute value of the standard score of the height."},{"Start":"03:57.025 ","End":"04:00.200","Text":"When we\u0027re asked, in what does Steve deviate"},{"Start":"04:00.200 ","End":"04:03.098","Text":"more in comparison with the other recruits: height or weight?"},{"Start":"04:03.098 ","End":"04:05.650","Text":"Well, the average is weight?"},{"Start":"04:05.650 ","End":"04:07.200","Text":"In section B, we\u0027re asked,"},{"Start":"04:07.200 ","End":"04:11.525","Text":"how much does Steve weigh to make his weight the equivalent of his height?"},{"Start":"04:11.525 ","End":"04:14.000","Text":"Now, what do we mean by this question?"},{"Start":"04:14.000 ","End":"04:18.335","Text":"Well, first of all, we\u0027re looking for a weight value. That\u0027s y."},{"Start":"04:18.335 ","End":"04:23.375","Text":"Now, we want this new weight value to be"},{"Start":"04:23.375 ","End":"04:28.835","Text":"equivalent to his height with respect to the other recruits."},{"Start":"04:28.835 ","End":"04:32.085","Text":"Now, how do we calculate that?"},{"Start":"04:32.085 ","End":"04:35.625","Text":"Well, we\u0027ve calculated that by using the standard score."},{"Start":"04:35.625 ","End":"04:43.715","Text":"We know that his height is 0.5 standard deviations above the average height."},{"Start":"04:43.715 ","End":"04:46.900","Text":"This is what we want to weight to be, the new weight."},{"Start":"04:46.900 ","End":"04:50.990","Text":"We want this weight that we were looking for to be 0.5"},{"Start":"04:50.990 ","End":"04:56.295","Text":"standard deviations above the average weight of all the recruits."},{"Start":"04:56.295 ","End":"05:01.520","Text":"We have our new value here that we\u0027re looking for."},{"Start":"05:01.520 ","End":"05:04.880","Text":"We have to calculate the standard score,"},{"Start":"05:04.880 ","End":"05:07.040","Text":"that will be y minus 66."},{"Start":"05:07.040 ","End":"05:08.330","Text":"That\u0027s the average weight,"},{"Start":"05:08.330 ","End":"05:11.180","Text":"divided by the standard deviation, that\u0027s 8."},{"Start":"05:11.180 ","End":"05:15.020","Text":"We want that to be equal to 0.5."},{"Start":"05:15.020 ","End":"05:24.050","Text":"We want this new value to be 0.5 standard deviations above the average."},{"Start":"05:24.050 ","End":"05:26.340","Text":"Now, how do we do that?"},{"Start":"05:26.340 ","End":"05:29.158","Text":"Well, first of all, let\u0027s multiply both sides by 8,"},{"Start":"05:29.158 ","End":"05:32.615","Text":"so that\u0027ll be y minus 66,"},{"Start":"05:32.615 ","End":"05:34.505","Text":"that equals to 4,"},{"Start":"05:34.505 ","End":"05:38.030","Text":"and therefore y equals to 70."},{"Start":"05:38.030 ","End":"05:42.110","Text":"If Steve would weigh 70 kilograms,"},{"Start":"05:42.110 ","End":"05:47.895","Text":"that would be comparable to his height being 180."},{"Start":"05:47.895 ","End":"05:53.420","Text":"Why is that? Because both 180 centimeters and 70 kilograms,"},{"Start":"05:53.420 ","End":"06:01.530","Text":"they\u0027re both 0.5 standard deviations above the respected averages."}],"ID":13366}],"Thumbnail":null,"ID":64520},{"Name":"Measures of Relative Location - Percentiles in Categories","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"1m 52s","ChapterTopicVideoID":13255,"CourseChapterTopicPlaylistID":64521,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In this chapter, I\u0027ll be talking about descriptive statistics,"},{"Start":"00:02.850 ","End":"00:05.340","Text":"specifically measures of relative location."},{"Start":"00:05.340 ","End":"00:08.175","Text":"I\u0027ll be talking about percentiles in categories."},{"Start":"00:08.175 ","End":"00:11.370","Text":"Now, the objective here is to see where"},{"Start":"00:11.370 ","End":"00:15.880","Text":"an observation is located relative to other observations in the group."},{"Start":"00:15.880 ","End":"00:20.310","Text":"For example, if we\u0027re talking about a group of newborn babies, well,"},{"Start":"00:20.310 ","End":"00:22.709","Text":"I want to know what the weight of a specific"},{"Start":"00:22.709 ","End":"00:27.840","Text":"newborn in relation to other newborns or my salary."},{"Start":"00:27.840 ","End":"00:32.264","Text":"What\u0027s my salary in relation to other employees in the company?"},{"Start":"00:32.264 ","End":"00:33.990","Text":"Where is that located?"},{"Start":"00:33.990 ","End":"00:37.755","Text":"Now, this measure is called a percentile."},{"Start":"00:37.755 ","End":"00:42.480","Text":"The pth percentile is the specific value that"},{"Start":"00:42.480 ","End":"00:47.705","Text":"divides the data so that there are P percent of the data up to that value."},{"Start":"00:47.705 ","End":"00:51.830","Text":"Now, the pth percentile is denoted by X_P."},{"Start":"00:51.830 ","End":"00:57.935","Text":"Now, we can also see the percentile denoted as C_p."},{"Start":"00:57.935 ","End":"01:00.560","Text":"But here, I won\u0027t be using this at all."},{"Start":"01:00.560 ","End":"01:04.894","Text":"Now, for example, the 25th percentile,"},{"Start":"01:04.894 ","End":"01:06.560","Text":"or the bottom quartile,"},{"Start":"01:06.560 ","End":"01:10.910","Text":"is the value under which they are 1/4 of the observations,"},{"Start":"01:10.910 ","End":"01:14.975","Text":"and obviously, the other 75 percent of the observations are above it."},{"Start":"01:14.975 ","End":"01:20.295","Text":"Now, this is again denoted by X_0.25."},{"Start":"01:20.295 ","End":"01:22.700","Text":"Now, for example,"},{"Start":"01:22.700 ","End":"01:25.355","Text":"if we\u0027re talking about the height in a specific group,"},{"Start":"01:25.355 ","End":"01:29.360","Text":"then X_0.25 is the height where"},{"Start":"01:29.360 ","End":"01:35.149","Text":"25 percent of the group members are shorter than this value right here."},{"Start":"01:35.149 ","End":"01:40.610","Text":"Here in this chapter,"},{"Start":"01:40.610 ","End":"01:44.225","Text":"we\u0027ll be talking about finding percentiles in categories."},{"Start":"01:44.225 ","End":"01:48.290","Text":"That means that we\u0027ll learn how to find a percentile when"},{"Start":"01:48.290 ","End":"01:53.670","Text":"the data is presented to us in a frequency table by categories."}],"ID":13743},{"Watched":false,"Name":"Tutorial 2","Duration":"1m 21s","ChapterTopicVideoID":13256,"CourseChapterTopicPlaylistID":64521,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.320","Text":"The status and finding the percentile and categories is as follows."},{"Start":"00:04.320 ","End":"00:09.210","Text":"For the first step is to identify the relevant category of"},{"Start":"00:09.210 ","End":"00:15.150","Text":"the percentile and the location of the percentile is calculated like this."},{"Start":"00:15.150 ","End":"00:17.385","Text":"It\u0027s n times p divided by a 100."},{"Start":"00:17.385 ","End":"00:24.030","Text":"Now, once I know what the location is of the percentile well I can identify the category."},{"Start":"00:24.030 ","End":"00:27.720","Text":"Having identified the category, well,"},{"Start":"00:27.720 ","End":"00:34.140","Text":"all I have to do now is use this formula to calculate the exact value of the percentile."},{"Start":"00:34.140 ","End":"00:40.650","Text":"Now, this looks very similar to the expression that I use to calculate the median."},{"Start":"00:40.650 ","End":"00:44.225","Text":"Now, we have here all the components and the very similar,"},{"Start":"00:44.225 ","End":"00:47.960","Text":"well, the percentile is equal to the following."},{"Start":"00:47.960 ","End":"00:49.820","Text":"Well, it\u0027s L sub 0."},{"Start":"00:49.820 ","End":"00:52.939","Text":"Well, that\u0027s the lower bound of the category where the percentile"},{"Start":"00:52.939 ","End":"00:56.540","Text":"is plus n times p divided by a 100."},{"Start":"00:56.540 ","End":"01:02.450","Text":"That\u0027s the location of the percentile minus big F of x sub m minus 1."},{"Start":"01:02.450 ","End":"01:07.610","Text":"Well, that\u0027s the cumulative frequency of the preceding category,"},{"Start":"01:07.610 ","End":"01:12.020","Text":"divided by the frequency of the category where"},{"Start":"01:12.020 ","End":"01:16.990","Text":"we found the percentile times the width of the category where we have the percentile."},{"Start":"01:16.990 ","End":"01:22.440","Text":"Let\u0027s just take a look at an example and see how we can calculate this."}],"ID":13744},{"Watched":false,"Name":"Tutorial 3","Duration":"5m 21s","ChapterTopicVideoID":13257,"CourseChapterTopicPlaylistID":64521,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.329","Text":"How do I find the percentile when I have categories?"},{"Start":"00:03.329 ","End":"00:05.340","Text":"Well, the first thing that I need to do is I need to"},{"Start":"00:05.340 ","End":"00:08.520","Text":"identify the relevant category of the percentile"},{"Start":"00:08.520 ","End":"00:14.730","Text":"and we have to know where the percentile is located."},{"Start":"00:14.730 ","End":"00:20.565","Text":"The location of the percentile is calculated at n times p divided by 100."},{"Start":"00:20.565 ","End":"00:23.340","Text":"Now, once I know the location of the percentile,"},{"Start":"00:23.340 ","End":"00:25.665","Text":"well then, I can identify the category."},{"Start":"00:25.665 ","End":"00:28.185","Text":"Having identified the category,"},{"Start":"00:28.185 ","End":"00:30.300","Text":"you will then go to step 2,"},{"Start":"00:30.300 ","End":"00:36.055","Text":"and this is the expression to calculate the exact value of the percentile."},{"Start":"00:36.055 ","End":"00:42.180","Text":"Now, this looks very similar to the expression used to calculate the median."},{"Start":"00:42.180 ","End":"00:45.230","Text":"We have the same components. They\u0027re very similar."},{"Start":"00:45.230 ","End":"00:48.350","Text":"We have here L_0,"},{"Start":"00:48.350 ","End":"00:54.560","Text":"which is the lower bound of the category where we found the percentile,"},{"Start":"00:54.560 ","End":"00:56.790","Text":"plus n times p divided by 100."},{"Start":"00:56.790 ","End":"00:59.300","Text":"Well, that\u0027s the location of the percentile."},{"Start":"00:59.300 ","End":"01:03.050","Text":"Minus F of x_n minus 1."},{"Start":"01:03.050 ","End":"01:07.055","Text":"Well, that\u0027s the cumulative frequency of"},{"Start":"01:07.055 ","End":"01:12.459","Text":"the category preceding the category where we find the percentile,"},{"Start":"01:12.459 ","End":"01:16.640","Text":"divided by the frequency of the category where we"},{"Start":"01:16.640 ","End":"01:21.575","Text":"found the percentile times the width of the category where the percentile is."},{"Start":"01:21.575 ","End":"01:25.710","Text":"Let\u0027s just take a look at an example."},{"Start":"01:28.190 ","End":"01:30.710","Text":"Here in this example,"},{"Start":"01:30.710 ","End":"01:32.330","Text":"we have a table that displays"},{"Start":"01:32.330 ","End":"01:36.575","Text":"the probability distribution of employee salaries in a given company."},{"Start":"01:36.575 ","End":"01:42.510","Text":"We have let\u0027s say, 140 employees that earn between $4,000 and $6,000,"},{"Start":"01:42.510 ","End":"01:47.930","Text":"128 employees who earn between $6,000 and $10,000 and so on and so forth."},{"Start":"01:47.930 ","End":"01:53.285","Text":"The first thing that we want to do is we want to calculate"},{"Start":"01:53.285 ","End":"01:58.970","Text":"the position of the percentile that we\u0027re asked to calculate, the 40th percentile."},{"Start":"01:58.970 ","End":"02:02.255","Text":"What\u0027s the position of the 40th percentile?"},{"Start":"02:02.255 ","End":"02:06.890","Text":"That\u0027s n times p divided by 100."},{"Start":"02:06.890 ","End":"02:08.930","Text":"Now, what\u0027s n in our case?"},{"Start":"02:08.930 ","End":"02:13.010","Text":"Well, that\u0027s the sum of the frequencies here,"},{"Start":"02:13.010 ","End":"02:16.539","Text":"so n here, that equals to 400,"},{"Start":"02:16.539 ","End":"02:21.965","Text":"so that equals to 400 times 40 divided by 100,"},{"Start":"02:21.965 ","End":"02:23.773","Text":"that equals to 160,"},{"Start":"02:23.773 ","End":"02:29.795","Text":"so what\u0027s the category that has the 160th employee?"},{"Start":"02:29.795 ","End":"02:32.105","Text":"Well, in order to calculate that,"},{"Start":"02:32.105 ","End":"02:36.725","Text":"I need to find the cumulative frequency."},{"Start":"02:36.725 ","End":"02:39.200","Text":"Well, the cumulative frequency here,"},{"Start":"02:39.200 ","End":"02:40.865","Text":"what that would be 140."},{"Start":"02:40.865 ","End":"02:44.000","Text":"140 plus 128, that\u0027s 268,"},{"Start":"02:44.000 ","End":"02:46.940","Text":"268 plus 60, that\u0027s 328,"},{"Start":"02:46.940 ","End":"02:50.680","Text":"328 plus 54, that\u0027s 382,"},{"Start":"02:50.680 ","End":"02:55.580","Text":"and 382 plus 18, well, that\u0027s 400."},{"Start":"02:55.580 ","End":"03:00.350","Text":"Where is employee number 160?"},{"Start":"03:00.350 ","End":"03:03.470","Text":"Well, it\u0027s not in the first category because in the first category,"},{"Start":"03:03.470 ","End":"03:06.530","Text":"we have only the first 140 categories."},{"Start":"03:06.530 ","End":"03:11.990","Text":"The next category has employees up to 268."},{"Start":"03:11.990 ","End":"03:19.970","Text":"This then would be the category that has the 160th employee."},{"Start":"03:19.970 ","End":"03:24.605","Text":"We know that 160th employee is the 40th percentile."},{"Start":"03:24.605 ","End":"03:32.715","Text":"How do we calculate the percentiles?"},{"Start":"03:32.715 ","End":"03:35.150","Text":"What\u0027s the exact value of the percentile?"},{"Start":"03:35.150 ","End":"03:42.620","Text":"Well, that equals to l_0 plus n times p divided by 100 minus F of"},{"Start":"03:42.620 ","End":"03:52.295","Text":"x_n minus 1 divided by f of x_n times L_1 minus L_0,"},{"Start":"03:52.295 ","End":"03:57.470","Text":"that\u0027s the expression used to calculate the exact value of the percentile."},{"Start":"03:57.470 ","End":"04:00.690","Text":"Now in our case, well, what\u0027s L_0?"},{"Start":"04:00.690 ","End":"04:06.920","Text":"L_0 is the lower bound of the category that has our percentile."},{"Start":"04:06.920 ","End":"04:10.040","Text":"Well, that\u0027s this thing right here, that\u0027s 6,000."},{"Start":"04:10.040 ","End":"04:16.760","Text":"That\u0027ll be equal to 6,000 plus n times p divided by 100, that\u0027s 160."},{"Start":"04:16.760 ","End":"04:18.380","Text":"We\u0027ve calculated that here,"},{"Start":"04:18.380 ","End":"04:21.620","Text":"so that\u0027s 160 minus."},{"Start":"04:21.620 ","End":"04:25.580","Text":"Now, what\u0027s the cumulative frequency of the preceding category?"},{"Start":"04:25.580 ","End":"04:27.079","Text":"Well, here\u0027s our category."},{"Start":"04:27.079 ","End":"04:29.430","Text":"That\u0027s the preceding category right here."},{"Start":"04:29.430 ","End":"04:35.990","Text":"The cumulative frequency is this thing right here That\u0027s 140 divided by"},{"Start":"04:35.990 ","End":"04:44.000","Text":"the frequency of this category, that\u0027s 128 times."},{"Start":"04:44.000 ","End":"04:48.440","Text":"Well, what\u0027s the width of this category right here?"},{"Start":"04:48.440 ","End":"04:53.160","Text":"Well, that\u0027s 10,000 minus 6,000 Now,"},{"Start":"04:53.160 ","End":"04:55.070","Text":"once I calculate all of this,"},{"Start":"04:55.070 ","End":"05:00.730","Text":"that comes out to 6,625."},{"Start":"05:00.730 ","End":"05:04.265","Text":"This is the 40th percentile."},{"Start":"05:04.265 ","End":"05:08.030","Text":"This equals to x_0.4."},{"Start":"05:08.030 ","End":"05:14.355","Text":"That means that 40 percent of the employees have salaries that are less than this value."},{"Start":"05:14.355 ","End":"05:17.735","Text":"Obviously, 60 percent of the employees have"},{"Start":"05:17.735 ","End":"05:22.410","Text":"salaries that are above this value right here."}],"ID":13745},{"Watched":false,"Name":"Tutorial 4","Duration":"2m 49s","ChapterTopicVideoID":13258,"CourseChapterTopicPlaylistID":64521,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.410","Text":"On section b,"},{"Start":"00:01.410 ","End":"00:06.795","Text":"we\u0027re asked what percentage of the employees earn less than $5,000?"},{"Start":"00:06.795 ","End":"00:11.280","Text":"On section a, we were asked to calculate the 40th percentile."},{"Start":"00:11.280 ","End":"00:13.290","Text":"We\u0027re asked to calculate"},{"Start":"00:13.290 ","End":"00:18.990","Text":"the x value where 40 percent of the employees were below this value."},{"Start":"00:18.990 ","End":"00:21.315","Text":"Now here, we\u0027re not asked that."},{"Start":"00:21.315 ","End":"00:26.550","Text":"That\u0027s the difference. We\u0027re given the x value and we want to know what is"},{"Start":"00:26.550 ","End":"00:35.760","Text":"the percentage of the population that\u0027s below or that has less than $5000."},{"Start":"00:35.760 ","End":"00:38.290","Text":"How do I calculate that?"},{"Start":"00:38.420 ","End":"00:44.765","Text":"If we want to derive the percentage of observations below a certain value,"},{"Start":"00:44.765 ","End":"00:47.105","Text":"then we use the following formula."},{"Start":"00:47.105 ","End":"00:49.550","Text":"That\u0027s the percentage."},{"Start":"00:49.550 ","End":"00:52.150","Text":"That equals to x,"},{"Start":"00:52.150 ","End":"00:53.540","Text":"what we\u0027re looking for, right?"},{"Start":"00:53.540 ","End":"01:01.340","Text":"What we\u0027re given minus the lower bound of the category divided by the width of"},{"Start":"01:01.340 ","End":"01:05.450","Text":"the category times the frequency at the category"},{"Start":"01:05.450 ","End":"01:10.430","Text":"plus the cumulative frequency of the preceding category."},{"Start":"01:10.430 ","End":"01:15.275","Text":"All that has to be multiplied by 100 divided by"},{"Start":"01:15.275 ","End":"01:21.470","Text":"n. Let\u0027s take a look at how we use this formula right here."},{"Start":"01:21.470 ","End":"01:25.175","Text":"This is the expression that we have to use."},{"Start":"01:25.175 ","End":"01:32.640","Text":"We\u0027re asked, what\u0027s the percentage of employees that earn less than $5,000?"},{"Start":"01:32.640 ","End":"01:34.680","Text":"Where is $5,000?"},{"Start":"01:34.680 ","End":"01:36.970","Text":"In which category is $5000?"},{"Start":"01:36.970 ","End":"01:39.860","Text":"That\u0027s in this category right here."},{"Start":"01:39.860 ","End":"01:43.710","Text":"That\u0027s this category right here."},{"Start":"01:43.710 ","End":"01:46.950","Text":"Let\u0027s just plug in the numbers."},{"Start":"01:46.950 ","End":"01:49.905","Text":"We have $5,000."},{"Start":"01:49.905 ","End":"01:56.015","Text":"5,000 minus the lower bound of the category that has the $5,000."},{"Start":"01:56.015 ","End":"01:57.730","Text":"That\u0027s 4,000, right?"},{"Start":"01:57.730 ","End":"02:02.565","Text":"4,000 divided by the width of the category."},{"Start":"02:02.565 ","End":"02:08.640","Text":"That\u0027s 6,000 minus 4,000 times the frequency,"},{"Start":"02:08.640 ","End":"02:13.455","Text":"and the frequency here is 140 plus"},{"Start":"02:13.455 ","End":"02:20.615","Text":"the cumulative frequency of the preceding category."},{"Start":"02:20.615 ","End":"02:26.545","Text":"There\u0027s no preceding category here because this is the first category, so that\u0027s 0."},{"Start":"02:26.545 ","End":"02:34.810","Text":"This, we have to multiply it by 100 divided by 400. N is 400."},{"Start":"02:34.810 ","End":"02:40.380","Text":"Now that comes out to 17.5 percent."},{"Start":"02:40.380 ","End":"02:50.080","Text":"That means that 17.5 percent of the employees earn less than $5,000."}],"ID":13746},{"Watched":false,"Name":"Exercise 1 - Parts a-b","Duration":"8m 22s","ChapterTopicVideoID":13259,"CourseChapterTopicPlaylistID":64521,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"In this question we\u0027re given a table that displays"},{"Start":"00:02.220 ","End":"00:06.480","Text":"the probability distribution of the salaries in thousands of dollars in a company."},{"Start":"00:06.480 ","End":"00:10.469","Text":"Now, we\u0027re asked to calculate the 60th percentile."},{"Start":"00:10.469 ","End":"00:13.860","Text":"Let\u0027s take a look at the table right here. We have 2 columns."},{"Start":"00:13.860 ","End":"00:18.059","Text":"The first column is the variable x and that\u0027s salary."},{"Start":"00:18.059 ","End":"00:21.855","Text":"Now, we have the salary that\u0027s divided into categories."},{"Start":"00:21.855 ","End":"00:24.060","Text":"The first category is from 6-10,"},{"Start":"00:24.060 ","End":"00:27.805","Text":"that means that this is from $6,000-$10,000."},{"Start":"00:27.805 ","End":"00:33.665","Text":"The second category is from $10,000-$15,000, and so on and so forth."},{"Start":"00:33.665 ","End":"00:37.790","Text":"The second column is the cumulative frequency,"},{"Start":"00:37.790 ","End":"00:40.310","Text":"that\u0027s F of x."},{"Start":"00:40.310 ","End":"00:44.000","Text":"That means that in the first category we have 48 employees,"},{"Start":"00:44.000 ","End":"00:47.180","Text":"but in the second category,"},{"Start":"00:47.180 ","End":"00:49.860","Text":"we do not have 100 employees."},{"Start":"00:49.860 ","End":"00:53.390","Text":"We have 100 employees up to the second category,"},{"Start":"00:53.390 ","End":"00:58.678","Text":"and 120 employees up to the third category,"},{"Start":"00:58.678 ","End":"01:00.205","Text":"and so on and so forth."},{"Start":"01:00.205 ","End":"01:04.505","Text":"The first thing that we want to do is we want to calculate f of x,"},{"Start":"01:04.505 ","End":"01:08.645","Text":"which is the frequency for each category."},{"Start":"01:08.645 ","End":"01:10.280","Text":"In the first category,"},{"Start":"01:10.280 ","End":"01:12.380","Text":"that\u0027ll be easy, that\u0027s 48."},{"Start":"01:12.380 ","End":"01:14.824","Text":"Now what about in the second category?"},{"Start":"01:14.824 ","End":"01:17.195","Text":"Well, from 48 to 100,"},{"Start":"01:17.195 ","End":"01:18.530","Text":"how many employees do I have?"},{"Start":"01:18.530 ","End":"01:23.900","Text":"Well, I have 52. That\u0027s the frequency in the second category."},{"Start":"01:23.900 ","End":"01:27.140","Text":"Now, how many employees do I have in the third category?"},{"Start":"01:27.140 ","End":"01:30.535","Text":"Well, that\u0027s from 100 to 120, that\u0027s 20."},{"Start":"01:30.535 ","End":"01:33.795","Text":"From 120 to 132, that\u0027s 12."},{"Start":"01:33.795 ","End":"01:37.965","Text":"From 132 to 136, that\u0027s 4."},{"Start":"01:37.965 ","End":"01:42.705","Text":"All in all, I have 136 employees."},{"Start":"01:42.705 ","End":"01:47.720","Text":"The next thing that I want to do is I want to calculate the position of"},{"Start":"01:47.720 ","End":"01:54.310","Text":"the 60th percentile in order to see in what category is it placed."},{"Start":"01:54.310 ","End":"01:57.360","Text":"Well, let\u0027s calculate that."},{"Start":"01:57.360 ","End":"01:59.420","Text":"Well, the position is this,"},{"Start":"01:59.420 ","End":"02:02.029","Text":"that\u0027s n times p divided by 100."},{"Start":"02:02.029 ","End":"02:05.210","Text":"That\u0027s the position of the 60th percentile."},{"Start":"02:05.210 ","End":"02:07.095","Text":"Now, in our case,"},{"Start":"02:07.095 ","End":"02:11.490","Text":"that\u0027ll be n. That\u0027ll be 136 times p. Well,"},{"Start":"02:11.490 ","End":"02:14.850","Text":"p is 60 divided by 100."},{"Start":"02:14.850 ","End":"02:19.020","Text":"That equals to 81.6."},{"Start":"02:19.020 ","End":"02:24.720","Text":"The 60th percentile is in the 81.6th place."},{"Start":"02:24.720 ","End":"02:29.060","Text":"Now obviously, we don\u0027t have a 0.6 place here,"},{"Start":"02:29.060 ","End":"02:36.890","Text":"but this then is the value of the position of the 60th percentile."},{"Start":"02:36.890 ","End":"02:38.620","Text":"Now where is that located?"},{"Start":"02:38.620 ","End":"02:42.634","Text":"Well, up to the first category we have 48 employees,"},{"Start":"02:42.634 ","End":"02:46.910","Text":"but up to the second category we have 100 employees."},{"Start":"02:46.910 ","End":"02:53.310","Text":"The 81.6th place would be in the second category right here."},{"Start":"02:53.840 ","End":"02:57.425","Text":"Having calculated the position,"},{"Start":"02:57.425 ","End":"03:02.090","Text":"let\u0027s now calculate the exact value of the 60th percentile."},{"Start":"03:02.090 ","End":"03:04.790","Text":"That\u0027s X_0.6."},{"Start":"03:04.790 ","End":"03:11.000","Text":"Now that equals to L_0 plus n times p"},{"Start":"03:11.000 ","End":"03:17.428","Text":"divided by 100 minus F of x_m minus 1,"},{"Start":"03:17.428 ","End":"03:23.955","Text":"divided by f of x_m times L_1 minus L_0."},{"Start":"03:23.955 ","End":"03:26.310","Text":"Now, again, what does that mean?"},{"Start":"03:26.310 ","End":"03:34.130","Text":"Well, L_0 is the lower bound of the category where we found our 60th percentile."},{"Start":"03:34.130 ","End":"03:37.385","Text":"Well, that\u0027s 10, that\u0027s this value right here."},{"Start":"03:37.385 ","End":"03:40.545","Text":"That equals to 10 plus,"},{"Start":"03:40.545 ","End":"03:42.750","Text":"now n times p divided by 100, well,"},{"Start":"03:42.750 ","End":"03:51.740","Text":"that\u0027s 81.6, minus the cumulative frequency of the preceding category."},{"Start":"03:51.740 ","End":"03:54.620","Text":"Well, this is a category where we found our percentile."},{"Start":"03:54.620 ","End":"03:57.695","Text":"So this is a preceding category."},{"Start":"03:57.695 ","End":"04:03.343","Text":"This would be the cumulative frequency, that\u0027s 48,"},{"Start":"04:03.343 ","End":"04:07.340","Text":"divided by f of x and m. That means"},{"Start":"04:07.340 ","End":"04:11.720","Text":"that this is the frequency of the category where we found our percentile."},{"Start":"04:11.720 ","End":"04:15.065","Text":"Well, the frequency is 52."},{"Start":"04:15.065 ","End":"04:22.000","Text":"That\u0027ll be here. Times L_1 minus L_0."},{"Start":"04:22.000 ","End":"04:24.695","Text":"Well, that\u0027s the upper bound minus the lower bound."},{"Start":"04:24.695 ","End":"04:28.595","Text":"That\u0027s the width of this category, 15 minus 10."},{"Start":"04:28.595 ","End":"04:33.900","Text":"Now, all this comes out to 13.23."},{"Start":"04:34.010 ","End":"04:38.700","Text":"Now, that\u0027s a dollar value in thousands of dollars."},{"Start":"04:38.700 ","End":"04:41.180","Text":"We have here, okay,"},{"Start":"04:41.180 ","End":"04:47.885","Text":"$13.23 thousand and that\u0027s our 60th percentile."},{"Start":"04:47.885 ","End":"04:49.434","Text":"In this section we\u0027re asked,"},{"Start":"04:49.434 ","End":"04:51.610","Text":"what\u0027s the top 10th percentile?"},{"Start":"04:51.610 ","End":"04:54.874","Text":"Well, what is the top 10th percentile?"},{"Start":"04:54.874 ","End":"04:58.235","Text":"That\u0027s the x value, the salary,"},{"Start":"04:58.235 ","End":"05:04.940","Text":"where 10 percent of the employees have a salary above that value."},{"Start":"05:04.940 ","End":"05:15.215","Text":"That means also that 90 percent of the employees have a salary below that value."},{"Start":"05:15.215 ","End":"05:20.900","Text":"Well, that\u0027s exactly the definition of the 90th percentile."},{"Start":"05:20.900 ","End":"05:24.950","Text":"The 90th percentile is the x value or"},{"Start":"05:24.950 ","End":"05:31.550","Text":"the salary value where 90 percent of the employees have a salary below that value,"},{"Start":"05:31.550 ","End":"05:37.295","Text":"so the top 10th percentile is equal to the 90th percentile."},{"Start":"05:37.295 ","End":"05:40.520","Text":"The first thing that we need to do is we need to calculate"},{"Start":"05:40.520 ","End":"05:43.520","Text":"the place of the 90th percentile."},{"Start":"05:43.520 ","End":"05:47.810","Text":"We want to know in which category can we find the 90th percentile."},{"Start":"05:47.810 ","End":"05:50.150","Text":"Well, how do we calculate that?"},{"Start":"05:50.150 ","End":"05:52.865","Text":"That\u0027s n times p divided by 100."},{"Start":"05:52.865 ","End":"05:56.095","Text":"Well, n here is 136."},{"Start":"05:56.095 ","End":"05:59.390","Text":"Here we\u0027re looking for the 90th percentile."},{"Start":"05:59.390 ","End":"06:02.728","Text":"That\u0027ll be 90 divided by 100,"},{"Start":"06:02.728 ","End":"06:08.180","Text":"and that equals to 122.4."},{"Start":"06:08.180 ","End":"06:13.849","Text":"We know that the 90th percentile is in the 122.4th place."},{"Start":"06:13.849 ","End":"06:17.759","Text":"Now, where is that here in our distribution?"},{"Start":"06:17.759 ","End":"06:24.885","Text":"Well, we see that up to the third category we have 120 people,"},{"Start":"06:24.885 ","End":"06:29.615","Text":"and up to the fourth category we have 132 people."},{"Start":"06:29.615 ","End":"06:37.500","Text":"That means that the 122.4th place that\u0027s in this category right here."},{"Start":"06:37.970 ","End":"06:41.225","Text":"Once I\u0027ve identified what the category is,"},{"Start":"06:41.225 ","End":"06:48.305","Text":"let\u0027s calculate the exact value of X_0.9 or the 90th percentile."},{"Start":"06:48.305 ","End":"06:55.445","Text":"Well, that equals to L_0 plus n times p divided by 100 minus F of"},{"Start":"06:55.445 ","End":"07:04.710","Text":"x_m minus 1 divided by f of x_m times L_1 minus L_0."},{"Start":"07:05.120 ","End":"07:09.365","Text":"Well, let\u0027s take a look at the numbers here."},{"Start":"07:09.365 ","End":"07:10.850","Text":"Well, what\u0027s L_0?"},{"Start":"07:10.850 ","End":"07:14.330","Text":"Well, that\u0027s the lower bound of"},{"Start":"07:14.330 ","End":"07:18.469","Text":"the category where we found our 90th percentile. That\u0027s right here."},{"Start":"07:18.469 ","End":"07:21.420","Text":"The lower bound is 20,"},{"Start":"07:21.420 ","End":"07:23.839","Text":"plus n times p divided by 100,"},{"Start":"07:23.839 ","End":"07:25.040","Text":"where we calculated that,"},{"Start":"07:25.040 ","End":"07:32.045","Text":"that\u0027s a 122.4 minus the cumulative frequency of the preceding category."},{"Start":"07:32.045 ","End":"07:33.830","Text":"Well, this is the preceding category,"},{"Start":"07:33.830 ","End":"07:35.780","Text":"so the cumulative frequency here,"},{"Start":"07:35.780 ","End":"07:44.675","Text":"that\u0027s 120 divided by the frequency of the category where we found our percentile,"},{"Start":"07:44.675 ","End":"07:49.220","Text":"and that\u0027s 12 times the width of the category."},{"Start":"07:49.220 ","End":"07:50.360","Text":"Well, that\u0027s the upper bound,"},{"Start":"07:50.360 ","End":"07:54.170","Text":"that\u0027s 30 minus the lower bound, that\u0027s 20."},{"Start":"07:54.170 ","End":"07:58.400","Text":"Now, that comes out to 22."},{"Start":"07:58.400 ","End":"08:00.770","Text":"Now, this number right here,"},{"Start":"08:00.770 ","End":"08:03.830","Text":"that\u0027s a dollar figure in thousands of dollars,"},{"Start":"08:03.830 ","End":"08:05.225","Text":"so what we have here."},{"Start":"08:05.225 ","End":"08:12.720","Text":"Then X_0.9, or the 90th percentile, is $22,000."},{"Start":"08:12.720 ","End":"08:18.625","Text":"This is the value where 10 percent of the employees earn above this value,"},{"Start":"08:18.625 ","End":"08:23.370","Text":"and 90 percent of the employees earn below this value."}],"ID":13747},{"Watched":false,"Name":"Exercise 1 - Parts c-d","Duration":"6m 1s","ChapterTopicVideoID":13260,"CourseChapterTopicPlaylistID":64521,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.710","Text":"In this section we\u0027re given that"},{"Start":"00:01.710 ","End":"00:05.295","Text":"the highest 20 percent of the salaries are earned by senior employees,"},{"Start":"00:05.295 ","End":"00:09.090","Text":"and we\u0027re asked, what\u0027s the minimum salary for a senior employee?"},{"Start":"00:09.090 ","End":"00:12.660","Text":"Let\u0027s try to understand this question."},{"Start":"00:12.660 ","End":"00:16.530","Text":"We\u0027re given that the highest 20 percent of the salaries are earned by senior employees,"},{"Start":"00:16.530 ","End":"00:24.060","Text":"that means that 20 percent of the employees earn above a specific value,"},{"Start":"00:24.060 ","End":"00:28.635","Text":"so we\u0027re looking for the top 20 percent."},{"Start":"00:28.635 ","End":"00:36.270","Text":"The salary value where 20 percent of the employees earn above that value."},{"Start":"00:36.270 ","End":"00:42.715","Text":"But that means that 80 percent of the employees earn below the specific value,"},{"Start":"00:42.715 ","End":"00:48.409","Text":"so that\u0027s exactly the definition of the 80th percentile."},{"Start":"00:48.409 ","End":"00:52.310","Text":"Where 80 percent of the employees earn below"},{"Start":"00:52.310 ","End":"00:57.785","Text":"this value and 20 percent of the employees earn above this value."},{"Start":"00:57.785 ","End":"00:59.690","Text":"Now, in order to calculate this,"},{"Start":"00:59.690 ","End":"01:03.155","Text":"the first thing that we need to do is we need to calculate its position."},{"Start":"01:03.155 ","End":"01:04.490","Text":"Now how do we do that?"},{"Start":"01:04.490 ","End":"01:07.820","Text":"Well, that\u0027s n times p divided by 100,"},{"Start":"01:07.820 ","End":"01:14.330","Text":"so that equals to 136 times 80 divided by 100."},{"Start":"01:14.330 ","End":"01:18.300","Text":"That equals to 108.8."},{"Start":"01:18.940 ","End":"01:22.535","Text":"Now, where is that located?"},{"Start":"01:22.535 ","End":"01:25.790","Text":"We see that up to the second category,"},{"Start":"01:25.790 ","End":"01:27.560","Text":"we have 100 employees,"},{"Start":"01:27.560 ","End":"01:30.160","Text":"and up to the third category,"},{"Start":"01:30.160 ","End":"01:33.560","Text":"we have 120 employees, so 100,"},{"Start":"01:33.560 ","End":"01:40.315","Text":"the position 108.8 falls in this category right here in the third category."},{"Start":"01:40.315 ","End":"01:46.639","Text":"Having identified that, let\u0027s calculate the exact value of the 80th percentile."},{"Start":"01:46.639 ","End":"01:55.640","Text":"That\u0027s X_0.8, that equals to L_0 plus n times p divided by 100 minus"},{"Start":"01:55.640 ","End":"02:00.620","Text":"big F of x of m minus 1 divided by"},{"Start":"02:00.620 ","End":"02:06.765","Text":"small f of X_m times L_1 minus L_0."},{"Start":"02:06.765 ","End":"02:08.835","Text":"Now let\u0027s just plug in the numbers."},{"Start":"02:08.835 ","End":"02:15.070","Text":"L_0, that\u0027s the lower bound of the category where the percentile is,"},{"Start":"02:15.070 ","End":"02:18.539","Text":"so that\u0027s 15 plus,"},{"Start":"02:18.539 ","End":"02:20.670","Text":"well, n times p divided by 100."},{"Start":"02:20.670 ","End":"02:24.355","Text":"We\u0027ve calculated that, that\u0027s 108.8."},{"Start":"02:24.355 ","End":"02:26.630","Text":"What\u0027s this with us?"},{"Start":"02:26.630 ","End":"02:29.720","Text":"The cumulative frequency of the preceding category."},{"Start":"02:29.720 ","End":"02:31.655","Text":"Well, here\u0027s the preceding category,"},{"Start":"02:31.655 ","End":"02:34.880","Text":"and the cumulative distribution is this thing right here."},{"Start":"02:34.880 ","End":"02:37.430","Text":"The cumulative frequency is this thing right here."},{"Start":"02:37.430 ","End":"02:41.150","Text":"That\u0027ll be minus 100 divided by,"},{"Start":"02:41.150 ","End":"02:43.655","Text":"well, small f of X_m,"},{"Start":"02:43.655 ","End":"02:45.950","Text":"small f, that\u0027s the frequency,"},{"Start":"02:45.950 ","End":"02:50.505","Text":"of the category right here, and that\u0027s 20,"},{"Start":"02:50.505 ","End":"02:53.475","Text":"times, now,"},{"Start":"02:53.475 ","End":"02:54.840","Text":"L_1 minus L_0,"},{"Start":"02:54.840 ","End":"02:57.380","Text":"well that\u0027s the width of the category."},{"Start":"02:57.380 ","End":"03:01.490","Text":"That\u0027s the upper bound, 20 minus 15."},{"Start":"03:01.490 ","End":"03:06.360","Text":"That equals to 17.2."},{"Start":"03:06.360 ","End":"03:09.185","Text":"Now, this is a dollar figure in thousands."},{"Start":"03:09.185 ","End":"03:15.770","Text":"The 80th percentile is $17.2 thousand,"},{"Start":"03:15.770 ","End":"03:20.509","Text":"but that\u0027s the minimum salary for a senior employee,"},{"Start":"03:20.509 ","End":"03:28.125","Text":"and we know that senior employees are in the top 20 percent of salaries."},{"Start":"03:28.125 ","End":"03:35.840","Text":"This then would be the minimum value for the top 10 or top 20 salaries."},{"Start":"03:35.840 ","End":"03:41.825","Text":"In this section, we\u0027re asked what percentage of employees earn less than $7,000."},{"Start":"03:41.825 ","End":"03:45.830","Text":"Well, this is a slightly different question than what we had in the preceding sections."},{"Start":"03:45.830 ","End":"03:48.125","Text":"In the preceding sections,"},{"Start":"03:48.125 ","End":"03:55.965","Text":"we were given the percentage of employees that were below a specific dollar amount,"},{"Start":"03:55.965 ","End":"03:59.045","Text":"and we were asked to calculate that dollar amount."},{"Start":"03:59.045 ","End":"04:02.480","Text":"Here, we\u0027re doing the exact opposite."},{"Start":"04:02.480 ","End":"04:05.915","Text":"We\u0027re given the dollar amount and we\u0027re asked to calculate"},{"Start":"04:05.915 ","End":"04:10.800","Text":"the percentage of employees who earn less than that dollar amount."},{"Start":"04:10.800 ","End":"04:12.485","Text":"How do we do that?"},{"Start":"04:12.485 ","End":"04:18.545","Text":"Well, here we\u0027re looking for the percentage of 7."},{"Start":"04:18.545 ","End":"04:23.105","Text":"P_7, percentage of $7,000."},{"Start":"04:23.105 ","End":"04:26.074","Text":"Now that equals to the following."},{"Start":"04:26.074 ","End":"04:34.080","Text":"Well, that\u0027s X minus L_0 divided by L_1 minus L_0 times f"},{"Start":"04:34.080 ","End":"04:43.260","Text":"of X_m plus F of X_m minus 1,"},{"Start":"04:43.260 ","End":"04:48.110","Text":"and all that, we have to multiply that by 100 divided"},{"Start":"04:48.110 ","End":"04:53.195","Text":"by n in order to calculate the percentage."},{"Start":"04:53.195 ","End":"04:56.180","Text":"Now, in our case X here,"},{"Start":"04:56.180 ","End":"05:00.950","Text":"what that equals to 7, because we\u0027re dealing in thousands of dollars,"},{"Start":"05:00.950 ","End":"05:02.215","Text":"so this is 7."},{"Start":"05:02.215 ","End":"05:03.940","Text":"Now, where\u0027s that situated?"},{"Start":"05:03.940 ","End":"05:08.740","Text":"Well $7,000 that\u0027s situated in this category right here."},{"Start":"05:08.740 ","End":"05:12.920","Text":"What\u0027s the lower bound of this category?"},{"Start":"05:12.920 ","End":"05:17.705","Text":"Would that equals to 6 divided by L_1 minus L_0?"},{"Start":"05:17.705 ","End":"05:22.100","Text":"Well, the upper bound is 10 and the lower bound, as we said,"},{"Start":"05:22.100 ","End":"05:28.450","Text":"is 6, times the frequency of this category with a frequency that\u0027s 48."},{"Start":"05:28.450 ","End":"05:33.020","Text":"Plus, well, the cumulative frequency of"},{"Start":"05:33.020 ","End":"05:35.360","Text":"the preceding category where there\u0027s"},{"Start":"05:35.360 ","End":"05:37.910","Text":"no preceding category because it\u0027s the first category,"},{"Start":"05:37.910 ","End":"05:39.670","Text":"so that equals to 0,"},{"Start":"05:39.670 ","End":"05:47.450","Text":"and now we have to multiply all this by 100 divided by n. Now n is 136,"},{"Start":"05:47.450 ","End":"05:53.120","Text":"so this comes out to 8.82 percent,"},{"Start":"05:53.120 ","End":"06:02.130","Text":"so 8.82 percent of employees earned less than $7,000."}],"ID":13748},{"Watched":false,"Name":"Exercise 1 - Parts e-f","Duration":"4m 31s","ChapterTopicVideoID":13261,"CourseChapterTopicPlaylistID":64521,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.405","Text":"In this section we\u0027re asked what percentage of the employees earn more than $25,000."},{"Start":"00:06.405 ","End":"00:10.260","Text":"When we calculate the percentage,"},{"Start":"00:10.260 ","End":"00:16.755","Text":"we calculate the percentage of employees that earn less than a specific dollar amount."},{"Start":"00:16.755 ","End":"00:22.425","Text":"Here we\u0027re asked about the percentage of employees that earn more than $25,000."},{"Start":"00:22.425 ","End":"00:23.625","Text":"How do we solve that?"},{"Start":"00:23.625 ","End":"00:26.160","Text":"Well, we have to do this."},{"Start":"00:26.160 ","End":"00:34.875","Text":"We have 100 percent of the employees minus the percentage that earn less than $25,000."},{"Start":"00:34.875 ","End":"00:37.995","Text":"That we\u0027re dealing in thousands of dollars."},{"Start":"00:37.995 ","End":"00:44.155","Text":"That will give us the percentage of employees that earn more than $25,000."},{"Start":"00:44.155 ","End":"00:49.425","Text":"What we need to do is we need to calculate P_25."},{"Start":"00:49.425 ","End":"00:54.570","Text":"That equals to x minus L0"},{"Start":"00:54.570 ","End":"01:00.945","Text":"divided by L1 minus L0 times f of x_m,"},{"Start":"01:00.945 ","End":"01:07.350","Text":"small for f of x_m plus big F of x_m minus 1."},{"Start":"01:07.350 ","End":"01:15.180","Text":"All that we have to multiply by 100 divided by n. Let\u0027s just plug in the numbers."},{"Start":"01:15.180 ","End":"01:21.210","Text":"X here, that\u0027s $25,000, that\u0027s 25 minus."},{"Start":"01:21.210 ","End":"01:23.670","Text":"Now, where is $25,000?"},{"Start":"01:23.670 ","End":"01:25.935","Text":"In what category can we find this?"},{"Start":"01:25.935 ","End":"01:28.860","Text":"That\u0027s right here between 20 and 30."},{"Start":"01:28.860 ","End":"01:33.360","Text":"That\u0027s right here. What\u0027s the lower bound of this category?"},{"Start":"01:33.360 ","End":"01:37.320","Text":"That\u0027s 20 divided by the width of the category."},{"Start":"01:37.320 ","End":"01:40.430","Text":"That\u0027s the upper bound minus the lower bound that\u0027s"},{"Start":"01:40.430 ","End":"01:43.640","Text":"20 times the frequency of this category."},{"Start":"01:43.640 ","End":"01:46.765","Text":"That\u0027s 12 plus."},{"Start":"01:46.765 ","End":"01:49.155","Text":"Now, what do we have here?"},{"Start":"01:49.155 ","End":"01:55.065","Text":"That\u0027s the cumulative frequency of the preceding category."},{"Start":"01:55.065 ","End":"01:56.810","Text":"Here\u0027s the preceding category,"},{"Start":"01:56.810 ","End":"01:58.880","Text":"and here\u0027s the cumulative frequency."},{"Start":"01:58.880 ","End":"02:00.485","Text":"That\u0027s 120."},{"Start":"02:00.485 ","End":"02:08.910","Text":"All of this I need to multiply that by 100 divided by n. Now n is 136."},{"Start":"02:08.910 ","End":"02:17.600","Text":"That equals to 92.64 percent."},{"Start":"02:17.600 ","End":"02:26.280","Text":"Here this number gives me the percentage of employees that earn less than $25,000,"},{"Start":"02:26.280 ","End":"02:30.905","Text":"but we want how many employees or what\u0027s the percentage of employees that earned more?"},{"Start":"02:30.905 ","End":"02:34.010","Text":"What I need to do now is go like this."},{"Start":"02:34.010 ","End":"02:37.920","Text":"That\u0027ll be 100 percent minus,"},{"Start":"02:37.920 ","End":"02:39.315","Text":"just like we did here,"},{"Start":"02:39.315 ","End":"02:47.400","Text":"92.64 percent and that equals to 7.36 percent."},{"Start":"02:47.400 ","End":"02:55.820","Text":"Now I know that 7.36 percent of the employees earn more than $25,000."},{"Start":"02:55.820 ","End":"02:58.760","Text":"In this section we\u0027re asked what percentage of the employees earned"},{"Start":"02:58.760 ","End":"03:02.530","Text":"between $7,000 and $25,000?"},{"Start":"03:02.530 ","End":"03:06.190","Text":"Let\u0027s just take a look at the salary axis here."},{"Start":"03:06.190 ","End":"03:11.835","Text":"Here we have $25,000 and here we have $7,000."},{"Start":"03:11.835 ","End":"03:16.950","Text":"We\u0027re looking for the percentage of employees that earned between $7,000 and $25,000."},{"Start":"03:19.070 ","End":"03:23.130","Text":"How do I calculate this percentage?"},{"Start":"03:23.590 ","End":"03:31.880","Text":"P_25 is the percentage of employees that earn less than $25,000."},{"Start":"03:31.880 ","End":"03:34.680","Text":"This thing right here."},{"Start":"03:36.890 ","End":"03:43.035","Text":"P_7 is the percentage of employees that earn less than $7,000."},{"Start":"03:43.035 ","End":"03:47.260","Text":"If I subtract P_7 from P_25,"},{"Start":"03:47.260 ","End":"03:54.030","Text":"that\u0027ll give me the percentage of employees that earn between $7,000 and $25,000."},{"Start":"03:56.060 ","End":"04:00.515","Text":"I\u0027ve calculated these values in the previous sections."},{"Start":"04:00.515 ","End":"04:05.935","Text":"P_25 that equal to 92.64."},{"Start":"04:05.935 ","End":"04:10.150","Text":"The percentage P_7,"},{"Start":"04:10.150 ","End":"04:13.295","Text":"that equals to 8.82."},{"Start":"04:13.295 ","End":"04:19.500","Text":"That equals to 83.82."},{"Start":"04:19.500 ","End":"04:21.165","Text":"This is percent."},{"Start":"04:21.165 ","End":"04:32.230","Text":"I know now that 83.82 percent of the employees earned between $7,000 and $25,000."}],"ID":13749},{"Watched":false,"Name":"Exercise 2","Duration":"4m 15s","ChapterTopicVideoID":13262,"CourseChapterTopicPlaylistID":64521,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.150","Text":"In this question, we\u0027re given 400 people took an exam."},{"Start":"00:03.150 ","End":"00:07.290","Text":"Now, it\u0027s known that the bottom 10th percentile is the mark 60,"},{"Start":"00:07.290 ","End":"00:09.705","Text":"the top quarter is a mark 80,"},{"Start":"00:09.705 ","End":"00:13.635","Text":"and the probability distribution of the mark is symmetrical."},{"Start":"00:13.635 ","End":"00:16.050","Text":"We\u0027re asked to fill in the missing frequencies."},{"Start":"00:16.050 ","End":"00:17.895","Text":"Here we have a frequency table."},{"Start":"00:17.895 ","End":"00:20.340","Text":"The first column, that\u0027s my variable,"},{"Start":"00:20.340 ","End":"00:21.990","Text":"that\u0027s X, and that\u0027s the mark."},{"Start":"00:21.990 ","End":"00:25.375","Text":"Now, the mark is divided into categories from 50-60."},{"Start":"00:25.375 ","End":"00:26.775","Text":"That\u0027s the first category."},{"Start":"00:26.775 ","End":"00:30.690","Text":"Second category is from 60-70 and so on and so forth."},{"Start":"00:30.690 ","End":"00:35.295","Text":"Now, we\u0027re asked to fill in these frequencies for each category."},{"Start":"00:35.295 ","End":"00:37.905","Text":"Let\u0027s just write down what we know."},{"Start":"00:37.905 ","End":"00:40.685","Text":"We know that we have 400 people,"},{"Start":"00:40.685 ","End":"00:45.000","Text":"and that\u0027s right here, and we know that the bottom 10th percentile is 60."},{"Start":"00:45.000 ","End":"00:47.160","Text":"So X_ 0.1,"},{"Start":"00:47.160 ","End":"00:49.034","Text":"that\u0027s the 10th percentile,"},{"Start":"00:49.034 ","End":"00:50.910","Text":"that equals to 60."},{"Start":"00:50.910 ","End":"00:54.995","Text":"We also know that the top quartile is the mark 80."},{"Start":"00:54.995 ","End":"00:56.225","Text":"Now what\u0027s the top quartile?"},{"Start":"00:56.225 ","End":"01:01.145","Text":"That\u0027s the last 25 percent of the population,"},{"Start":"01:01.145 ","End":"01:06.530","Text":"so 25 percent of the people might have a mark of 80 or more."},{"Start":"01:06.530 ","End":"01:12.185","Text":"That means that 75 percent of the people have a mark of 80 or less."},{"Start":"01:12.185 ","End":"01:17.430","Text":"But that\u0027s exactly the definition of the 75th percentile."},{"Start":"01:17.650 ","End":"01:21.710","Text":"So X_0.75, that equals to 80."},{"Start":"01:21.710 ","End":"01:25.835","Text":"Now, so here we have the 10th percentile,"},{"Start":"01:25.835 ","End":"01:28.720","Text":"here we have the 75th percentile."},{"Start":"01:28.720 ","End":"01:34.220","Text":"What else do we know? Well, we know that the distribution of the marks is symmetrical."},{"Start":"01:34.220 ","End":"01:37.970","Text":"Let\u0027s now calculate these frequencies."},{"Start":"01:37.970 ","End":"01:40.930","Text":"We know that n equals to 400."},{"Start":"01:40.930 ","End":"01:43.400","Text":"Let\u0027s take a look at this thing right here."},{"Start":"01:43.400 ","End":"01:45.200","Text":"What\u0027s the 10th percentile?"},{"Start":"01:45.200 ","End":"01:53.855","Text":"That\u0027s 10 percent of the people who received a mark of 60 or less."},{"Start":"01:53.855 ","End":"01:56.570","Text":"Now, what\u0027s 10 percent of the people?"},{"Start":"01:56.570 ","End":"01:59.990","Text":"Well, that\u0027s 0.1, that\u0027s 10 percent times 400."},{"Start":"01:59.990 ","End":"02:01.640","Text":"That equals to 40."},{"Start":"02:01.640 ","End":"02:06.230","Text":"We have 40 people that received the mark of 60 or less."},{"Start":"02:06.230 ","End":"02:08.480","Text":"Now let\u0027s take a look at the categories here."},{"Start":"02:08.480 ","End":"02:13.160","Text":"The first category, what are the bounds of the first category?"},{"Start":"02:13.160 ","End":"02:15.010","Text":"That\u0027s between 50 and 60."},{"Start":"02:15.010 ","End":"02:20.156","Text":"That means that these 40 people,"},{"Start":"02:20.156 ","End":"02:22.385","Text":"they received the mark of 60 or less,"},{"Start":"02:22.385 ","End":"02:25.685","Text":"that means that they\u0027re right here in this category right here."},{"Start":"02:25.685 ","End":"02:29.135","Text":"Now, because the distribution is symmetrical,"},{"Start":"02:29.135 ","End":"02:32.510","Text":"then the same frequency,"},{"Start":"02:32.510 ","End":"02:35.600","Text":"the frequency in that bottom category"},{"Start":"02:35.600 ","End":"02:38.870","Text":"here would be equal to the frequency in that top category."},{"Start":"02:38.870 ","End":"02:42.140","Text":"That\u0027s what makes this distribution symmetrical."},{"Start":"02:42.140 ","End":"02:45.120","Text":"Here that will be 40 as well."},{"Start":"02:45.230 ","End":"02:48.695","Text":"What about this thing right here?"},{"Start":"02:48.695 ","End":"02:51.425","Text":"Well, that\u0027s the 75th percentile."},{"Start":"02:51.425 ","End":"02:57.095","Text":"That means that that 75 percent of the population received a mark of 80 or less."},{"Start":"02:57.095 ","End":"03:05.375","Text":"That means that 75 percent of the people are in these 3 categories."},{"Start":"03:05.375 ","End":"03:12.565","Text":"But that means that 25 percent of the people are in these categories right here."},{"Start":"03:12.565 ","End":"03:15.450","Text":"What\u0027s 25 percent of 400?"},{"Start":"03:15.450 ","End":"03:18.420","Text":"Well that\u0027s 0.25 times 400,"},{"Start":"03:18.420 ","End":"03:20.085","Text":"and that equals to a 100."},{"Start":"03:20.085 ","End":"03:23.660","Text":"A 100 people are in these 2 categories."},{"Start":"03:23.660 ","End":"03:27.980","Text":"Now, we know that we have 40 people in the top category."},{"Start":"03:27.980 ","End":"03:31.700","Text":"That means that here we need to have 60 people"},{"Start":"03:31.700 ","End":"03:35.810","Text":"in order for these 2 categories to be equal to 100 people."},{"Start":"03:35.810 ","End":"03:39.830","Text":"Now, again, because the distribution is symmetrical,"},{"Start":"03:39.830 ","End":"03:43.640","Text":"then here that\u0027ll be 60 as well."},{"Start":"03:43.640 ","End":"03:47.495","Text":"This is what makes this distribution symmetrical."},{"Start":"03:47.495 ","End":"03:52.515","Text":"We have 40 people in the top and bottom categories,"},{"Start":"03:52.515 ","End":"03:56.255","Text":"and then in the next categories we have 60 people."},{"Start":"03:56.255 ","End":"03:58.520","Text":"Now we have a total of 400 people."},{"Start":"03:58.520 ","End":"04:00.245","Text":"How many people do we have now?"},{"Start":"04:00.245 ","End":"04:02.360","Text":"Well, we have 100 people in"},{"Start":"04:02.360 ","End":"04:06.785","Text":"the bottom 2 categories and 100 people in the top 2 categories."},{"Start":"04:06.785 ","End":"04:09.065","Text":"That means that in the middle category,"},{"Start":"04:09.065 ","End":"04:11.300","Text":"we have 200 people."},{"Start":"04:11.300 ","End":"04:16.650","Text":"There we go. These are the frequencies that we were looking for."}],"ID":13750},{"Watched":false,"Name":"Exercise 3 - Parts a-b","Duration":"7m 32s","ChapterTopicVideoID":13263,"CourseChapterTopicPlaylistID":64521,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.400","Text":"In this question, we\u0027re given a histogram that describes"},{"Start":"00:02.400 ","End":"00:07.658","Text":"the probability distribution of the height in centimeters in a given group."},{"Start":"00:07.658 ","End":"00:09.150","Text":"We have this histogram."},{"Start":"00:09.150 ","End":"00:11.850","Text":"The first thing that we want to do is want to transform"},{"Start":"00:11.850 ","End":"00:16.665","Text":"this histogram into a frequency table by categories."},{"Start":"00:16.665 ","End":"00:18.660","Text":"Here\u0027s a frequency table."},{"Start":"00:18.660 ","End":"00:20.040","Text":"We see the categories here,"},{"Start":"00:20.040 ","End":"00:24.260","Text":"155-160 centimeters, that\u0027s this category right here."},{"Start":"00:24.260 ","End":"00:28.055","Text":"160-165, that\u0027s this category right here."},{"Start":"00:28.055 ","End":"00:31.950","Text":"165-170, that\u0027s this category."},{"Start":"00:31.950 ","End":"00:33.607","Text":"Now, 170-180,"},{"Start":"00:33.607 ","End":"00:37.500","Text":"that\u0027s this category right here and 180-190,"},{"Start":"00:37.500 ","End":"00:39.345","Text":"well that\u0027s that category."},{"Start":"00:39.345 ","End":"00:46.190","Text":"Now the frequency in a histogram is denoted by the area under the curve."},{"Start":"00:46.190 ","End":"00:48.785","Text":"And we can see here we have a key here,"},{"Start":"00:48.785 ","End":"00:51.220","Text":"that 1 square,"},{"Start":"00:51.220 ","End":"00:53.750","Text":"that equals to 5 people."},{"Start":"00:53.750 ","End":"00:55.820","Text":"So in the first category,"},{"Start":"00:55.820 ","End":"00:57.230","Text":"we have 1 square,"},{"Start":"00:57.230 ","End":"00:59.615","Text":"so that equals to 5 people."},{"Start":"00:59.615 ","End":"01:04.130","Text":"Now, what about here in this category from 160-165?"},{"Start":"01:04.130 ","End":"01:06.410","Text":"Well here we have 2 squares,"},{"Start":"01:06.410 ","End":"01:10.595","Text":"1 and 2, so that equals to 10 people."},{"Start":"01:10.595 ","End":"01:13.990","Text":"And in the category from 165-170,"},{"Start":"01:13.990 ","End":"01:15.260","Text":"That\u0027s 1, 2, and 3,,"},{"Start":"01:15.260 ","End":"01:18.155","Text":"3 times 5 is 15."},{"Start":"01:18.155 ","End":"01:20.555","Text":"In the next category,"},{"Start":"01:20.555 ","End":"01:22.050","Text":"well we have 1, 2, 3,"},{"Start":"01:22.050 ","End":"01:23.730","Text":"4, 5, 6, 7, 8."},{"Start":"01:23.730 ","End":"01:26.330","Text":"8 times 5 is 40."},{"Start":"01:26.330 ","End":"01:28.730","Text":"In the last category that\u0027s 1,"},{"Start":"01:28.730 ","End":"01:30.770","Text":"2, 3, 4, 5, 6."},{"Start":"01:30.770 ","End":"01:33.975","Text":"6 times 5, that\u0027s 30."},{"Start":"01:33.975 ","End":"01:41.720","Text":"That means that we have here n and that\u0027ll be equal to 100."},{"Start":"01:41.720 ","End":"01:43.405","Text":"We have 100 people."},{"Start":"01:43.405 ","End":"01:47.345","Text":"Before we get to answering questions,"},{"Start":"01:47.345 ","End":"01:50.585","Text":"let\u0027s just calculate the cumulative frequency as well."},{"Start":"01:50.585 ","End":"01:52.055","Text":"Here they\u0027ll be 5,"},{"Start":"01:52.055 ","End":"01:56.820","Text":"that\u0027s 15, that\u0027ll be 30, that\u0027ll be 70."},{"Start":"01:56.820 ","End":"02:01.880","Text":"70 plus 30, that\u0027s 100, so we\u0027re all set."},{"Start":"02:01.880 ","End":"02:07.977","Text":"The first question is to calculate the bottom tenth percentile,"},{"Start":"02:07.977 ","End":"02:13.070","Text":"so we\u0027re looking at x_0.1,"},{"Start":"02:13.070 ","End":"02:14.750","Text":"that\u0027s the 10th percentile."},{"Start":"02:14.750 ","End":"02:17.090","Text":"Well, in order to do that, first of all,"},{"Start":"02:17.090 ","End":"02:19.670","Text":"we have to calculate the position,"},{"Start":"02:19.670 ","End":"02:23.375","Text":"the location of the 10th percentile."},{"Start":"02:23.375 ","End":"02:27.860","Text":"How do we do that without the n times p divided by 100?"},{"Start":"02:27.860 ","End":"02:35.930","Text":"Well, n is 100 times p or p is 10 divided by 100,"},{"Start":"02:35.930 ","End":"02:38.520","Text":"and that equals to 10."},{"Start":"02:38.590 ","End":"02:43.820","Text":"Here let\u0027s look at the cumulative frequency here."},{"Start":"02:43.820 ","End":"02:45.515","Text":"In the first category,"},{"Start":"02:45.515 ","End":"02:49.220","Text":"we have only 5 people and in the second category,"},{"Start":"02:49.220 ","End":"02:51.170","Text":"we have up to 15 people,"},{"Start":"02:51.170 ","End":"02:53.525","Text":"so person number 10,"},{"Start":"02:53.525 ","End":"02:57.023","Text":"that\u0027s in this category right here."},{"Start":"02:57.023 ","End":"03:03.530","Text":"That equals 2L_0 plus n times p"},{"Start":"03:03.530 ","End":"03:11.240","Text":"divided by 100 minus big F at x_m minus 1,"},{"Start":"03:11.240 ","End":"03:16.280","Text":"divided by the frequency small f at"},{"Start":"03:16.280 ","End":"03:21.950","Text":"x_m times L_1 minus L_0,"},{"Start":"03:21.950 ","End":"03:27.020","Text":"the width of the category where we found the percentile."},{"Start":"03:27.020 ","End":"03:29.585","Text":"Let\u0027s just plug in the numbers."},{"Start":"03:29.585 ","End":"03:35.090","Text":"We know that the 10th percentile is in this category right here."},{"Start":"03:35.090 ","End":"03:36.920","Text":"That\u0027ll be equal to this,"},{"Start":"03:36.920 ","End":"03:40.310","Text":"the L_0, that\u0027s the lower bound of this category."},{"Start":"03:40.310 ","End":"03:45.200","Text":"That\u0027s 160 plus n times p divided by 100."},{"Start":"03:45.200 ","End":"03:49.085","Text":"Well, we\u0027ve calculated that that\u0027s 10 minus"},{"Start":"03:49.085 ","End":"03:55.640","Text":"the cumulative frequency of the previous or the preceding category."},{"Start":"03:55.640 ","End":"03:59.015","Text":"Well, there\u0027s the preceding category and the cumulative frequency here,"},{"Start":"03:59.015 ","End":"04:07.355","Text":"that\u0027s 5 divided by the frequency of the category that has the percentile."},{"Start":"04:07.355 ","End":"04:09.890","Text":"That\u0027s this thing, that\u0027s 10."},{"Start":"04:09.890 ","End":"04:16.670","Text":"Now we have to multiply this by the width of this category, that\u0027s 165."},{"Start":"04:16.670 ","End":"04:20.870","Text":"That\u0027s the upper bound minus the lower bound, it\u0027s 160."},{"Start":"04:20.870 ","End":"04:28.100","Text":"Now, all this comes out to 162.5."},{"Start":"04:28.100 ","End":"04:36.770","Text":"This then is the bottom 10th percentile, 162.5 centimeters."},{"Start":"04:36.770 ","End":"04:40.100","Text":"In this section, we\u0027re asked to calculate the 30th percentile."},{"Start":"04:40.100 ","End":"04:43.145","Text":"Here is our frequency table."},{"Start":"04:43.145 ","End":"04:46.730","Text":"Now, what\u0027s the position of the 30th percentile?"},{"Start":"04:46.730 ","End":"04:48.875","Text":"That\u0027s the first thing that we need to do."},{"Start":"04:48.875 ","End":"04:53.224","Text":"The position is n times p divided by 100."},{"Start":"04:53.224 ","End":"04:56.060","Text":"Well, n here that equals to 100,"},{"Start":"04:56.060 ","End":"04:59.090","Text":"p is 30 divided by 100,"},{"Start":"04:59.090 ","End":"05:01.855","Text":"and that equals to 30."},{"Start":"05:01.855 ","End":"05:06.083","Text":"Now we go to the cumulative frequency"},{"Start":"05:06.083 ","End":"05:10.370","Text":"and we see where position number 30 is and what category?"},{"Start":"05:10.370 ","End":"05:12.590","Text":"Well, here\u0027s 5."},{"Start":"05:12.590 ","End":"05:14.029","Text":"That\u0027s the first category."},{"Start":"05:14.029 ","End":"05:17.300","Text":"The next category goes up to 15 people,"},{"Start":"05:17.300 ","End":"05:20.420","Text":"and the next category goes up to 30 people."},{"Start":"05:20.420 ","End":"05:22.790","Text":"There we go. This is what we want."},{"Start":"05:22.790 ","End":"05:24.965","Text":"So without any calculations,"},{"Start":"05:24.965 ","End":"05:28.700","Text":"x_0.3 or the 30th percentile,"},{"Start":"05:28.700 ","End":"05:30.110","Text":"we have that here,"},{"Start":"05:30.110 ","End":"05:33.920","Text":"that equals to 170 centimeters."},{"Start":"05:33.920 ","End":"05:36.365","Text":"We didn\u0027t have to do any calculations."},{"Start":"05:36.365 ","End":"05:42.080","Text":"But for those who really insist in doing the calculations, let\u0027s do that."},{"Start":"05:42.080 ","End":"05:45.900","Text":"Well, x_0.3 or the 30th percentile,"},{"Start":"05:45.900 ","End":"05:52.820","Text":"that\u0027s L_0 plus n times p divided by 100 minus big F of"},{"Start":"05:52.820 ","End":"06:02.285","Text":"x_m minus 1 divided by small f of x and m times L_1 minus L_0."},{"Start":"06:02.285 ","End":"06:06.506","Text":"So this then would be our category."},{"Start":"06:06.506 ","End":"06:09.140","Text":"What\u0027s the lower bound of this category?"},{"Start":"06:09.140 ","End":"06:15.260","Text":"That\u0027s 165 plus now np times np divided by a 100,"},{"Start":"06:15.260 ","End":"06:20.870","Text":"that\u0027s 30 minus the cumulative frequency of the preceding category."},{"Start":"06:20.870 ","End":"06:27.540","Text":"Well, this is a preceding category right here and the cumulative frequency is 15 divided"},{"Start":"06:27.540 ","End":"06:31.970","Text":"by the frequency of the category"},{"Start":"06:31.970 ","End":"06:36.500","Text":"where we have the percentile that\u0027s right here and this is the frequency right here."},{"Start":"06:36.500 ","End":"06:41.060","Text":"So that\u0027s also 15 times the width of the category."},{"Start":"06:41.060 ","End":"06:43.670","Text":"Well, that\u0027s 170, that\u0027s the upper bound,"},{"Start":"06:43.670 ","End":"06:47.015","Text":"minus 165, that\u0027s the lower bound."},{"Start":"06:47.015 ","End":"06:49.235","Text":"Now, if we do the math,"},{"Start":"06:49.235 ","End":"06:52.940","Text":"that comes out to 170 centimeters as well."},{"Start":"06:52.940 ","End":"06:55.640","Text":"As we can see, without doing any calculations,"},{"Start":"06:55.640 ","End":"06:59.060","Text":"just using straight logic or with the calculations,"},{"Start":"06:59.060 ","End":"07:00.860","Text":"we come to the same conclusion."},{"Start":"07:00.860 ","End":"07:04.310","Text":"Now, for those of you who have a sharp pie,"},{"Start":"07:04.310 ","End":"07:09.245","Text":"we see here in the next category that we also have 170."},{"Start":"07:09.245 ","End":"07:11.750","Text":"I\u0027ll leave that to you."},{"Start":"07:11.750 ","End":"07:14.660","Text":"If you want to use this category or"},{"Start":"07:14.660 ","End":"07:20.120","Text":"this category in order to calculate the 30th percentile,"},{"Start":"07:20.120 ","End":"07:22.625","Text":"we\u0027ll still get the same number."},{"Start":"07:22.625 ","End":"07:24.770","Text":"That\u0027s the spoiler alert right here."},{"Start":"07:24.770 ","End":"07:28.399","Text":"If you use this as the category for the 30th percentile,"},{"Start":"07:28.399 ","End":"07:33.210","Text":"you\u0027ll still get 170 centimeters."}],"ID":13751},{"Watched":false,"Name":"Exercise 3 - Parts c-d","Duration":"4m 24s","ChapterTopicVideoID":13264,"CourseChapterTopicPlaylistID":64521,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"In this section, we\u0027re asked to calculate the value over"},{"Start":"00:02.730 ","End":"00:05.790","Text":"which 20 percent of the observations are found."},{"Start":"00:05.790 ","End":"00:12.270","Text":"That means that we have a value that 20 percent of the observations are above that value,"},{"Start":"00:12.270 ","End":"00:16.845","Text":"but 80 percent of the observations are below that value."},{"Start":"00:16.845 ","End":"00:24.360","Text":"That means, we\u0027re looking for the 80th percentile or x_0.8."},{"Start":"00:24.650 ","End":"00:27.105","Text":"If that\u0027s the case,"},{"Start":"00:27.105 ","End":"00:28.950","Text":"where is this located?"},{"Start":"00:28.950 ","End":"00:30.165","Text":"What\u0027s the location here?"},{"Start":"00:30.165 ","End":"00:32.890","Text":"That\u0027s n times p divided by 100."},{"Start":"00:32.890 ","End":"00:36.740","Text":"That equals to 100 times 80 divided by 100,"},{"Start":"00:36.740 ","End":"00:41.345","Text":"and that equals to 80. Where\u0027s that?"},{"Start":"00:41.345 ","End":"00:44.614","Text":"Again, let\u0027s go to the cumulative frequency here."},{"Start":"00:44.614 ","End":"00:48.735","Text":"Here in this category,"},{"Start":"00:48.735 ","End":"00:52.385","Text":"we only have the first 70 people out of 100."},{"Start":"00:52.385 ","End":"00:57.430","Text":"That means that person number 80 has to be in this category right here."},{"Start":"00:57.430 ","End":"01:06.725","Text":"Now we can go ahead and calculate the exact value of the 80th percentile, so x_0.8."},{"Start":"01:06.725 ","End":"01:13.085","Text":"That equals to L_0 plus np divided by a 100 minus F of"},{"Start":"01:13.085 ","End":"01:21.315","Text":"x_m minus 1 divided by small f of x_m times L_1 minus L_0."},{"Start":"01:21.315 ","End":"01:23.295","Text":"Let\u0027s plug in the numbers."},{"Start":"01:23.295 ","End":"01:30.640","Text":"L_0 is the lower bound of the category where the percentile is located. What\u0027s a 180?"},{"Start":"01:30.640 ","End":"01:33.780","Text":"Plus now n times p divided by a 100,"},{"Start":"01:33.780 ","End":"01:40.190","Text":"that\u0027s 80 minus the cumulative frequency of the preceding category."},{"Start":"01:40.190 ","End":"01:42.020","Text":"This is the preceding category right here,"},{"Start":"01:42.020 ","End":"01:44.390","Text":"and the cumulative frequency is 70."},{"Start":"01:44.390 ","End":"01:51.515","Text":"That\u0027s 70 divided by the frequency of the category where the percentile is located."},{"Start":"01:51.515 ","End":"01:54.200","Text":"The frequency here is 30,"},{"Start":"01:54.200 ","End":"01:57.650","Text":"times the width of this category."},{"Start":"01:57.650 ","End":"02:02.150","Text":"The upper bound is 190 minus the lower bound, that\u0027s 180."},{"Start":"02:02.150 ","End":"02:11.010","Text":"That equals to 183.33 and that\u0027s in centimeters."},{"Start":"02:11.010 ","End":"02:14.070","Text":"This is the value,"},{"Start":"02:14.070 ","End":"02:18.740","Text":"this is the height over which 20 percent of the observations are found."},{"Start":"02:18.740 ","End":"02:22.880","Text":"In this section, we\u0027re asked to calculate the percentage of the observations"},{"Start":"02:22.880 ","End":"02:27.095","Text":"below 158 centimeters. This is different."},{"Start":"02:27.095 ","End":"02:28.580","Text":"In the previous sections,"},{"Start":"02:28.580 ","End":"02:31.340","Text":"we were asked to find the percentile,"},{"Start":"02:31.340 ","End":"02:33.500","Text":"which was the x value."},{"Start":"02:33.500 ","End":"02:39.920","Text":"Here, we\u0027re asked to calculate the percentage and we\u0027re given the x value here."},{"Start":"02:39.920 ","End":"02:41.375","Text":"In order to do that,"},{"Start":"02:41.375 ","End":"02:44.570","Text":"we have to use the following expressions."},{"Start":"02:44.570 ","End":"02:48.635","Text":"That\u0027s the percentage at 158,"},{"Start":"02:48.635 ","End":"02:52.305","Text":"that equals to x."},{"Start":"02:52.305 ","End":"02:59.720","Text":"This is our x, minus L_0 divided by L_1 minus L_0 times,"},{"Start":"02:59.720 ","End":"03:07.785","Text":"f of x_m plus F at x_n minus 1,"},{"Start":"03:07.785 ","End":"03:11.030","Text":"and all this, we have to multiply that by"},{"Start":"03:11.030 ","End":"03:15.980","Text":"100 divided by n. Let\u0027s just plug in the numbers."},{"Start":"03:15.980 ","End":"03:21.796","Text":"X here, that\u0027s 158 minus L_0."},{"Start":"03:21.796 ","End":"03:25.110","Text":"158, where\u0027s that situated?"},{"Start":"03:25.110 ","End":"03:26.860","Text":"In what category is that situated?"},{"Start":"03:26.860 ","End":"03:28.240","Text":"That\u0027s here, that\u0027s right here,"},{"Start":"03:28.240 ","End":"03:30.940","Text":"between 155 and 160,"},{"Start":"03:30.940 ","End":"03:32.680","Text":"that\u0027s the first category."},{"Start":"03:32.680 ","End":"03:34.870","Text":"What\u0027s the lower bound of this category?"},{"Start":"03:34.870 ","End":"03:39.670","Text":"That\u0027s 155 divided by the width of this category."},{"Start":"03:39.670 ","End":"03:45.280","Text":"That\u0027s 160 minus 155 times the frequency of that category."},{"Start":"03:45.280 ","End":"03:47.530","Text":"The frequency is equal to 5."},{"Start":"03:47.530 ","End":"03:54.070","Text":"Plus, the cumulative frequency of the previous or the preceding category."},{"Start":"03:54.070 ","End":"03:59.270","Text":"There\u0027s no preceding category because this is the first category, so that\u0027s 0."},{"Start":"03:59.270 ","End":"04:04.905","Text":"All that has to be multiplied by 100 divided by n,"},{"Start":"04:04.905 ","End":"04:08.220","Text":"n is 100, so that equals to 1."},{"Start":"04:08.220 ","End":"04:14.070","Text":"All this equals to 3 and that\u0027s 3 percent."},{"Start":"04:14.070 ","End":"04:19.640","Text":"The percentage of the observations below 158 centimeters,"},{"Start":"04:19.640 ","End":"04:25.770","Text":"only 3% of the observations are below 158 centimeters."}],"ID":13752},{"Watched":false,"Name":"Exercise 3 - Parts e-f","Duration":"4m 34s","ChapterTopicVideoID":13265,"CourseChapterTopicPlaylistID":64521,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"In this section, we\u0027re asked to calculate the percentage of"},{"Start":"00:02.580 ","End":"00:05.894","Text":"the observations above 185 centimeters."},{"Start":"00:05.894 ","End":"00:07.185","Text":"Well, up to now,"},{"Start":"00:07.185 ","End":"00:14.715","Text":"we can calculate the percentage of the observations below a specific x value right here."},{"Start":"00:14.715 ","End":"00:20.040","Text":"How do we calculate the percentage of observations above a specific value?"},{"Start":"00:20.040 ","End":"00:23.715","Text":"Well, all we have to do is we have to get that 100 percent"},{"Start":"00:23.715 ","End":"00:27.750","Text":"of the observations we have to take away from that P,"},{"Start":"00:27.750 ","End":"00:31.035","Text":"the percentage at 185."},{"Start":"00:31.035 ","End":"00:34.500","Text":"The percentage of observations below 185."},{"Start":"00:34.500 ","End":"00:40.770","Text":"This difference will give me the percentage of observations above 185 centimeters."},{"Start":"00:40.770 ","End":"00:44.250","Text":"Let\u0027s calculate P at 185."},{"Start":"00:44.250 ","End":"00:49.970","Text":"P at 185, that equals to the following."},{"Start":"00:49.970 ","End":"00:56.315","Text":"Well, that\u0027s x minus L_0 divided by L_1 minus L_0"},{"Start":"00:56.315 ","End":"01:04.185","Text":"times f of x_m plus f and x at n minus 1."},{"Start":"01:04.185 ","End":"01:12.200","Text":"All this, I have to multiply by 100 divided by n. That\u0027s the expression that I\u0027m using."},{"Start":"01:12.200 ","End":"01:14.375","Text":"Well, let\u0027s just plug in the numbers."},{"Start":"01:14.375 ","End":"01:17.045","Text":"X here, but that\u0027s 185."},{"Start":"01:17.045 ","End":"01:21.005","Text":"Now, where\u0027s 185 located here?"},{"Start":"01:21.005 ","End":"01:23.225","Text":"In which category is it located?"},{"Start":"01:23.225 ","End":"01:25.730","Text":"What\u0027s located right here in the last category,"},{"Start":"01:25.730 ","End":"01:28.435","Text":"between 180 and 190."},{"Start":"01:28.435 ","End":"01:32.570","Text":"What\u0027s the lower bound of this category right here?"},{"Start":"01:32.570 ","End":"01:39.720","Text":"Well, that\u0027s 180, so that has to be minus 180 divided by L_1 minus L_0."},{"Start":"01:39.720 ","End":"01:41.865","Text":"That\u0027s the width of the category."},{"Start":"01:41.865 ","End":"01:43.400","Text":"That\u0027ll be the upper bound,"},{"Start":"01:43.400 ","End":"01:45.100","Text":"190 minus 180,"},{"Start":"01:45.100 ","End":"01:51.170","Text":"that\u0027s the lower bound times the frequency of this category. Well, that\u0027s 30."},{"Start":"01:51.170 ","End":"01:58.160","Text":"Plus the cumulative frequency of the preceding category."},{"Start":"01:58.160 ","End":"02:02.165","Text":"Well, here\u0027s the preceding category right here and the cumulative frequency,"},{"Start":"02:02.165 ","End":"02:04.550","Text":"but that\u0027s right here, that\u0027s 70."},{"Start":"02:04.550 ","End":"02:11.840","Text":"All this, I have to multiply it by 100 divided by n. Now n is a 100."},{"Start":"02:11.840 ","End":"02:15.890","Text":"That equals to what?"},{"Start":"02:15.890 ","End":"02:18.455","Text":"That equals to 85."},{"Start":"02:18.455 ","End":"02:20.750","Text":"Now, that\u0027s 85 percent."},{"Start":"02:20.750 ","End":"02:26.257","Text":"Now, this isn\u0027t the percentage of observations above 185 centimeters,"},{"Start":"02:26.257 ","End":"02:32.060","Text":"that\u0027s the percentage of observations below 185 centimeters."},{"Start":"02:32.060 ","End":"02:33.455","Text":"What do we have to do?"},{"Start":"02:33.455 ","End":"02:35.120","Text":"We have to do exactly this."},{"Start":"02:35.120 ","End":"02:41.300","Text":"We have to take 100 percent and we have to subtract from that 85 percent."},{"Start":"02:41.300 ","End":"02:45.620","Text":"Now that equals to 15 percent."},{"Start":"02:45.620 ","End":"02:53.735","Text":"This then is the percentage of observations that are above 185 centimeters."},{"Start":"02:53.735 ","End":"02:57.170","Text":"In this section, we\u0027re asked to calculate the percentage of the observations"},{"Start":"02:57.170 ","End":"03:01.235","Text":"between 170, 185 centimeters."},{"Start":"03:01.235 ","End":"03:03.665","Text":"Let\u0027s just see where we\u0027re at here."},{"Start":"03:03.665 ","End":"03:07.130","Text":"This then would be 185 centimeters."},{"Start":"03:07.130 ","End":"03:10.380","Text":"In here, we have 170 centimeters,"},{"Start":"03:10.380 ","End":"03:15.380","Text":"so we\u0027re asked to calculate the percentage of observations between these 2 values."},{"Start":"03:15.380 ","End":"03:19.865","Text":"Now, we know that P at 185, well,"},{"Start":"03:19.865 ","End":"03:26.840","Text":"that gives me the percentage of observations below 185 centimeters."},{"Start":"03:26.840 ","End":"03:33.290","Text":"P at 170 gives me the percentage of observations below 170."},{"Start":"03:33.290 ","End":"03:35.715","Text":"If I subtract the 2,"},{"Start":"03:35.715 ","End":"03:37.205","Text":"if I take the difference,"},{"Start":"03:37.205 ","End":"03:42.430","Text":"that\u0027ll give me the percentage of observations between the 2 values."},{"Start":"03:42.430 ","End":"03:44.750","Text":"What\u0027s P At 185?"},{"Start":"03:44.750 ","End":"03:50.030","Text":"We\u0027ve calculated that in the previous section, that\u0027s 85 percent."},{"Start":"03:50.030 ","End":"03:52.505","Text":"Now, what about P at 170?"},{"Start":"03:52.505 ","End":"03:55.160","Text":"Well, in section b above,"},{"Start":"03:55.160 ","End":"04:01.080","Text":"we calculated the 30th percentile X_0.3,"},{"Start":"04:01.080 ","End":"04:05.535","Text":"that equal to 170 centimeters."},{"Start":"04:05.535 ","End":"04:09.350","Text":"What does that mean? That means that 30 percent of"},{"Start":"04:09.350 ","End":"04:13.730","Text":"the population are below 170 centimeters."},{"Start":"04:13.730 ","End":"04:16.295","Text":"But that\u0027s exactly this thing right here."},{"Start":"04:16.295 ","End":"04:19.940","Text":"P at 170, that equals to 30 percent."},{"Start":"04:19.940 ","End":"04:22.370","Text":"Now let\u0027s take the difference."},{"Start":"04:22.370 ","End":"04:26.210","Text":"That equals to 55 percent."},{"Start":"04:26.210 ","End":"04:35.520","Text":"This then would be the percentage of observations between 170 and 185 centimeters."}],"ID":13753}],"Thumbnail":null,"ID":64521},{"Name":"Percentiles in a Discrete Frequencies Table","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"40s","ChapterTopicVideoID":12892,"CourseChapterTopicPlaylistID":64522,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"In this chapter, I\u0027ll be talking about Descriptive Statistics,"},{"Start":"00:02.730 ","End":"00:04.980","Text":"specifically Measures of Relative Location."},{"Start":"00:04.980 ","End":"00:09.105","Text":"I\u0027ll be talking about Percentiles in Discrete Frequency Tables."},{"Start":"00:09.105 ","End":"00:14.640","Text":"Now, the p^th percentile is the value under which p percent of the data is found,"},{"Start":"00:14.640 ","End":"00:18.885","Text":"and the p^th percentile is denoted as X_p."},{"Start":"00:18.885 ","End":"00:24.465","Text":"Now, how do we calculate the Percentile from Data in a Discrete Frequency Table?"},{"Start":"00:24.465 ","End":"00:29.820","Text":"Well, the percentile is the value for which the relative cumulative frequency,"},{"Start":"00:29.820 ","End":"00:31.340","Text":"that\u0027s in percentages,"},{"Start":"00:31.340 ","End":"00:36.600","Text":"is greater than or equal to p percent for the first time."},{"Start":"00:36.600 ","End":"00:38.775","Text":"This sounds like a lot to chew,"},{"Start":"00:38.775 ","End":"00:41.470","Text":"so let\u0027s take a look at an example."}],"ID":13399},{"Watched":false,"Name":"Tutorial 2","Duration":"4m 17s","ChapterTopicVideoID":12893,"CourseChapterTopicPlaylistID":64522,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.125","Text":"In this example, there are 250 customers in a specific bank."},{"Start":"00:04.125 ","End":"00:08.130","Text":"Now, the number of savings plans has been counted for each customer,"},{"Start":"00:08.130 ","End":"00:11.520","Text":"and this is displayed in the following table."},{"Start":"00:11.520 ","End":"00:13.740","Text":"In the first column here,"},{"Start":"00:13.740 ","End":"00:15.180","Text":"we have our variable X."},{"Start":"00:15.180 ","End":"00:16.740","Text":"That\u0027s the number of savings plans;"},{"Start":"00:16.740 ","End":"00:19.020","Text":"0, 1, 2, 3, and 4."},{"Start":"00:19.020 ","End":"00:21.210","Text":"Here in the second column,"},{"Start":"00:21.210 ","End":"00:23.160","Text":"we have the frequency."},{"Start":"00:23.160 ","End":"00:27.585","Text":"100 clients here have 0 savings plans,"},{"Start":"00:27.585 ","End":"00:31.050","Text":"75 clients have 1 saving plan,"},{"Start":"00:31.050 ","End":"00:34.740","Text":"25 clients have 2 savings plans, and so on and so forth."},{"Start":"00:34.740 ","End":"00:39.225","Text":"Now, let\u0027s calculate the cumulative frequency here."},{"Start":"00:39.225 ","End":"00:43.695","Text":"That\u0027ll be 100, that\u0027ll be 175,"},{"Start":"00:43.695 ","End":"00:49.620","Text":"that\u0027s 200, 225, and 250."},{"Start":"00:49.620 ","End":"00:55.045","Text":"We have here 250 clients, that\u0027s what we\u0027re given."},{"Start":"00:55.045 ","End":"00:58.565","Text":"Now, what about the relative cumulative frequency?"},{"Start":"00:58.565 ","End":"01:03.740","Text":"Well, this is calculated by taking the value of the cumulative frequency for"},{"Start":"01:03.740 ","End":"01:10.360","Text":"each X value and dividing it by the total number of observations that we have."},{"Start":"01:10.360 ","End":"01:12.695","Text":"In our case, that will be 250."},{"Start":"01:12.695 ","End":"01:18.800","Text":"Here, the relative cumulative frequency for this X value here,"},{"Start":"01:18.800 ","End":"01:23.390","Text":"well, that\u0027ll be 100 divided by 250."},{"Start":"01:23.390 ","End":"01:25.670","Text":"Now, we want to make this a percentage,"},{"Start":"01:25.670 ","End":"01:28.160","Text":"so we\u0027ll multiply that by 100."},{"Start":"01:28.160 ","End":"01:31.390","Text":"That\u0027ll be equal to 40 percent."},{"Start":"01:31.390 ","End":"01:34.380","Text":"What about this value right here?"},{"Start":"01:34.380 ","End":"01:40.160","Text":"That\u0027ll be 175 divided by 250 times 100,"},{"Start":"01:40.160 ","End":"01:44.310","Text":"and that equals to 70 percent."},{"Start":"01:44.750 ","End":"01:50.910","Text":"The next value that\u0027ll be 200 divided by 250 times 100,"},{"Start":"01:50.910 ","End":"01:52.950","Text":"that\u0027ll be 80 percent,"},{"Start":"01:52.950 ","End":"02:00.315","Text":"and that\u0027ll be 225 divided by 250 times 100, that\u0027s 90 percent."},{"Start":"02:00.315 ","End":"02:06.305","Text":"Here we have 250 divided by 250 times 100."},{"Start":"02:06.305 ","End":"02:09.595","Text":"That equals to 100 percent."},{"Start":"02:09.595 ","End":"02:16.070","Text":"We\u0027ve completed this table right here and we\u0027re asked to find the 25th percentile."},{"Start":"02:16.070 ","End":"02:23.105","Text":"Now, the 25th percentile is defined as the X value,"},{"Start":"02:23.105 ","End":"02:27.110","Text":"where 25 percent of the observations are below that value."},{"Start":"02:27.110 ","End":"02:31.055","Text":"When we have a discrete frequency table, well,"},{"Start":"02:31.055 ","End":"02:40.765","Text":"that\u0027s the X value where I encounter for the first time 25 percent."},{"Start":"02:40.765 ","End":"02:47.180","Text":"Now, let\u0027s take a look at the relative cumulative frequency here."},{"Start":"02:47.180 ","End":"02:48.980","Text":"Well, in the first category,"},{"Start":"02:48.980 ","End":"02:51.860","Text":"we\u0027ve already reached 40 percent."},{"Start":"02:51.860 ","End":"02:59.270","Text":"When we look at X_0.25 or the 25th percentile,"},{"Start":"02:59.270 ","End":"03:01.680","Text":"that\u0027s 25 percent,"},{"Start":"03:01.700 ","End":"03:04.200","Text":"now what\u0027s the x value here?"},{"Start":"03:04.200 ","End":"03:07.555","Text":"That equals to 0. Now,"},{"Start":"03:07.555 ","End":"03:10.100","Text":"what about this question right here,"},{"Start":"03:10.100 ","End":"03:14.750","Text":"we\u0027re asked to find the value over which 20 percent of the cases are found."},{"Start":"03:14.750 ","End":"03:17.390","Text":"That means that there is an X value where we"},{"Start":"03:17.390 ","End":"03:21.815","Text":"have 20 percent of the observations above that value."},{"Start":"03:21.815 ","End":"03:26.630","Text":"Well, that means that 80 percent are below that value."},{"Start":"03:26.630 ","End":"03:35.155","Text":"We\u0027re looking at X_0.8, the 80th percentile."},{"Start":"03:35.155 ","End":"03:39.270","Text":"That means X_80 percent."},{"Start":"03:39.270 ","End":"03:40.865","Text":"Now, what does that equal to?"},{"Start":"03:40.865 ","End":"03:45.650","Text":"Well, again, we have to look at the relative cumulative frequency and"},{"Start":"03:45.650 ","End":"03:50.650","Text":"see where we encounter 80 percent for the first time."},{"Start":"03:50.650 ","End":"03:53.390","Text":"Well, we encounter right here."},{"Start":"03:53.390 ","End":"03:55.160","Text":"We even have the exact value here."},{"Start":"03:55.160 ","End":"03:57.050","Text":"Now, what\u0027s the x value here?"},{"Start":"03:57.050 ","End":"03:58.910","Text":"Well, that\u0027s 2."},{"Start":"03:58.910 ","End":"04:07.425","Text":"That means that 80 percent of the clients have 0,1,"},{"Start":"04:07.425 ","End":"04:17.980","Text":"or 2 savings plans and 20 percent of the clients have 3 or 4 savings plans."}],"ID":13400},{"Watched":false,"Name":"Exercise 1","Duration":"3m 59s","ChapterTopicVideoID":12894,"CourseChapterTopicPlaylistID":64522,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.160","Text":"In this question, we\u0027re given a table that displays"},{"Start":"00:02.160 ","End":"00:05.385","Text":"the probability distribution of a discrete variable."},{"Start":"00:05.385 ","End":"00:08.640","Text":"Here\u0027s our discrete variable x,"},{"Start":"00:08.640 ","End":"00:11.580","Text":"and it has these values from 0 to 4."},{"Start":"00:11.580 ","End":"00:16.230","Text":"We have the frequency of f. We have 10,"},{"Start":"00:16.230 ","End":"00:19.560","Text":"40, 30, 15, and 5 for each one of the values here."},{"Start":"00:19.560 ","End":"00:23.475","Text":"Now let\u0027s calculate the cumulative frequency here."},{"Start":"00:23.475 ","End":"00:26.715","Text":"Well, that\u0027ll be 10, that\u0027ll be 50,"},{"Start":"00:26.715 ","End":"00:30.915","Text":"80, 95, and 100."},{"Start":"00:30.915 ","End":"00:32.088","Text":"So n here,"},{"Start":"00:32.088 ","End":"00:33.810","Text":"that equals to 100."},{"Start":"00:33.810 ","End":"00:36.930","Text":"Now, what about the relative cumulative frequency?"},{"Start":"00:36.930 ","End":"00:41.470","Text":"Well, that\u0027ll be 10 divided by 100,"},{"Start":"00:41.470 ","End":"00:47.735","Text":"that\u0027ll be n, times 100 to make this a percentage."},{"Start":"00:47.735 ","End":"00:50.570","Text":"Now, that equals to 10 percent."},{"Start":"00:50.570 ","End":"00:52.625","Text":"Now since n equals 100,"},{"Start":"00:52.625 ","End":"00:55.460","Text":"then all these are also percentages."},{"Start":"00:55.460 ","End":"00:58.010","Text":"Here that\u0027ll be 50 percent,"},{"Start":"00:58.010 ","End":"01:00.205","Text":"that\u0027ll be 80 percent,"},{"Start":"01:00.205 ","End":"01:02.640","Text":"that\u0027ll be 95 percent,"},{"Start":"01:02.640 ","End":"01:05.760","Text":"and here, that\u0027ll be 100 percent."},{"Start":"01:05.760 ","End":"01:10.160","Text":"Now we\u0027re asked to find the 60th percentile."},{"Start":"01:10.160 ","End":"01:16.800","Text":"That\u0027s the first time that I encounter 60 percent or over 60 percent."},{"Start":"01:16.800 ","End":"01:23.040","Text":"I\u0027m looking at x_0.6 or x at 60 percent."},{"Start":"01:23.040 ","End":"01:27.710","Text":"Well, let\u0027s take a look at the relative cumulative frequency right here."},{"Start":"01:27.710 ","End":"01:28.910","Text":"Well, it\u0027s not here."},{"Start":"01:28.910 ","End":"01:31.355","Text":"It\u0027s not here, here\u0027s 50 percent."},{"Start":"01:31.355 ","End":"01:33.080","Text":"Then I jumped to 80 percent."},{"Start":"01:33.080 ","End":"01:37.040","Text":"This is the first time where I encountered 60 percent or over."},{"Start":"01:37.040 ","End":"01:40.610","Text":"What\u0027s the x value here? That\u0027s 2."},{"Start":"01:40.610 ","End":"01:45.115","Text":"The 60th percentile, that equals to 2."},{"Start":"01:45.115 ","End":"01:47.300","Text":"What about the 40th percentile?"},{"Start":"01:47.300 ","End":"01:51.725","Text":"X_0.4 or X_40 percent."},{"Start":"01:51.725 ","End":"01:57.500","Text":"Now, again, we have to take a look at the relative cumulative frequency."},{"Start":"01:57.500 ","End":"01:59.315","Text":"Well, it\u0027s not in this category."},{"Start":"01:59.315 ","End":"02:02.250","Text":"Here we encounter 50 percent."},{"Start":"02:02.250 ","End":"02:06.020","Text":"This is the first time where we encounter 40 percent or over."},{"Start":"02:06.020 ","End":"02:08.285","Text":"Now, that was the x-value here,"},{"Start":"02:08.285 ","End":"02:09.770","Text":"that equals to 1."},{"Start":"02:09.770 ","End":"02:13.760","Text":"So X_0.4 or the 40th percentile,"},{"Start":"02:13.760 ","End":"02:15.755","Text":"that equals to 1."},{"Start":"02:15.755 ","End":"02:18.320","Text":"Now, what about the top 10th percentile?"},{"Start":"02:18.320 ","End":"02:24.230","Text":"That means that 10 percent of the population is above this value,"},{"Start":"02:24.230 ","End":"02:30.095","Text":"this X value, but that means that 90 percent is below this value."},{"Start":"02:30.095 ","End":"02:36.735","Text":"So we\u0027re looking at X_0.9 or X at 90 percent."},{"Start":"02:36.735 ","End":"02:39.515","Text":"We\u0027re at the 90th percentile."},{"Start":"02:39.515 ","End":"02:40.970","Text":"Now, what does that equal to?"},{"Start":"02:40.970 ","End":"02:45.530","Text":"Well, again, let\u0027s take a look at the relative cumulative distribution."},{"Start":"02:45.530 ","End":"02:48.295","Text":"It\u0027s not 10 percent, 50, 80,"},{"Start":"02:48.295 ","End":"02:51.980","Text":"but here we encounter 90 percent for the first time."},{"Start":"02:51.980 ","End":"02:53.930","Text":"Where we have 95 percent,"},{"Start":"02:53.930 ","End":"02:55.370","Text":"it can\u0027t be 80 percent,"},{"Start":"02:55.370 ","End":"02:57.005","Text":"so it must be here."},{"Start":"02:57.005 ","End":"03:01.170","Text":"What\u0027s the X value here? That equals to 3."},{"Start":"03:01.370 ","End":"03:04.770","Text":"Now, what about the inter-quartile range,"},{"Start":"03:04.770 ","End":"03:14.765","Text":"well the interquartile range is defined as the 75th percentile minus the 25th percentile."},{"Start":"03:14.765 ","End":"03:18.170","Text":"That\u0027s the definition of the inter-quartile range."},{"Start":"03:18.170 ","End":"03:24.725","Text":"Let\u0027s calculate X_0.75 for the 75th percentile."},{"Start":"03:24.725 ","End":"03:27.650","Text":"Well, again, we have to go to"},{"Start":"03:27.650 ","End":"03:33.410","Text":"the relative cumulative frequency and we have to look at 75 percent."},{"Start":"03:33.410 ","End":"03:36.710","Text":"It\u0027s not here, it\u0027s not here. It\u0027s here."},{"Start":"03:36.710 ","End":"03:40.565","Text":"This is the first time when we encounter 75 percent or over."},{"Start":"03:40.565 ","End":"03:43.190","Text":"The value here, the x-value is 2,"},{"Start":"03:43.190 ","End":"03:46.295","Text":"and what about the 25th percentile?"},{"Start":"03:46.295 ","End":"03:48.215","Text":"Well, it\u0027s not here."},{"Start":"03:48.215 ","End":"03:50.780","Text":"It has to be here in the second category."},{"Start":"03:50.780 ","End":"03:52.160","Text":"Now, what\u0027s the x value here?"},{"Start":"03:52.160 ","End":"03:57.905","Text":"That\u0027s 1. The inter-quartile range is 2 minus 1."},{"Start":"03:57.905 ","End":"04:00.480","Text":"That equals to 1."}],"ID":13401},{"Watched":false,"Name":"Exercise 2","Duration":"4m 41s","ChapterTopicVideoID":12895,"CourseChapterTopicPlaylistID":64522,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.700","Text":"In this question, we are given a table that displays the number"},{"Start":"00:02.700 ","End":"00:05.910","Text":"of cars owned by a family in a certain community."},{"Start":"00:05.910 ","End":"00:08.490","Text":"This is our frequency table."},{"Start":"00:08.490 ","End":"00:10.800","Text":"In the first row, we have a variable,"},{"Start":"00:10.800 ","End":"00:13.590","Text":"that\u0027s the number of cars from 1 to 5."},{"Start":"00:13.590 ","End":"00:15.735","Text":"That\u0027s the values for the variable."},{"Start":"00:15.735 ","End":"00:17.190","Text":"Now, what\u0027s the frequency,"},{"Start":"00:17.190 ","End":"00:20.955","Text":"the number of families that have a certain number of cars?"},{"Start":"00:20.955 ","End":"00:24.975","Text":"Well, here we have 65 families who have 1 car,"},{"Start":"00:24.975 ","End":"00:27.930","Text":"here we have 150 families who have 2 cars,"},{"Start":"00:27.930 ","End":"00:29.340","Text":"and so on and so forth."},{"Start":"00:29.340 ","End":"00:32.494","Text":"We\u0027re asked to calculate these percentiles."},{"Start":"00:32.494 ","End":"00:37.320","Text":"The first thing that I want to do is I want to calculate the cumulative frequency."},{"Start":"00:37.320 ","End":"00:39.135","Text":"That\u0027s this row right here."},{"Start":"00:39.135 ","End":"00:46.440","Text":"That\u0027ll be 65 plus 150, that\u0027s 215."},{"Start":"00:46.440 ","End":"00:52.260","Text":"That\u0027ll be 435, that\u0027ll be 575,"},{"Start":"00:52.260 ","End":"00:55.305","Text":"and that\u0027ll be 630."},{"Start":"00:55.305 ","End":"01:00.095","Text":"All in all, I have 630 families."},{"Start":"01:00.095 ","End":"01:02.120","Text":"Now, this row right here, well,"},{"Start":"01:02.120 ","End":"01:05.165","Text":"that\u0027s the relative cumulative frequency."},{"Start":"01:05.165 ","End":"01:07.910","Text":"What\u0027s that? Well, that\u0027s the cumulative frequency"},{"Start":"01:07.910 ","End":"01:11.465","Text":"divided by n. Let\u0027s just calculate that."},{"Start":"01:11.465 ","End":"01:16.010","Text":"That\u0027ll be 65 divided by 630,"},{"Start":"01:16.010 ","End":"01:18.605","Text":"and we want that in percentage, right?"},{"Start":"01:18.605 ","End":"01:19.817","Text":"We want that to be a percentage,"},{"Start":"01:19.817 ","End":"01:22.370","Text":"so we have to multiply that by 100."},{"Start":"01:22.370 ","End":"01:27.335","Text":"Now that comes out to 10.3 percent,"},{"Start":"01:27.335 ","End":"01:30.740","Text":"so we do this for all the other values here,"},{"Start":"01:30.740 ","End":"01:35.775","Text":"and here, that\u0027ll come out to 34.1 percent."},{"Start":"01:35.775 ","End":"01:40.020","Text":"Here, that\u0027ll be 69.0 percent."},{"Start":"01:40.020 ","End":"01:45.405","Text":"Here, that\u0027ll be 91.3 percent and this will be 100 percent."},{"Start":"01:45.405 ","End":"01:50.284","Text":"We\u0027re asked now to calculate the bottom 10th percentile."},{"Start":"01:50.284 ","End":"01:53.195","Text":"That\u0027s the X value where 10 percent of"},{"Start":"01:53.195 ","End":"01:57.797","Text":"the families are below that value, so that\u0027s X_0.1."},{"Start":"01:57.797 ","End":"02:03.739","Text":"Well, we go to this row right here and we try to find"},{"Start":"02:03.739 ","End":"02:11.405","Text":"the value here where we meet this 10 percent for the first time."},{"Start":"02:11.405 ","End":"02:15.795","Text":"Well, here we have 10.3 percent, so that\u0027s it."},{"Start":"02:15.795 ","End":"02:19.100","Text":"This is the category right here,"},{"Start":"02:19.100 ","End":"02:20.840","Text":"so what\u0027s the X-value?"},{"Start":"02:20.840 ","End":"02:26.330","Text":"Well, the X-value is 1 so the bottom 10th percentile or X_0.1,"},{"Start":"02:26.330 ","End":"02:27.980","Text":"that equals to 1."},{"Start":"02:27.980 ","End":"02:30.259","Text":"Now, what about the 30th percentile?"},{"Start":"02:30.259 ","End":"02:34.200","Text":"Well, that\u0027s X_0.3."},{"Start":"02:34.200 ","End":"02:39.215","Text":"We\u0027re looking at these values right here and we\u0027re looking to find"},{"Start":"02:39.215 ","End":"02:45.575","Text":"the value here where we meet 0.3 or 30 percent for the first time."},{"Start":"02:45.575 ","End":"02:49.850","Text":"Well, it\u0027s not here because here we only get up to 10.3 percent,"},{"Start":"02:49.850 ","End":"02:53.990","Text":"but it is here because right here we get up to 34.1 percent,"},{"Start":"02:53.990 ","End":"02:56.540","Text":"we need only 30 percent."},{"Start":"02:56.540 ","End":"02:58.810","Text":"What\u0027s the x value here?"},{"Start":"02:58.810 ","End":"03:03.475","Text":"That\u0027s 2, so the 30th percentile equals 2."},{"Start":"03:03.475 ","End":"03:09.395","Text":"Now, what about the value where 20 percent of the observations are above that value?"},{"Start":"03:09.395 ","End":"03:13.160","Text":"Well, if 20 percent of the observations are above that value,"},{"Start":"03:13.160 ","End":"03:16.080","Text":"that means that 80 percent are below that value."},{"Start":"03:16.080 ","End":"03:20.310","Text":"Well, that\u0027s a definition of the 80th percentile,"},{"Start":"03:20.310 ","End":"03:24.925","Text":"so we\u0027re looking for x_0.8."},{"Start":"03:24.925 ","End":"03:26.390","Text":"Now, what does that equal to?"},{"Start":"03:26.390 ","End":"03:29.285","Text":"Well, we\u0027re going to have to look for 80 percent."},{"Start":"03:29.285 ","End":"03:31.880","Text":"Where did we meet 80 percent for the first time?"},{"Start":"03:31.880 ","End":"03:35.009","Text":"Well, not here, not here, not here,"},{"Start":"03:35.009 ","End":"03:36.060","Text":"it\u0027s 69 percent,"},{"Start":"03:36.060 ","End":"03:37.170","Text":"but the next category,"},{"Start":"03:37.170 ","End":"03:39.825","Text":"that\u0027s 91.3 percent so it\u0027s here."},{"Start":"03:39.825 ","End":"03:43.140","Text":"What\u0027s the X-value here, well that\u0027s 4?"},{"Start":"03:43.140 ","End":"03:46.760","Text":"The 80th percentile equals 4."},{"Start":"03:46.760 ","End":"03:49.310","Text":"Now, what about the upper quartile?"},{"Start":"03:49.310 ","End":"03:53.810","Text":"The upper quartile, that means that 25 percent of"},{"Start":"03:53.810 ","End":"03:58.670","Text":"the families are above this specific X value."},{"Start":"03:58.670 ","End":"04:01.590","Text":"Well, if 25 percent are above this value,"},{"Start":"04:01.590 ","End":"04:04.700","Text":"that means that 75 percent are below this value."},{"Start":"04:04.700 ","End":"04:09.425","Text":"That means that we\u0027re looking at x sub 0.75,"},{"Start":"04:09.425 ","End":"04:11.660","Text":"or the 75th percentile."},{"Start":"04:11.660 ","End":"04:16.520","Text":"Well, let\u0027s take a look at this row right here."},{"Start":"04:16.520 ","End":"04:18.770","Text":"It\u0027s not here, not here, and not here, but it is here, right?"},{"Start":"04:18.770 ","End":"04:22.550","Text":"It\u0027s not 69 percent but it is here at 91.3 percent."},{"Start":"04:22.550 ","End":"04:24.919","Text":"So 75 percent,"},{"Start":"04:24.919 ","End":"04:29.225","Text":"that\u0027s in this category right here and the X value is 4."},{"Start":"04:29.225 ","End":"04:34.095","Text":"Here we go, the 75th percentile is 4,"},{"Start":"04:34.095 ","End":"04:36.435","Text":"the 80th percentile is 4,"},{"Start":"04:36.435 ","End":"04:42.280","Text":"the 30th percentile is 2 and the 10th percentile is 1."}],"ID":13402}],"Thumbnail":null,"ID":64522},{"Name":"Linear Transformations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"2m 56s","ChapterTopicVideoID":12896,"CourseChapterTopicPlaylistID":64523,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.500","Text":"In this chapter I\u0027ll be talking about"},{"Start":"00:01.500 ","End":"00:05.025","Text":"descriptive statistics, specifically linear transformations."},{"Start":"00:05.025 ","End":"00:07.320","Text":"Now, what\u0027s a linear transformation?"},{"Start":"00:07.320 ","End":"00:09.330","Text":"Well, that\u0027s a change."},{"Start":"00:09.330 ","End":"00:11.954","Text":"That\u0027s a change in the random variable."},{"Start":"00:11.954 ","End":"00:16.020","Text":"When we have a random variable X and we want to change"},{"Start":"00:16.020 ","End":"00:19.890","Text":"that into a new variable Y by either adding,"},{"Start":"00:19.890 ","End":"00:22.650","Text":"subtracting, multiplying, dividing it."},{"Start":"00:22.650 ","End":"00:24.420","Text":"Now, how do we make that change?"},{"Start":"00:24.420 ","End":"00:27.435","Text":"Well, again, we take the random variable and we"},{"Start":"00:27.435 ","End":"00:32.430","Text":"multiply it by a certain constant, or we divide it."},{"Start":"00:32.430 ","End":"00:37.460","Text":"Then we add or subtract this by another constant."},{"Start":"00:37.460 ","End":"00:43.280","Text":"A linear transformation is defined as the addition or subtraction and,"},{"Start":"00:43.280 ","End":"00:49.565","Text":"or the multiplication or division of a random variable by constant for all observations."},{"Start":"00:49.565 ","End":"00:51.080","Text":"This would be the template,"},{"Start":"00:51.080 ","End":"00:54.275","Text":"y equals to ax plus b."},{"Start":"00:54.275 ","End":"00:59.345","Text":"Now, we usually see it in this form but we also"},{"Start":"00:59.345 ","End":"01:05.150","Text":"see this type of transformation where we have y equals bx plus a."},{"Start":"01:05.150 ","End":"01:08.765","Text":"Now, it really doesn\u0027t matter which template you use, right?"},{"Start":"01:08.765 ","End":"01:11.921","Text":"I\u0027ll use this template right here,"},{"Start":"01:11.921 ","End":"01:13.745","Text":"y equals ax plus b."},{"Start":"01:13.745 ","End":"01:16.740","Text":"But if you use this template, well,"},{"Start":"01:16.740 ","End":"01:20.420","Text":"that\u0027s fine, all you need to do is switch between a and b,"},{"Start":"01:20.420 ","End":"01:27.755","Text":"where here b is the x multiplier and a would be a constant that you add and subtract."},{"Start":"01:27.755 ","End":"01:34.265","Text":"Here, a would be the x multiplier and b would be the constant that you add and subtract."},{"Start":"01:34.265 ","End":"01:35.870","Text":"I\u0027ll be using, as I said,"},{"Start":"01:35.870 ","End":"01:40.670","Text":"this template right here where a would be the x multiplier."},{"Start":"01:40.670 ","End":"01:46.265","Text":"Now, we also want to know what happens to the various measures."},{"Start":"01:46.265 ","End":"01:48.620","Text":"We talked about various measures."},{"Start":"01:48.620 ","End":"01:51.590","Text":"We talked about measures of centralization,"},{"Start":"01:51.590 ","End":"01:54.920","Text":"which would be the mid-range, the mode,"},{"Start":"01:54.920 ","End":"01:56.630","Text":"the average, and the median,"},{"Start":"01:56.630 ","End":"01:58.595","Text":"or measures of dispersion,"},{"Start":"01:58.595 ","End":"02:02.310","Text":"which would be the distribution range,"},{"Start":"02:02.310 ","End":"02:05.210","Text":"the standard deviation or the variance,"},{"Start":"02:05.210 ","End":"02:07.790","Text":"or the measures of relative location which"},{"Start":"02:07.790 ","End":"02:10.520","Text":"would be the percentiles to the standard score."},{"Start":"02:10.520 ","End":"02:19.400","Text":"Well, again, we want to know what happens to these measures when we transform x into y."},{"Start":"02:19.400 ","End":"02:23.930","Text":"Now, there are also various work stages we have"},{"Start":"02:23.930 ","End":"02:28.630","Text":"to follow in order to calculate these measures."},{"Start":"02:28.630 ","End":"02:31.220","Text":"Here we have 4 stages."},{"Start":"02:31.220 ","End":"02:34.535","Text":"We have to verify that a linear transformation is involved."},{"Start":"02:34.535 ","End":"02:37.130","Text":"We have to formulate the transformation rule"},{"Start":"02:37.130 ","End":"02:39.800","Text":"according to the information provided in the question,"},{"Start":"02:39.800 ","End":"02:44.495","Text":"we have to simplify the rule and identify the values of a and b."},{"Start":"02:44.495 ","End":"02:46.810","Text":"Then, and only then,"},{"Start":"02:46.810 ","End":"02:51.515","Text":"do we substitute in the above formulas according to the measures involved."},{"Start":"02:51.515 ","End":"02:54.290","Text":"This may sound a little complicated."},{"Start":"02:54.290 ","End":"02:57.630","Text":"Let\u0027s just go ahead and do an example."}],"ID":13371},{"Watched":false,"Name":"Tutorial 2","Duration":"5m 46s","ChapterTopicVideoID":12897,"CourseChapterTopicPlaylistID":64523,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.019","Text":"In this example, we\u0027re given that the average salary of employees is $9,000,"},{"Start":"00:04.019 ","End":"00:05.948","Text":"and the range is $6,000."},{"Start":"00:05.948 ","End":"00:10.140","Text":"We\u0027re asked to calculate the above measures after the salaries have"},{"Start":"00:10.140 ","End":"00:14.741","Text":"been raised by 10 percent and then penalized by $100."},{"Start":"00:14.741 ","End":"00:17.340","Text":"The first thing that we need to do is look at"},{"Start":"00:17.340 ","End":"00:21.630","Text":"the steps that are involved in the linear transformation."},{"Start":"00:21.630 ","End":"00:26.010","Text":"Here are the steps and the first step right"},{"Start":"00:26.010 ","End":"00:29.805","Text":"here is to verify that the linear transformation is involved."},{"Start":"00:29.805 ","End":"00:31.080","Text":"Now, what does that mean?"},{"Start":"00:31.080 ","End":"00:35.580","Text":"Well, let\u0027s take a look at a random variable."},{"Start":"00:35.580 ","End":"00:39.600","Text":"Let\u0027s define X as a random variable and that\u0027ll be"},{"Start":"00:39.600 ","End":"00:45.900","Text":"the wages before we do anything with them."},{"Start":"00:45.900 ","End":"00:50.165","Text":"Let\u0027s define y as the new random variable,"},{"Start":"00:50.165 ","End":"00:52.010","Text":"which are the wages,"},{"Start":"00:52.010 ","End":"00:56.250","Text":"but after the change."},{"Start":"00:56.250 ","End":"00:58.580","Text":"What do we want to do?"},{"Start":"00:58.580 ","End":"01:02.900","Text":"Well, for each employee that receives a salary,"},{"Start":"01:02.900 ","End":"01:09.560","Text":"we\u0027re taking the salary and we\u0027re changing it according to some rule right here."},{"Start":"01:09.560 ","End":"01:14.210","Text":"That\u0027s the definition of a linear transformation for all the observations,"},{"Start":"01:14.210 ","End":"01:20.720","Text":"were changing these observations from the old value to a new value."},{"Start":"01:20.720 ","End":"01:22.805","Text":"Now, what else are we given?"},{"Start":"01:22.805 ","End":"01:24.290","Text":"Well, we\u0027re given that X bar,"},{"Start":"01:24.290 ","End":"01:26.126","Text":"the average of the old wages,"},{"Start":"01:26.126 ","End":"01:33.807","Text":"that was $9,000 and the range R sub x that equals to $6,000."},{"Start":"01:33.807 ","End":"01:36.585","Text":"Here, we\u0027ve done number 1,"},{"Start":"01:36.585 ","End":"01:37.985","Text":"That\u0027s 1 right here."},{"Start":"01:37.985 ","End":"01:39.560","Text":"Now, what about number 2,"},{"Start":"01:39.560 ","End":"01:41.660","Text":"formulating the transformation rule"},{"Start":"01:41.660 ","End":"01:44.675","Text":"according to the information provided in the question."},{"Start":"01:44.675 ","End":"01:47.955","Text":"Well, what are we doing here? What\u0027s the rule?"},{"Start":"01:47.955 ","End":"01:49.770","Text":"We\u0027re taking the salaries,"},{"Start":"01:49.770 ","End":"01:51.780","Text":"we have to raise them by 10 percent,"},{"Start":"01:51.780 ","End":"01:55.160","Text":"then penalize this new salaries by $100."},{"Start":"01:55.160 ","End":"01:57.890","Text":"For example, if we have a salary,"},{"Start":"01:57.890 ","End":"02:03.045","Text":"let\u0027s say $8,000,"},{"Start":"02:03.045 ","End":"02:05.070","Text":"by raising it by 10 percent,"},{"Start":"02:05.070 ","End":"02:07.905","Text":"then we get $8,800,"},{"Start":"02:07.905 ","End":"02:11.595","Text":"and then we have to take away $100."},{"Start":"02:11.595 ","End":"02:16.810","Text":"What would happen? The new salary would be $8,700."},{"Start":"02:16.810 ","End":"02:20.045","Text":"What happens if we take a salary of $6,000?"},{"Start":"02:20.045 ","End":"02:22.575","Text":"Well, we have to raise that by 10 percent."},{"Start":"02:22.575 ","End":"02:25.455","Text":"We get $6,600 and again,"},{"Start":"02:25.455 ","End":"02:28.325","Text":"we have to take away $100,"},{"Start":"02:28.325 ","End":"02:32.060","Text":"so that\u0027ll be $6,500."},{"Start":"02:32.060 ","End":"02:33.920","Text":"Now, what are we doing here?"},{"Start":"02:33.920 ","End":"02:35.330","Text":"We have a template,"},{"Start":"02:35.330 ","End":"02:39.140","Text":"y equals to aX plus b."},{"Start":"02:39.140 ","End":"02:43.685","Text":"How do we formulate the transformation?"},{"Start":"02:43.685 ","End":"02:45.500","Text":"Well, let\u0027s take a look at this."},{"Start":"02:45.500 ","End":"02:51.410","Text":"We have X our old wages and we have to raise the old wages by 10 percent."},{"Start":"02:51.410 ","End":"02:56.190","Text":"That means we have to multiply X by 1.1."},{"Start":"02:56.290 ","End":"03:03.320","Text":"That\u0027s what we need to do in order to increase X by 10 percent."},{"Start":"03:03.320 ","End":"03:08.870","Text":"But then, we have to penalize this new value by $100."},{"Start":"03:08.870 ","End":"03:11.780","Text":"We have to take away $100."},{"Start":"03:11.780 ","End":"03:16.280","Text":"That\u0027ll be our transformation rule,"},{"Start":"03:16.280 ","End":"03:22.775","Text":"that\u0027ll be our new variable right here and this is the rule that transforms X into y."},{"Start":"03:22.775 ","End":"03:24.575","Text":"So this is number 2."},{"Start":"03:24.575 ","End":"03:26.395","Text":"We\u0027ve done number 2 here."},{"Start":"03:26.395 ","End":"03:28.155","Text":"Now, what\u0027s number 3?"},{"Start":"03:28.155 ","End":"03:32.630","Text":"3 means that I take this and I simplify the rule."},{"Start":"03:32.630 ","End":"03:36.455","Text":"In here, this really doesn\u0027t need any simplification here,"},{"Start":"03:36.455 ","End":"03:39.575","Text":"but we need to identify a and b."},{"Start":"03:39.575 ","End":"03:42.735","Text":"So 3 here, what would be a?"},{"Start":"03:42.735 ","End":"03:47.670","Text":"Well, what\u0027s the X multiplier, that\u0027ll be 1.1."},{"Start":"03:47.670 ","End":"03:51.725","Text":"What\u0027s b? What do we add and or subtract from this?"},{"Start":"03:51.725 ","End":"03:53.663","Text":"Well, that\u0027s minus 100,"},{"Start":"03:53.663 ","End":"03:59.100","Text":"we\u0027re subtracting 100 so b would be minus 100."},{"Start":"04:00.380 ","End":"04:04.375","Text":"We\u0027ve done number 3 so what\u0027s number 4?"},{"Start":"04:04.375 ","End":"04:07.070","Text":"Well, we need to substitute in the above formulas"},{"Start":"04:07.070 ","End":"04:10.020","Text":"according to the measures involved where we have a and b."},{"Start":"04:10.020 ","End":"04:11.615","Text":"Now, what are the measures?"},{"Start":"04:11.615 ","End":"04:15.050","Text":"Well, the measures here are average and range."},{"Start":"04:15.050 ","End":"04:16.775","Text":"Let\u0027s start with the average."},{"Start":"04:16.775 ","End":"04:19.040","Text":"What\u0027s y-bar?"},{"Start":"04:19.040 ","End":"04:22.370","Text":"Well, according to the transformation rule,"},{"Start":"04:22.370 ","End":"04:28.085","Text":"y-bar equals to a times x-bar plus b."},{"Start":"04:28.085 ","End":"04:30.455","Text":"Well, in our case,"},{"Start":"04:30.455 ","End":"04:32.790","Text":"that\u0027ll be equal to the following,"},{"Start":"04:32.790 ","End":"04:34.990","Text":"y-bar would be equal to a,"},{"Start":"04:34.990 ","End":"04:39.110","Text":"that\u0027s 1.1 times x-bar."},{"Start":"04:39.110 ","End":"04:43.910","Text":"Well, x-bar is $9,000 plus b,"},{"Start":"04:43.910 ","End":"04:45.350","Text":"where b is minus 100,"},{"Start":"04:45.350 ","End":"04:48.110","Text":"so that\u0027ll be minus $100."},{"Start":"04:48.110 ","End":"04:53.045","Text":"Now, that\u0027ll be equal to $9,900 minus 100,"},{"Start":"04:53.045 ","End":"04:56.675","Text":"and that equals to $9,800."},{"Start":"04:56.675 ","End":"05:01.085","Text":"This then would be the new average after"},{"Start":"05:01.085 ","End":"05:05.945","Text":"we raise the salaries by 10 percent and then penalize the salaries by $100."},{"Start":"05:05.945 ","End":"05:10.425","Text":"Now, what about the range R sub x or R sub y?"},{"Start":"05:10.425 ","End":"05:11.925","Text":"R sub y,"},{"Start":"05:11.925 ","End":"05:16.805","Text":"that equals to the absolute value of x times R sub x."},{"Start":"05:16.805 ","End":"05:20.100","Text":"In our case, R sub y,"},{"Start":"05:20.100 ","End":"05:23.270","Text":"well, that\u0027ll be equal to the absolute value of a."},{"Start":"05:23.270 ","End":"05:27.620","Text":"Now, a is 1.1 times R sub x."},{"Start":"05:27.620 ","End":"05:36.530","Text":"Well, R sub x is $6,000 and that equals to $6,600. There we go."},{"Start":"05:36.530 ","End":"05:39.350","Text":"This would be the new range right here,"},{"Start":"05:39.350 ","End":"05:47.040","Text":"$6,600 and the new average would be $9,800 after the transformation."}],"ID":13372},{"Watched":false,"Name":"Exercise 1","Duration":"4m 35s","ChapterTopicVideoID":12898,"CourseChapterTopicPlaylistID":64523,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.980","Text":"In this question, we\u0027re given a certain series of"},{"Start":"00:01.980 ","End":"00:05.339","Text":"data that has the following measures: the average,"},{"Start":"00:05.339 ","End":"00:06.780","Text":"X-bar, equals to 80,"},{"Start":"00:06.780 ","End":"00:08.760","Text":"the standard deviation equals to 15,"},{"Start":"00:08.760 ","End":"00:10.920","Text":"and the mode equals to 70."},{"Start":"00:10.920 ","End":"00:17.865","Text":"Now, it\u0027s decided to multiply every observation by 4 and then subtract 5 from the result."},{"Start":"00:17.865 ","End":"00:22.680","Text":"We\u0027re asked to calculate the above measures after the transformation."},{"Start":"00:22.680 ","End":"00:29.370","Text":"Let\u0027s first look at the 4 steps that we have to go through."},{"Start":"00:29.370 ","End":"00:32.220","Text":"Here we have the 4 stages."},{"Start":"00:32.220 ","End":"00:37.085","Text":"The first stage then is to verify that a linear transformation is involved."},{"Start":"00:37.085 ","End":"00:44.275","Text":"Well, let\u0027s define x as the old variable."},{"Start":"00:44.275 ","End":"00:49.890","Text":"What we want to do is we want to transform that into y, the new variable."},{"Start":"00:49.890 ","End":"00:51.125","Text":"Now, how do we do that?"},{"Start":"00:51.125 ","End":"00:54.650","Text":"By multiplication or division of"},{"Start":"00:54.650 ","End":"00:59.590","Text":"the old variable and then the addition or subtraction of a constant."},{"Start":"00:59.590 ","End":"01:02.915","Text":"This then would be step number 1."},{"Start":"01:02.915 ","End":"01:08.795","Text":"We know that we have an old variable and we have to transform that into a new variable."},{"Start":"01:08.795 ","End":"01:11.015","Text":"Now, what\u0027s the second step?"},{"Start":"01:11.015 ","End":"01:14.570","Text":"The second step is to formulate the transformation rule."},{"Start":"01:14.570 ","End":"01:17.570","Text":"Well, what\u0027s the rule here?"},{"Start":"01:17.570 ","End":"01:20.600","Text":"Well, for every observation of x,"},{"Start":"01:20.600 ","End":"01:27.945","Text":"we have to multiply every observation by 4 and then to subtract 5."},{"Start":"01:27.945 ","End":"01:29.970","Text":"What do we need to do here?"},{"Start":"01:29.970 ","End":"01:32.780","Text":"Well, we have to take x,"},{"Start":"01:32.780 ","End":"01:38.960","Text":"we have to multiply x by 4 and then we have to take away 5,"},{"Start":"01:38.960 ","End":"01:43.040","Text":"and that will bring us to the new variable y."},{"Start":"01:43.040 ","End":"01:47.285","Text":"Now, that\u0027s 2. Now what about 3?"},{"Start":"01:47.285 ","End":"01:52.670","Text":"Well, 3 says that we need to simplify the rule and identify the values of a and b."},{"Start":"01:52.670 ","End":"01:58.845","Text":"Well, our template is y equals to ax plus b."},{"Start":"01:58.845 ","End":"02:03.065","Text":"We really don\u0027t need to simplify this anymore."},{"Start":"02:03.065 ","End":"02:05.705","Text":"We can identify what a and what b is."},{"Start":"02:05.705 ","End":"02:08.795","Text":"A would be the x multiplier, so a here,"},{"Start":"02:08.795 ","End":"02:12.475","Text":"that would be equal to 4 and b,"},{"Start":"02:12.475 ","End":"02:16.250","Text":"well that\u0027s equal to minus 5."},{"Start":"02:16.880 ","End":"02:19.250","Text":"What\u0027s number 4?"},{"Start":"02:19.250 ","End":"02:24.380","Text":"Well, we substitute these guys into the above formulas according to the measures."},{"Start":"02:24.380 ","End":"02:25.430","Text":"What are the measures?"},{"Start":"02:25.430 ","End":"02:27.050","Text":"The measures are the average,"},{"Start":"02:27.050 ","End":"02:30.050","Text":"the standard deviation, and the mode."},{"Start":"02:30.050 ","End":"02:34.515","Text":"Let\u0027s get to work. Let\u0027s start with the mode."},{"Start":"02:34.515 ","End":"02:38.375","Text":"The mode, that\u0027s M_O of y."},{"Start":"02:38.375 ","End":"02:40.310","Text":"Now, what does that equal to?"},{"Start":"02:40.310 ","End":"02:47.250","Text":"That equals to a times M_O of x plus b."},{"Start":"02:47.660 ","End":"02:51.510","Text":"That\u0027s the transformation template that\u0027s used."},{"Start":"02:51.510 ","End":"02:53.810","Text":"What do we have here in our case?"},{"Start":"02:53.810 ","End":"02:58.250","Text":"A equals to 4 times the mode of x."},{"Start":"02:58.250 ","End":"03:01.350","Text":"Well, the mode of x, that\u0027s 70."},{"Start":"03:01.550 ","End":"03:03.810","Text":"That\u0027s times 70,"},{"Start":"03:03.810 ","End":"03:06.660","Text":"4 times 70, plus b. Now, what\u0027s b?"},{"Start":"03:06.660 ","End":"03:09.060","Text":"B is minus 5."},{"Start":"03:09.060 ","End":"03:16.025","Text":"4 times 70 would be 280 minus 5,"},{"Start":"03:16.025 ","End":"03:19.865","Text":"and that equals to 275."},{"Start":"03:19.865 ","End":"03:24.220","Text":"Now, what about the average?"},{"Start":"03:24.220 ","End":"03:27.770","Text":"The average y-bar, well,"},{"Start":"03:27.770 ","End":"03:32.750","Text":"that equals to a times X-bar plus b."},{"Start":"03:32.750 ","End":"03:36.810","Text":"In our case, a again would be 4 times,"},{"Start":"03:36.810 ","End":"03:38.130","Text":"now, what\u0027s X-bar?"},{"Start":"03:38.130 ","End":"03:42.330","Text":"Well, X-bar is 80, plus b."},{"Start":"03:42.330 ","End":"03:43.650","Text":"Well, b is minus 5,"},{"Start":"03:43.650 ","End":"03:45.555","Text":"so that\u0027ll be minus 5,"},{"Start":"03:45.555 ","End":"03:47.010","Text":"and that equals to the following."},{"Start":"03:47.010 ","End":"03:48.981","Text":"Now, 4 times 80,"},{"Start":"03:48.981 ","End":"03:51.600","Text":"that equals to what?"},{"Start":"03:51.600 ","End":"04:00.395","Text":"That equals to 320 minus 5 and that equals to 315."},{"Start":"04:00.395 ","End":"04:03.190","Text":"Now, the standard deviation,"},{"Start":"04:03.190 ","End":"04:05.695","Text":"S_y, well,"},{"Start":"04:05.695 ","End":"04:07.085","Text":"that equals to what?"},{"Start":"04:07.085 ","End":"04:12.140","Text":"That equals to the absolute value of a times the standard deviation of x."},{"Start":"04:12.140 ","End":"04:15.340","Text":"Now, again, in our case a is 4,"},{"Start":"04:15.340 ","End":"04:19.850","Text":"so that\u0027s the absolute value of 4 times the standard deviation of x."},{"Start":"04:19.850 ","End":"04:23.290","Text":"Well, the standard deviation of x here, that\u0027s 15."},{"Start":"04:23.290 ","End":"04:28.815","Text":"4 times 15, that equals to 60."},{"Start":"04:28.815 ","End":"04:31.805","Text":"There we have it. This is the standard deviation,"},{"Start":"04:31.805 ","End":"04:33.664","Text":"this would be the average,"},{"Start":"04:33.664 ","End":"04:36.750","Text":"and this would be the mode."}],"ID":13373},{"Watched":false,"Name":"Exercise 2","Duration":"5m 2s","ChapterTopicVideoID":12899,"CourseChapterTopicPlaylistID":64523,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"In this question, we\u0027re given that the average salary in a certain company is"},{"Start":"00:03.180 ","End":"00:07.440","Text":"$40 per hour with a standard deviation of $5 per hour."},{"Start":"00:07.440 ","End":"00:11.685","Text":"Management decided to raise all the salaries by 10 percent,"},{"Start":"00:11.685 ","End":"00:13.770","Text":"but this didn\u0027t satisfy the employee."},{"Start":"00:13.770 ","End":"00:18.915","Text":"After further negotiations, they received an additional $2 per hour."},{"Start":"00:18.915 ","End":"00:25.584","Text":"We\u0027re asked what are the average and variance of the hourly rate after all the changes?"},{"Start":"00:25.584 ","End":"00:29.270","Text":"First, let\u0027s go into our 4 stages and"},{"Start":"00:29.270 ","End":"00:33.815","Text":"see how we can calculate the new average in variance."},{"Start":"00:33.815 ","End":"00:36.950","Text":"These then are the stages."},{"Start":"00:36.950 ","End":"00:40.625","Text":"The first 1 is to verify that a linear transformation is involved."},{"Start":"00:40.625 ","End":"00:46.580","Text":"Well, that\u0027s defined x as the old wages per hour."},{"Start":"00:46.580 ","End":"00:50.330","Text":"That\u0027s the dollar per hour value."},{"Start":"00:50.330 ","End":"00:52.340","Text":"That\u0027s the random variable rate here,"},{"Start":"00:52.340 ","End":"00:53.855","Text":"and that\u0027s the old 1."},{"Start":"00:53.855 ","End":"01:02.360","Text":"We want to go from here to y and we\u0027ll define that as the new dollars per hour,"},{"Start":"01:02.360 ","End":"01:04.085","Text":"and that\u0027s the new wage,"},{"Start":"01:04.085 ","End":"01:07.970","Text":"and that\u0027s after the change."},{"Start":"01:07.970 ","End":"01:10.025","Text":"Now, what else are we given?"},{"Start":"01:10.025 ","End":"01:14.270","Text":"We\u0027re given that x bar equals to 40,"},{"Start":"01:14.270 ","End":"01:19.060","Text":"and we\u0027re given that the standard deviation that equals to 5."},{"Start":"01:19.060 ","End":"01:24.545","Text":"That\u0027s right here, that\u0027s $40 per hour and $5 per hour."},{"Start":"01:24.545 ","End":"01:29.420","Text":"Now, let\u0027s go to number 2."},{"Start":"01:29.420 ","End":"01:30.650","Text":"We want to formulate"},{"Start":"01:30.650 ","End":"01:34.970","Text":"the transformation rule according to the information provided in the question."},{"Start":"01:34.970 ","End":"01:36.740","Text":"What\u0027s the transformation rule?"},{"Start":"01:36.740 ","End":"01:40.955","Text":"Well, we first have to raise all the salaries by 10 percent,"},{"Start":"01:40.955 ","End":"01:44.869","Text":"and then we have to tack on another $2 per hour."},{"Start":"01:44.869 ","End":"01:47.195","Text":"Let\u0027s say for example,"},{"Start":"01:47.195 ","End":"01:50.540","Text":"if we had $40 per hour,"},{"Start":"01:50.540 ","End":"01:52.280","Text":"what we want to do,"},{"Start":"01:52.280 ","End":"01:54.725","Text":"is we want to add 10 percent to that."},{"Start":"01:54.725 ","End":"02:02.165","Text":"Well, that\u0027s multiplying this value by 1.1 so that\u0027ll be 1.1 times 40 plus 2."},{"Start":"02:02.165 ","End":"02:03.695","Text":"That\u0027s what we need to do here."},{"Start":"02:03.695 ","End":"02:05.735","Text":"If we had $50,"},{"Start":"02:05.735 ","End":"02:11.630","Text":"then I would multiply this by 1.1 and then again add $2."},{"Start":"02:11.630 ","End":"02:16.775","Text":"If I had $60, then that\u0027ll be 1.1 times 60 plus 2."},{"Start":"02:16.775 ","End":"02:20.960","Text":"I think you get the general idea here."},{"Start":"02:20.960 ","End":"02:23.240","Text":"Let\u0027s do number 2."},{"Start":"02:23.240 ","End":"02:26.640","Text":"How do we formulate the transformation rule?"},{"Start":"02:26.640 ","End":"02:29.460","Text":"Well, in number 2,"},{"Start":"02:29.460 ","End":"02:36.152","Text":"we have x and we have to multiply each value of x by 1.1"},{"Start":"02:36.152 ","End":"02:42.710","Text":"and then we have to add on another $2 and that equals to y."},{"Start":"02:42.710 ","End":"02:47.435","Text":"This is how we get from the old wage to the new wage."},{"Start":"02:47.435 ","End":"02:50.090","Text":"We\u0027ve done that."},{"Start":"02:50.090 ","End":"02:52.010","Text":"Now what about number 3?"},{"Start":"02:52.010 ","End":"02:56.090","Text":"We want to simplify the rule and identify the values for a and b."},{"Start":"02:56.090 ","End":"02:57.515","Text":"Well, what does that mean?"},{"Start":"02:57.515 ","End":"02:58.790","Text":"Well, for 3,"},{"Start":"02:58.790 ","End":"02:59.870","Text":"we use a template,"},{"Start":"02:59.870 ","End":"03:03.054","Text":"y equals to ax plus b,"},{"Start":"03:03.054 ","End":"03:06.290","Text":"that\u0027s a transformation template."},{"Start":"03:06.290 ","End":"03:09.380","Text":"Do we need to simplify this anymore?"},{"Start":"03:09.380 ","End":"03:11.570","Text":"Well no, this is pretty much simplified."},{"Start":"03:11.570 ","End":"03:15.650","Text":"All we need to do is we need to identify the values for a and b."},{"Start":"03:15.650 ","End":"03:18.335","Text":"Well, a, that\u0027s our x multiplier."},{"Start":"03:18.335 ","End":"03:21.830","Text":"Well here, that equals to 1.1."},{"Start":"03:21.830 ","End":"03:23.590","Text":"What about b?"},{"Start":"03:23.590 ","End":"03:25.775","Text":"That equals to 2?"},{"Start":"03:25.775 ","End":"03:28.505","Text":"That\u0027s the constant that we\u0027re using here."},{"Start":"03:28.505 ","End":"03:33.570","Text":"That\u0027s 2. Great. We\u0027ve done 3 what about 4?"},{"Start":"03:33.570 ","End":"03:36.455","Text":"Now we need to take this information right here,"},{"Start":"03:36.455 ","End":"03:40.940","Text":"and we need to substitute in the above formulas according to the measures involved."},{"Start":"03:40.940 ","End":"03:42.395","Text":"What are the measures involved?"},{"Start":"03:42.395 ","End":"03:45.184","Text":"Well, we are asked about the average and variance."},{"Start":"03:45.184 ","End":"03:48.825","Text":"For the average y bar,"},{"Start":"03:48.825 ","End":"03:52.435","Text":"well that equals to 8 times x bar plus b."},{"Start":"03:52.435 ","End":"03:56.840","Text":"In our case, y bar would be equal to the following."},{"Start":"03:56.840 ","End":"04:00.890","Text":"Well a here that equals to 1.1 times x-bar."},{"Start":"04:00.890 ","End":"04:04.550","Text":"Well x-bar equals to 40 plus b."},{"Start":"04:04.550 ","End":"04:06.005","Text":"Well b equals to 2."},{"Start":"04:06.005 ","End":"04:07.985","Text":"What is that equal?"},{"Start":"04:07.985 ","End":"04:10.220","Text":"That equals to 46,"},{"Start":"04:10.220 ","End":"04:13.535","Text":"and that\u0027s dollars per hour."},{"Start":"04:13.535 ","End":"04:16.685","Text":"Now, what about the variance?"},{"Start":"04:16.685 ","End":"04:25.160","Text":"The variance of y that equals to a squared times the variance of x."},{"Start":"04:25.160 ","End":"04:31.250","Text":"In our case, the variance of y that equals to a squared,"},{"Start":"04:31.250 ","End":"04:38.210","Text":"well that\u0027s 1.1. a equals 1.1 squared times the variance of x."},{"Start":"04:38.210 ","End":"04:43.895","Text":"Well, the standard deviation of x is 5 so we have here times 5 squared."},{"Start":"04:43.895 ","End":"04:49.865","Text":"Now that equals to 30.25."},{"Start":"04:49.865 ","End":"04:51.875","Text":"Now, what are the units here?"},{"Start":"04:51.875 ","End":"04:55.760","Text":"The units here are dollars squared per hour."},{"Start":"04:55.760 ","End":"04:57.830","Text":"This is it."},{"Start":"04:57.830 ","End":"04:59.990","Text":"This would be the variance of y,"},{"Start":"04:59.990 ","End":"05:03.360","Text":"and this would be the average of y."}],"ID":13374},{"Watched":false,"Name":"Exercise 3","Duration":"5m ","ChapterTopicVideoID":12900,"CourseChapterTopicPlaylistID":64523,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.795","Text":"In this question, we\u0027re given that the median mark in an exam is 73,"},{"Start":"00:03.795 ","End":"00:06.000","Text":"the range of the marks is 40 points,"},{"Start":"00:06.000 ","End":"00:08.684","Text":"and the upper 10th percentile is 87."},{"Start":"00:08.684 ","End":"00:11.520","Text":"Now because the students marks in the exam were so low,"},{"Start":"00:11.520 ","End":"00:15.150","Text":"the teacher decided to boost the student\u0027s marks by 4 points,"},{"Start":"00:15.150 ","End":"00:20.955","Text":"and we\u0027re asked to calculate the median range and upper 10th percentile after the boost."},{"Start":"00:20.955 ","End":"00:27.870","Text":"Let\u0027s just see how we can calculate these measures by using the 4 stages."},{"Start":"00:27.870 ","End":"00:29.670","Text":"Here are the 4 stages."},{"Start":"00:29.670 ","End":"00:34.320","Text":"The first stage is to verify that a linear transformation is involved."},{"Start":"00:34.320 ","End":"00:44.474","Text":"Let\u0027s just define X as the marks before the boost and,"},{"Start":"00:44.474 ","End":"00:50.580","Text":"Y would be the marks after the boost."},{"Start":"00:50.580 ","End":"00:52.465","Text":"Now, what are we doing?"},{"Start":"00:52.465 ","End":"00:55.990","Text":"Well, we\u0027re taking each individual mark and we\u0027re transforming"},{"Start":"00:55.990 ","End":"01:00.160","Text":"it from its value to a new value."},{"Start":"01:00.160 ","End":"01:02.185","Text":"Now, how are we doing that?"},{"Start":"01:02.185 ","End":"01:06.385","Text":"Well, we\u0027re doing that by either multiplying or dividing by a constant,"},{"Start":"01:06.385 ","End":"01:09.940","Text":"or we\u0027re taking X and we\u0027re adding or subtracting a constant."},{"Start":"01:09.940 ","End":"01:13.955","Text":"That\u0027s what makes this a linear transformation."},{"Start":"01:13.955 ","End":"01:20.275","Text":"We verify then that we have a linear transformation from X to Y."},{"Start":"01:20.275 ","End":"01:22.195","Text":"Now, what\u0027s the second stage?"},{"Start":"01:22.195 ","End":"01:25.915","Text":"The second stage is to formulate the transformation rule."},{"Start":"01:25.915 ","End":"01:27.920","Text":"What did the teacher do?"},{"Start":"01:27.920 ","End":"01:31.820","Text":"The teacher decided to boost the students marked by 4 points."},{"Start":"01:31.820 ","End":"01:34.910","Text":"That means that if I had,"},{"Start":"01:34.910 ","End":"01:37.220","Text":"let\u0027s say, a mark of 60,"},{"Start":"01:37.220 ","End":"01:39.905","Text":"well, then it became 64."},{"Start":"01:39.905 ","End":"01:41.120","Text":"I added 4 points."},{"Start":"01:41.120 ","End":"01:43.565","Text":"Or if I had 30 points,"},{"Start":"01:43.565 ","End":"01:46.190","Text":"then now I would have 34."},{"Start":"01:46.190 ","End":"01:48.530","Text":"In general, if I have X points,"},{"Start":"01:48.530 ","End":"01:51.834","Text":"I would now have X plus 4."},{"Start":"01:51.834 ","End":"01:54.960","Text":"In stage number 2,"},{"Start":"01:54.960 ","End":"02:02.175","Text":"I take X and I add 4 points to it and that gives me the new mark Y."},{"Start":"02:02.175 ","End":"02:04.850","Text":"This would be the transformation rule."},{"Start":"02:04.850 ","End":"02:07.250","Text":"Now, what about 3?"},{"Start":"02:07.250 ","End":"02:11.150","Text":"We need to simplify the rule and identify the values of a and b."},{"Start":"02:11.150 ","End":"02:19.160","Text":"Well, for 3, the linear transformation template is y equals to a times X plus b."},{"Start":"02:19.160 ","End":"02:21.905","Text":"In order to simplify this,"},{"Start":"02:21.905 ","End":"02:25.160","Text":"I would need to write this out like this."},{"Start":"02:25.160 ","End":"02:28.880","Text":"Y would be equal to 1 times X plus 4."},{"Start":"02:28.880 ","End":"02:30.470","Text":"Now why did they do that?"},{"Start":"02:30.470 ","End":"02:35.855","Text":"Well now, I can clearly identify what are the values of a and b,"},{"Start":"02:35.855 ","End":"02:37.860","Text":"a would be equal to 1,"},{"Start":"02:37.860 ","End":"02:39.525","Text":"that\u0027s the X multiplier,"},{"Start":"02:39.525 ","End":"02:41.840","Text":"and b that equals to 4,"},{"Start":"02:41.840 ","End":"02:45.755","Text":"that\u0027s the constant that we\u0027re adding to this multiplication."},{"Start":"02:45.755 ","End":"02:49.100","Text":"Now, before we can go on to stage number 4,"},{"Start":"02:49.100 ","End":"02:52.415","Text":"let\u0027s just review the median mark,"},{"Start":"02:52.415 ","End":"02:56.240","Text":"md of x, that equals to 73."},{"Start":"02:56.240 ","End":"03:00.440","Text":"The range, R of x that equals to 40,"},{"Start":"03:00.440 ","End":"03:02.930","Text":"and the upper 10th percentile, well,"},{"Start":"03:02.930 ","End":"03:05.750","Text":"that\u0027s the 90th percentile,"},{"Start":"03:05.750 ","End":"03:08.910","Text":"and that equals to 87."},{"Start":"03:09.110 ","End":"03:12.845","Text":"Having done that, let\u0027s go to number 4."},{"Start":"03:12.845 ","End":"03:16.460","Text":"We need to substitute in the above formulas."},{"Start":"03:16.460 ","End":"03:19.460","Text":"According to the measures involved. What are the measures?"},{"Start":"03:19.460 ","End":"03:20.930","Text":"We\u0027re asked to calculate the median,"},{"Start":"03:20.930 ","End":"03:23.260","Text":"the range, and the upper 10th percentile."},{"Start":"03:23.260 ","End":"03:28.470","Text":"Let\u0027s get to work. Let\u0027s first do the median."},{"Start":"03:28.470 ","End":"03:31.860","Text":"The median of y, well,"},{"Start":"03:31.860 ","End":"03:37.530","Text":"that equals to 8 times the median of x plus b."},{"Start":"03:37.530 ","End":"03:43.065","Text":"In our case, a would be equal to 1 times the median of x."},{"Start":"03:43.065 ","End":"03:47.180","Text":"The median of x is 73 plus b."},{"Start":"03:47.180 ","End":"03:50.720","Text":"Well b equals 4, so that equals to 77."},{"Start":"03:50.720 ","End":"03:54.130","Text":"Now, what about the range of y?"},{"Start":"03:54.130 ","End":"04:01.130","Text":"Well, the range of y is the absolute value of a times the range of x."},{"Start":"04:01.130 ","End":"04:08.145","Text":"In our case, there\u0027ll be the absolute value of 1 times the range of x."},{"Start":"04:08.145 ","End":"04:10.560","Text":"Well, the range of x here, that\u0027s 40."},{"Start":"04:10.560 ","End":"04:13.275","Text":"That equals to 40."},{"Start":"04:13.275 ","End":"04:18.385","Text":"What about the upper 10th percentile?"},{"Start":"04:18.385 ","End":"04:22.760","Text":"We know that when we\u0027re dealing with percentiles, Y_ p,"},{"Start":"04:22.760 ","End":"04:29.390","Text":"that equals to a times X_ p plus b."},{"Start":"04:29.390 ","End":"04:33.110","Text":"In our case, Y_0.9,"},{"Start":"04:33.110 ","End":"04:34.774","Text":"that\u0027s a 90th percentile."},{"Start":"04:34.774 ","End":"04:36.340","Text":"That equals to a,"},{"Start":"04:36.340 ","End":"04:42.500","Text":"that equals to 1 times X_ p. The 90th percentile of x,"},{"Start":"04:42.500 ","End":"04:44.030","Text":"that equals to 87,"},{"Start":"04:44.030 ","End":"04:48.150","Text":"it\u0027s right here, plus b plus 4,"},{"Start":"04:48.150 ","End":"04:50.535","Text":"so that equals to 91."},{"Start":"04:50.535 ","End":"04:52.310","Text":"There we have it."},{"Start":"04:52.310 ","End":"04:55.490","Text":"Here we have the 90th percentile of y."},{"Start":"04:55.490 ","End":"04:58.415","Text":"That\u0027s the range of y,"},{"Start":"04:58.415 ","End":"05:01.530","Text":"and that\u0027s the median of y."}],"ID":13375},{"Watched":false,"Name":"Exercise 4","Duration":"4m 46s","ChapterTopicVideoID":12901,"CourseChapterTopicPlaylistID":64523,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"In this question, we\u0027re given 50 matchboxes that are sampled from"},{"Start":"00:03.660 ","End":"00:08.550","Text":"a production line producing matchboxes that contain 40 matches per box."},{"Start":"00:08.550 ","End":"00:11.055","Text":"The number of faulty matches is checked."},{"Start":"00:11.055 ","End":"00:16.410","Text":"An average of 3 faulty matches were found per matchbox with a standard deviation of 1.5,"},{"Start":"00:16.410 ","End":"00:17.820","Text":"and we\u0027re asked what are"},{"Start":"00:17.820 ","End":"00:22.110","Text":"the average and standard deviation of the number of good matches in a box?"},{"Start":"00:22.110 ","End":"00:27.825","Text":"Well, let\u0027s review the stages that we have to go through. Here we are."},{"Start":"00:27.825 ","End":"00:33.765","Text":"These are the stages and the first is is to verify that there\u0027s a linear transformation."},{"Start":"00:33.765 ","End":"00:36.375","Text":"Let\u0027s define our variable."},{"Start":"00:36.375 ","End":"00:43.665","Text":"We\u0027ll define X as the number of faulty matches"},{"Start":"00:43.665 ","End":"00:52.910","Text":"in a box and Y will define as the number of good matches in a box."},{"Start":"00:52.910 ","End":"00:55.700","Text":"Now, we want to take this random variable,"},{"Start":"00:55.700 ","End":"00:59.910","Text":"we want to transform that into this random variable"},{"Start":"00:59.910 ","End":"01:04.325","Text":"from the number of faulty matches in a box to the number of good matches in a box."},{"Start":"01:04.325 ","End":"01:07.400","Text":"Now, in order for this to be a transformation,"},{"Start":"01:07.400 ","End":"01:13.605","Text":"then we have to apply this to all the 50 matchboxes and assume we do that."},{"Start":"01:13.605 ","End":"01:16.130","Text":"The second thing is that we"},{"Start":"01:16.130 ","End":"01:19.550","Text":"need to make sure that we are dealing with a linear transformation."},{"Start":"01:19.550 ","End":"01:23.790","Text":"That means that we multiply a constant by"},{"Start":"01:23.790 ","End":"01:30.180","Text":"X and then we either add or subtract a constant to that multiplication."},{"Start":"01:30.180 ","End":"01:35.875","Text":"Having done that, we verify that we\u0027re dealing with a linear transformation."},{"Start":"01:35.875 ","End":"01:40.090","Text":"Now, what about formulating the transformation rule?"},{"Start":"01:40.090 ","End":"01:45.100","Text":"Well, let\u0027s say that I found 5 faulty matches,"},{"Start":"01:45.100 ","End":"01:47.395","Text":"well, how many good matches do I have in a box?"},{"Start":"01:47.395 ","End":"01:52.075","Text":"Well, I have a total of 40 matches in a box and if 5 of them are faulty,"},{"Start":"01:52.075 ","End":"01:55.165","Text":"that means that 35 are good."},{"Start":"01:55.165 ","End":"01:57.525","Text":"40 minus 5."},{"Start":"01:57.525 ","End":"02:01.060","Text":"If I have, let\u0027s say 10 faulty matches,"},{"Start":"02:01.060 ","End":"02:03.700","Text":"then that\u0027ll be 40 minus 10."},{"Start":"02:03.700 ","End":"02:07.165","Text":"That\u0027ll be the number of good matches in a box."},{"Start":"02:07.165 ","End":"02:11.170","Text":"If I found X faulty matches,"},{"Start":"02:11.170 ","End":"02:14.810","Text":"then I have to subtract that from a total number of"},{"Start":"02:14.810 ","End":"02:20.480","Text":"40 matches in a box and that\u0027ll give me the number of good matches."},{"Start":"02:20.480 ","End":"02:23.570","Text":"We\u0027ve done 1, we\u0027ve done 2."},{"Start":"02:23.570 ","End":"02:26.495","Text":"This is the formulation of the transformation rule."},{"Start":"02:26.495 ","End":"02:28.940","Text":"Now, what about stage number 3,"},{"Start":"02:28.940 ","End":"02:32.390","Text":"to simplify the rule and identify the values of a and b?"},{"Start":"02:32.390 ","End":"02:34.160","Text":"Well, for 3,"},{"Start":"02:34.160 ","End":"02:40.965","Text":"the transformation template would be y equals aX plus b."},{"Start":"02:40.965 ","End":"02:44.090","Text":"What we need to do is we need to write this in"},{"Start":"02:44.090 ","End":"02:47.960","Text":"such a way that\u0027ll conform to this template."},{"Start":"02:47.960 ","End":"02:55.729","Text":"Let\u0027s write this like this: Y would be equal to minus 1 times X plus 40."},{"Start":"02:55.729 ","End":"03:02.450","Text":"Now, if I\u0027ve written it like this then I can easily see that a here is equal to minus 1,"},{"Start":"03:02.450 ","End":"03:04.993","Text":"that\u0027s the X multiplier,"},{"Start":"03:04.993 ","End":"03:08.530","Text":"and b that equals to 40."},{"Start":"03:08.530 ","End":"03:12.290","Text":"Let\u0027s go on then to stage number 4,"},{"Start":"03:12.290 ","End":"03:16.520","Text":"to substitute in the above formulas according to the measures involved."},{"Start":"03:16.520 ","End":"03:18.575","Text":"Well, let\u0027s do that."},{"Start":"03:18.575 ","End":"03:20.240","Text":"But before we do that,"},{"Start":"03:20.240 ","End":"03:22.975","Text":"let\u0027s just remind ourselves that X-bar,"},{"Start":"03:22.975 ","End":"03:26.795","Text":"the average of the faulty matches, that equals to 3,"},{"Start":"03:26.795 ","End":"03:29.840","Text":"and the standard deviation of the faulty matches,"},{"Start":"03:29.840 ","End":"03:32.060","Text":"that equals to 1.5."},{"Start":"03:32.060 ","End":"03:36.790","Text":"Now we can go ahead and go to stage number 4."},{"Start":"03:36.790 ","End":"03:39.010","Text":"Well, in stage number 4,"},{"Start":"03:39.010 ","End":"03:43.265","Text":"we\u0027re dealing with the average and standard deviation."},{"Start":"03:43.265 ","End":"03:45.995","Text":"We\u0027re looking at Y-bar."},{"Start":"03:45.995 ","End":"03:49.940","Text":"Now, the template for the transformation of an average,"},{"Start":"03:49.940 ","End":"03:51.260","Text":"that equals to Y-bar,"},{"Start":"03:51.260 ","End":"03:55.085","Text":"that equals to a times X-bar plus b."},{"Start":"03:55.085 ","End":"03:56.915","Text":"Now, in our case,"},{"Start":"03:56.915 ","End":"03:59.610","Text":"Y-bar, that\u0027ll b equal to a."},{"Start":"03:59.610 ","End":"04:02.400","Text":"Now, a is minus 1 times X-bar."},{"Start":"04:02.400 ","End":"04:04.605","Text":"Well, X-bar is 3,"},{"Start":"04:04.605 ","End":"04:08.940","Text":"plus b, b is 40."},{"Start":"04:08.940 ","End":"04:10.780","Text":"Minus b plus 40,"},{"Start":"04:10.780 ","End":"04:13.135","Text":"that equals to 37."},{"Start":"04:13.135 ","End":"04:17.470","Text":"What about the standard deviation of Y?"},{"Start":"04:17.470 ","End":"04:19.960","Text":"Well, according to the transformation template,"},{"Start":"04:19.960 ","End":"04:25.215","Text":"that equals to the absolute value of a times the standard deviation of X."},{"Start":"04:25.215 ","End":"04:31.750","Text":"In our case, that\u0027ll be the absolute values minus 1 times the standard deviation of x."},{"Start":"04:31.750 ","End":"04:33.620","Text":"Well, that\u0027s 1.5."},{"Start":"04:33.620 ","End":"04:36.405","Text":"That equals to 1.5."},{"Start":"04:36.405 ","End":"04:39.210","Text":"We can see here that on average,"},{"Start":"04:39.210 ","End":"04:47.409","Text":"we have 37 good matches in a box with a standard deviation of 1.5 good matches."}],"ID":13376},{"Watched":false,"Name":"Exercise 5","Duration":"5m 42s","ChapterTopicVideoID":12902,"CourseChapterTopicPlaylistID":64523,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.460","Text":"In this question, we have a local phone company that"},{"Start":"00:02.460 ","End":"00:04.920","Text":"offered the following package: 30 dollars"},{"Start":"00:04.920 ","End":"00:10.470","Text":"fixed monthly subscription fees plus 10 cents for each outgoing call per minute."},{"Start":"00:10.470 ","End":"00:14.430","Text":"Now, a person checked his outgoing calls for a year found that he averaged"},{"Start":"00:14.430 ","End":"00:20.370","Text":"600 minutes of outgoing calls per month with a variance of 2500 minutes squared."},{"Start":"00:20.370 ","End":"00:23.740","Text":"In January, his standard score was 2."},{"Start":"00:23.740 ","End":"00:28.345","Text":"That means that he was 2 standard deviations away from the average."},{"Start":"00:28.345 ","End":"00:31.190","Text":"Now we\u0027re asked to calculate the average, the variance,"},{"Start":"00:31.190 ","End":"00:32.420","Text":"and the standard score for"},{"Start":"00:32.420 ","End":"00:35.390","Text":"the subscriber\u0027s monthly telephone bill if he"},{"Start":"00:35.390 ","End":"00:39.630","Text":"would have used the package proposed to him by the phone company."},{"Start":"00:40.010 ","End":"00:45.570","Text":"Let\u0027s take a look at the stages that we have to follow."},{"Start":"00:46.160 ","End":"00:48.530","Text":"Here are the stages."},{"Start":"00:48.530 ","End":"00:52.460","Text":"The first stage is to verify that the linear transformation is involved."},{"Start":"00:52.460 ","End":"00:54.470","Text":"Well, let\u0027s define a variable here."},{"Start":"00:54.470 ","End":"01:03.400","Text":"Let\u0027s define X as the number of minutes of outgoing calls."},{"Start":"01:03.680 ","End":"01:06.174","Text":"Let\u0027s define Y,"},{"Start":"01:06.174 ","End":"01:13.200","Text":"that would be the monthly cost."},{"Start":"01:13.200 ","End":"01:17.290","Text":"What we want to do is we want to take the number of minutes,"},{"Start":"01:17.290 ","End":"01:24.230","Text":"his usage, and want to transform that into a monthly cost."},{"Start":"01:24.830 ","End":"01:26.970","Text":"We did number 1."},{"Start":"01:26.970 ","End":"01:28.160","Text":"Let\u0027s go to number 2,"},{"Start":"01:28.160 ","End":"01:31.090","Text":"which is to formulate the transformation rule."},{"Start":"01:31.090 ","End":"01:33.190","Text":"Now, what\u0027s the transformation rule?"},{"Start":"01:33.190 ","End":"01:37.660","Text":"Well, every month he has to pay 30 dollars and for"},{"Start":"01:37.660 ","End":"01:44.155","Text":"each outgoing call he needs to pay 10 cents, that\u0027s 0.1 dollar."},{"Start":"01:44.155 ","End":"01:46.915","Text":"If he has an outgoing call,"},{"Start":"01:46.915 ","End":"01:51.190","Text":"he needs to pay 10 cents, that\u0027s 0.1."},{"Start":"01:51.190 ","End":"01:59.270","Text":"So 0.1 times X plus a constant of 30 dollars per month."},{"Start":"01:59.870 ","End":"02:03.545","Text":"This is the transformation rule."},{"Start":"02:03.545 ","End":"02:09.785","Text":"This takes the number of calls and transforms that into a monthly cost."},{"Start":"02:09.785 ","End":"02:13.325","Text":"This would be number 1, this is number 2."},{"Start":"02:13.325 ","End":"02:15.800","Text":"Let\u0027s go to number 3,"},{"Start":"02:15.800 ","End":"02:21.050","Text":"to simplify the rule and to identify the values of a and b."},{"Start":"02:21.690 ","End":"02:28.220","Text":"3 here, we do need to simplify the rules. Well, let\u0027s see."},{"Start":"02:28.220 ","End":"02:33.695","Text":"The template for the linear transformation is aX plus b that equals to Y."},{"Start":"02:33.695 ","End":"02:36.395","Text":"We don\u0027t really need to simplify anything,"},{"Start":"02:36.395 ","End":"02:40.265","Text":"but we do need to identify that a here,"},{"Start":"02:40.265 ","End":"02:43.325","Text":"that equals to 0.1,"},{"Start":"02:43.325 ","End":"02:47.735","Text":"and b that equals to 30."},{"Start":"02:47.735 ","End":"02:51.380","Text":"Excellent. Now let\u0027s go to stage number 4,"},{"Start":"02:51.380 ","End":"02:55.610","Text":"which is to substitute in the above formulas according to the measures involve."},{"Start":"02:55.610 ","End":"02:56.930","Text":"Well, first of all,"},{"Start":"02:56.930 ","End":"03:04.750","Text":"let\u0027s take a look at the data that\u0027s given to us about the calls."},{"Start":"03:04.750 ","End":"03:11.600","Text":"We know that X bar that\u0027s the average number of minutes in a specific month,"},{"Start":"03:11.600 ","End":"03:15.080","Text":"January, that was 600. What else?"},{"Start":"03:15.080 ","End":"03:17.828","Text":"Well, the variance of X,"},{"Start":"03:17.828 ","End":"03:25.115","Text":"that equal to 2500 minutes squared. What else?"},{"Start":"03:25.115 ","End":"03:29.495","Text":"Well, the standard score for January,"},{"Start":"03:29.495 ","End":"03:31.990","Text":"that equal to 2."},{"Start":"03:31.990 ","End":"03:39.090","Text":"Now we\u0027re ready to deal with the measures for Y."},{"Start":"03:39.090 ","End":"03:41.925","Text":"Now, let\u0027s do that."},{"Start":"03:41.925 ","End":"03:46.830","Text":"Here we go. Let\u0027s start with the average."},{"Start":"03:46.830 ","End":"03:50.780","Text":"The template for the transformation of"},{"Start":"03:50.780 ","End":"03:55.145","Text":"an average is Y bar that equals to a times X bar plus b."},{"Start":"03:55.145 ","End":"03:57.985","Text":"In our case, Y bar,"},{"Start":"03:57.985 ","End":"03:59.895","Text":"well, that equals to a."},{"Start":"03:59.895 ","End":"04:02.790","Text":"A is 0.1 times X bar,"},{"Start":"04:02.790 ","End":"04:04.830","Text":"X bar is 600,"},{"Start":"04:04.830 ","End":"04:07.560","Text":"that\u0027s right here plus b."},{"Start":"04:07.560 ","End":"04:10.650","Text":"Well, what\u0027s b? B is 30."},{"Start":"04:10.650 ","End":"04:17.135","Text":"That means that we\u0027re looking at 90 and the units here are in dollars."},{"Start":"04:17.135 ","End":"04:20.635","Text":"What about the variance of Y?"},{"Start":"04:20.635 ","End":"04:24.990","Text":"Well, the template for the transformation,"},{"Start":"04:24.990 ","End":"04:31.840","Text":"the variance, that equals to a squared times the variance of X."},{"Start":"04:31.840 ","End":"04:37.800","Text":"In our case, the variance for Y that equals to a squared. What\u0027s a square?"},{"Start":"04:37.800 ","End":"04:43.685","Text":"That\u0027s 0.1 squared times the variance of X."},{"Start":"04:43.685 ","End":"04:52.025","Text":"Well, the variance of X is 2500 and that equals to 25."},{"Start":"04:52.025 ","End":"04:56.065","Text":"Now, the units here are in dollars squared."},{"Start":"04:56.065 ","End":"04:59.775","Text":"What about the standard score?"},{"Start":"04:59.775 ","End":"05:04.870","Text":"Well, the standard score of Y,"},{"Start":"05:04.870 ","End":"05:14.435","Text":"that equals to a divided by the absolute value of a times the standard score of X."},{"Start":"05:14.435 ","End":"05:16.670","Text":"Now, in our case,"},{"Start":"05:16.670 ","End":"05:20.465","Text":"the standard score of Y was a,"},{"Start":"05:20.465 ","End":"05:27.560","Text":"that\u0027s 0.1 divided by the absolute value of 0.1 times the standard score of x."},{"Start":"05:27.560 ","End":"05:29.920","Text":"Well, the standard score of x is 2."},{"Start":"05:29.920 ","End":"05:34.605","Text":"This cancels each other out and this equals to 2."},{"Start":"05:34.605 ","End":"05:37.520","Text":"There we have it. That\u0027s the standard score,"},{"Start":"05:37.520 ","End":"05:43.890","Text":"that\u0027s the variance, and that\u0027s the average of Y, the monthly cost."}],"ID":13377},{"Watched":false,"Name":"Exercise 6 - Parts a-b","Duration":"5m 54s","ChapterTopicVideoID":12903,"CourseChapterTopicPlaylistID":64523,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In this question, we\u0027re asked to prove that if all the observations in"},{"Start":"00:02.940 ","End":"00:07.215","Text":"a probability distribution have undergone a linear transformation like this,"},{"Start":"00:07.215 ","End":"00:10.860","Text":"then the average and variance of all the observations after"},{"Start":"00:10.860 ","End":"00:15.210","Text":"the transformation will be these expressions right here."},{"Start":"00:15.210 ","End":"00:18.438","Text":"Let\u0027s start with the average y-bar,"},{"Start":"00:18.438 ","End":"00:19.665","Text":"that\u0027s the average of y,"},{"Start":"00:19.665 ","End":"00:24.330","Text":"that equals to the sum where i goes from 1 to n of"},{"Start":"00:24.330 ","End":"00:29.775","Text":"y_i divided by n. Now, what\u0027s y_i?"},{"Start":"00:29.775 ","End":"00:32.535","Text":"Well, that\u0027s this linear transformation right here."},{"Start":"00:32.535 ","End":"00:35.670","Text":"Let\u0027s replace y_i with this expression."},{"Start":"00:35.670 ","End":"00:41.865","Text":"That equals to the sum where i goes from 1 to n of a times X_i plus"},{"Start":"00:41.865 ","End":"00:48.995","Text":"b divided by n. Let\u0027s take a look at the numerator and expand that."},{"Start":"00:48.995 ","End":"00:53.310","Text":"Now, that equals to a times X_1 plus"},{"Start":"00:53.310 ","End":"00:59.240","Text":"b plus a times X_2 plus b plus,"},{"Start":"00:59.240 ","End":"01:00.770","Text":"and so on and so forth,"},{"Start":"01:00.770 ","End":"01:03.770","Text":"until a times X_n plus b,"},{"Start":"01:03.770 ","End":"01:07.665","Text":"and that has to be divided by"},{"Start":"01:07.665 ","End":"01:15.230","Text":"n. Let\u0027s take a look now at this expression right here,"},{"Start":"01:15.230 ","End":"01:18.385","Text":"the numerator, and let\u0027s rearrange this."},{"Start":"01:18.385 ","End":"01:26.280","Text":"We have a times X_1 plus a times X_2 plus,"},{"Start":"01:26.280 ","End":"01:27.450","Text":"and so on and so forth,"},{"Start":"01:27.450 ","End":"01:33.689","Text":"plus a times X_n plus b plus b,"},{"Start":"01:33.689 ","End":"01:36.505","Text":"and so on and so forth plus b."},{"Start":"01:36.505 ","End":"01:41.060","Text":"Now, all that needs to be divided by n. Now,"},{"Start":"01:41.060 ","End":"01:43.760","Text":"let\u0027s take a look again at the numerator."},{"Start":"01:43.760 ","End":"01:47.630","Text":"Let\u0027s take a look at the a times X_1 plus a times X_2."},{"Start":"01:47.630 ","End":"01:50.120","Text":"We can take a out as a common factor,"},{"Start":"01:50.120 ","End":"01:52.160","Text":"so that\u0027ll be a times,"},{"Start":"01:52.160 ","End":"01:53.839","Text":"now, what are we summing?"},{"Start":"01:53.839 ","End":"02:03.685","Text":"The Xs. That\u0027ll be X_1 plus X_2 plus X_n plus,"},{"Start":"02:03.685 ","End":"02:05.840","Text":"now, how many b\u0027s do I have here?"},{"Start":"02:05.840 ","End":"02:07.805","Text":"Well, have n b\u0027s,"},{"Start":"02:07.805 ","End":"02:17.790","Text":"and that has to be divided by n. That equals to a times the sum of X_i,"},{"Start":"02:17.790 ","End":"02:22.740","Text":"where i goes from 1 to n plus n"},{"Start":"02:22.740 ","End":"02:27.900","Text":"times b divided by n. We can write this out like this."},{"Start":"02:27.900 ","End":"02:32.070","Text":"That\u0027ll be a times the sum of X_i divided by n,"},{"Start":"02:32.070 ","End":"02:38.355","Text":"where i goes from 1 to n plus n times b divided by n. Now,"},{"Start":"02:38.355 ","End":"02:41.940","Text":"that equals to a,"},{"Start":"02:41.940 ","End":"02:44.370","Text":"now, the sum of X_i divided by n, well,"},{"Start":"02:44.370 ","End":"02:48.450","Text":"that\u0027s X-bar, and n times b divided by n,"},{"Start":"02:48.450 ","End":"02:50.250","Text":"well, n\u0027s cancel each other out,"},{"Start":"02:50.250 ","End":"02:51.970","Text":"and I have b."},{"Start":"02:51.970 ","End":"02:55.625","Text":"This is exactly what I was asked to prove,"},{"Start":"02:55.625 ","End":"03:00.780","Text":"that Y-bar equals to a times X-bar plus b, and here we go."},{"Start":"03:00.780 ","End":"03:03.695","Text":"Y-bar, after all this thing,"},{"Start":"03:03.695 ","End":"03:07.480","Text":"is equal to a times X-bar plus b."},{"Start":"03:07.480 ","End":"03:10.310","Text":"Let\u0027s continue with a variance of Y."},{"Start":"03:10.310 ","End":"03:11.900","Text":"Now, the variance of Y,"},{"Start":"03:11.900 ","End":"03:21.300","Text":"that\u0027s defined as the sum of y_i minus y-bar squared divided by n. Now,"},{"Start":"03:21.300 ","End":"03:24.500","Text":"since Y is a linear transformation of X,"},{"Start":"03:24.500 ","End":"03:25.955","Text":"that\u0027s given right here,"},{"Start":"03:25.955 ","End":"03:30.020","Text":"we can replace y_i with this expression right here."},{"Start":"03:30.020 ","End":"03:37.470","Text":"That equals to the sum of a times X_i plus b,"},{"Start":"03:37.470 ","End":"03:41.540","Text":"we replaced y_i with the transformation,"},{"Start":"03:41.540 ","End":"03:44.360","Text":"minus, now, what\u0027s Y-bar?"},{"Start":"03:44.360 ","End":"03:51.170","Text":"Well, we proved that Y-bar equals to a times X-bar plus b."},{"Start":"03:51.170 ","End":"03:54.710","Text":"We have to square"},{"Start":"03:54.710 ","End":"04:01.715","Text":"the difference and divide all that by n. Now,"},{"Start":"04:01.715 ","End":"04:04.880","Text":"we can see that the b\u0027s cancel each other out."},{"Start":"04:04.880 ","End":"04:06.245","Text":"What are we left with?"},{"Start":"04:06.245 ","End":"04:13.635","Text":"We\u0027re left with the sum of a times X_i minus a times X-bar"},{"Start":"04:13.635 ","End":"04:21.455","Text":"squared divided by n. Let\u0027s just take a as a common factor out."},{"Start":"04:21.455 ","End":"04:27.785","Text":"That\u0027ll be the sum of a times X_i minus X-bar,"},{"Start":"04:27.785 ","End":"04:30.144","Text":"all this needs to be squared,"},{"Start":"04:30.144 ","End":"04:36.140","Text":"divided by n. That equals to the sum"},{"Start":"04:36.140 ","End":"04:42.775","Text":"of a squared times X_i minus X-bar squared divided by N. Now,"},{"Start":"04:42.775 ","End":"04:46.240","Text":"let\u0027s just expand this numerator right here."},{"Start":"04:46.240 ","End":"04:51.055","Text":"That\u0027ll be a squared times X_1 minus X-bar squared"},{"Start":"04:51.055 ","End":"04:56.800","Text":"plus a squared times X_2 minus X-bar squared plus,"},{"Start":"04:56.800 ","End":"04:58.460","Text":"and so on and so forth,"},{"Start":"04:58.460 ","End":"05:03.699","Text":"plus a squared times X_n minus X-bar squared."},{"Start":"05:03.699 ","End":"05:06.880","Text":"Now, all this has to be divided by"},{"Start":"05:06.880 ","End":"05:12.745","Text":"n. Let\u0027s take again a squared now out as a common factor,"},{"Start":"05:12.745 ","End":"05:15.740","Text":"that\u0027ll be a squared, and what\u0027s left?"},{"Start":"05:15.740 ","End":"05:19.400","Text":"Well, X_1 minus X-bar plus X_2 minus X-bar,"},{"Start":"05:19.400 ","End":"05:26.050","Text":"that equals to the sum of X_i minus X-bar squared,"},{"Start":"05:26.050 ","End":"05:30.195","Text":"where i goes from 1 to n divided by n. Now,"},{"Start":"05:30.195 ","End":"05:34.460","Text":"that equals to a squared times,"},{"Start":"05:34.460 ","End":"05:38.210","Text":"the sum of X_i minus X-bar squared divided by n,"},{"Start":"05:38.210 ","End":"05:42.420","Text":"well, that\u0027s the variance of X."},{"Start":"05:42.420 ","End":"05:45.260","Text":"This is exactly what we were asked to prove."},{"Start":"05:45.260 ","End":"05:51.140","Text":"That the variance of Y is equal to a squared times the variance of X,"},{"Start":"05:51.140 ","End":"05:54.810","Text":"where Y is a linear transformation of X."}],"ID":13378}],"Thumbnail":null,"ID":64523},{"Name":"The Probability Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"4m 36s","ChapterTopicVideoID":12904,"CourseChapterTopicPlaylistID":64524,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:03.240","Text":"In this chapter, we\u0027ll be talking about"},{"Start":"00:03.240 ","End":"00:07.125","Text":"discrete random variables and the probability function."},{"Start":"00:07.125 ","End":"00:10.275","Text":"Now, what\u0027s a random variable?"},{"Start":"00:10.275 ","End":"00:12.495","Text":"A discrete random variable?"},{"Start":"00:12.495 ","End":"00:18.465","Text":"Well, it\u0027s a variable that can receive individual values with various probabilities."},{"Start":"00:18.465 ","End":"00:23.370","Text":"For example, let\u0027s define X as"},{"Start":"00:23.370 ","End":"00:29.830","Text":"a random variable having values when you throw a die."},{"Start":"00:29.830 ","End":"00:32.090","Text":"What are the values when you throw a die?"},{"Start":"00:32.090 ","End":"00:39.165","Text":"It can be either 1 or 2 or 3 or 4 or 5 or 6,"},{"Start":"00:39.165 ","End":"00:44.195","Text":"and what are the probabilities for each one of the values?"},{"Start":"00:44.195 ","End":"00:46.115","Text":"Well, if it\u0027s a fair die,"},{"Start":"00:46.115 ","End":"00:53.730","Text":"then the probability of X for each x will equal to 1/6."},{"Start":"00:53.730 ","End":"00:58.525","Text":"Another example. Let\u0027s say Y is when you toss a coin."},{"Start":"00:58.525 ","End":"01:02.105","Text":"Now, what is the values of tossing a coin?"},{"Start":"01:02.105 ","End":"01:05.165","Text":"Well, it\u0027s either heads or tails."},{"Start":"01:05.165 ","End":"01:12.315","Text":"What\u0027s the probability of each one of the values when you toss the coin?"},{"Start":"01:12.315 ","End":"01:14.685","Text":"Well, it\u0027s 1/2."},{"Start":"01:14.685 ","End":"01:18.740","Text":"You can have 1/2 for heads and 1/2 for tails if it\u0027s a fair coin."},{"Start":"01:18.740 ","End":"01:24.755","Text":"Now, let\u0027s take a look at an example for a roulette wheel."},{"Start":"01:24.755 ","End":"01:29.110","Text":"Now, let\u0027s assume that this is our roulette wheel,"},{"Start":"01:29.110 ","End":"01:33.765","Text":"and let\u0027s assume or let\u0027s define X as"},{"Start":"01:33.765 ","End":"01:40.350","Text":"the profit that 1 has when 1 turns a roulette wheel."},{"Start":"01:40.350 ","End":"01:47.765","Text":"Excellent. What are the values that X can have when he tosses a roulette wheel?"},{"Start":"01:47.765 ","End":"01:50.975","Text":"Well, he can win 10,"},{"Start":"01:50.975 ","End":"01:55.705","Text":"or he can win 20, or he can win 30."},{"Start":"01:55.705 ","End":"02:00.875","Text":"What are the probabilities for each one of the values?"},{"Start":"02:00.875 ","End":"02:04.105","Text":"Well, if X equals 10,"},{"Start":"02:04.105 ","End":"02:07.140","Text":"the probability is 1/4,"},{"Start":"02:07.140 ","End":"02:09.770","Text":"0.25. Why is that?"},{"Start":"02:09.770 ","End":"02:15.805","Text":"Because that\u0027s the area of the circle that\u0027s associated with the number 10."},{"Start":"02:15.805 ","End":"02:19.880","Text":"What\u0027s the probability of x equaling 20?"},{"Start":"02:19.880 ","End":"02:22.820","Text":"Well, again, that\u0027s 1/4."},{"Start":"02:22.820 ","End":"02:26.000","Text":"That\u0027s just like 10."},{"Start":"02:26.000 ","End":"02:30.095","Text":"This is the area associated with the number 20."},{"Start":"02:30.095 ","End":"02:34.125","Text":"What\u0027s the probability of x equaling 30?"},{"Start":"02:34.125 ","End":"02:37.660","Text":"Here it\u0027s 1/2. Why is that?"},{"Start":"02:37.660 ","End":"02:41.120","Text":"Because this area right here,"},{"Start":"02:41.120 ","End":"02:45.920","Text":"half of the circle, half is a roulette wheel is associated with the profits of 30."},{"Start":"02:45.920 ","End":"02:51.319","Text":"So when we want to write the probability function,"},{"Start":"02:51.319 ","End":"02:53.615","Text":"we do it in a table form,"},{"Start":"02:53.615 ","End":"02:56.730","Text":"and this is how we do it."},{"Start":"02:58.250 ","End":"03:03.715","Text":"Here, we write down the values of X."},{"Start":"03:03.715 ","End":"03:09.080","Text":"Now X can have a value of 10, 20, or 30."},{"Start":"03:09.080 ","End":"03:12.005","Text":"Now, what\u0027s the probability of X?"},{"Start":"03:12.005 ","End":"03:14.420","Text":"Well, we\u0027ve calculated that right here."},{"Start":"03:14.420 ","End":"03:18.810","Text":"That\u0027s 25 percent if X equals 10,"},{"Start":"03:18.810 ","End":"03:22.560","Text":"25 percent if X equals 20,"},{"Start":"03:22.560 ","End":"03:26.065","Text":"and 50 percent if X equals 30."},{"Start":"03:26.065 ","End":"03:28.940","Text":"Now the last thing that you have to do is make"},{"Start":"03:28.940 ","End":"03:35.375","Text":"sure that the sum of all the probabilities have to add up to 1."},{"Start":"03:35.375 ","End":"03:38.435","Text":"If they don\u0027t, you\u0027ve made a mistake."},{"Start":"03:38.435 ","End":"03:40.550","Text":"Let\u0027s take a look. There\u0027s 50,"},{"Start":"03:40.550 ","End":"03:44.330","Text":"75, 1, that\u0027s excellent."},{"Start":"03:44.330 ","End":"03:47.045","Text":"Let\u0027s just recap."},{"Start":"03:47.045 ","End":"03:48.935","Text":"First of all,"},{"Start":"03:48.935 ","End":"03:53.310","Text":"you have to identify the variable."},{"Start":"03:54.800 ","End":"04:01.530","Text":"Secondly, you have to see what its values are."},{"Start":"04:03.250 ","End":"04:14.849","Text":"The third step is to calculate the probabilities for each one of the values."},{"Start":"04:14.849 ","End":"04:18.355","Text":"4, let\u0027s not forget our check."},{"Start":"04:18.355 ","End":"04:24.270","Text":"You have to check that the sum equals 1."},{"Start":"04:24.680 ","End":"04:27.385","Text":"Now, in this chapter,"},{"Start":"04:27.385 ","End":"04:31.175","Text":"you\u0027ll be given a whole bunch of problems,"},{"Start":"04:31.175 ","End":"04:37.140","Text":"please solve them by yourselves before you see the solution videos."}],"ID":13379},{"Watched":false,"Name":"Exercise 1","Duration":"4m 9s","ChapterTopicVideoID":12905,"CourseChapterTopicPlaylistID":64524,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.600","Text":"In this question, we\u0027re given that the distribution of cards in"},{"Start":"00:03.600 ","End":"00:08.385","Text":"a given community is as follows: 50 families have no cars,"},{"Start":"00:08.385 ","End":"00:10.335","Text":"70 families have 1 car,"},{"Start":"00:10.335 ","End":"00:12.255","Text":"60 families have 2 cars,"},{"Start":"00:12.255 ","End":"00:14.730","Text":"and 20 families have 3 cars."},{"Start":"00:14.730 ","End":"00:18.285","Text":"A family is randomly selected from the community."},{"Start":"00:18.285 ","End":"00:23.190","Text":"We define X as the number of cars owned by the selected family."},{"Start":"00:23.190 ","End":"00:27.210","Text":"It will ask to construct the probability function of X."},{"Start":"00:27.210 ","End":"00:31.209","Text":"Well, the first thing that we need to know is"},{"Start":"00:31.209 ","End":"00:36.690","Text":"what\u0027s the total number of families in the community?"},{"Start":"00:36.690 ","End":"00:40.305","Text":"Well, let\u0027s just sum up these guys right here,"},{"Start":"00:40.305 ","End":"00:45.270","Text":"50 plus 70 plus 60 plus 20."},{"Start":"00:45.270 ","End":"00:50.390","Text":"Now n is the total number of families in the community."},{"Start":"00:50.390 ","End":"00:57.440","Text":"Again, that\u0027s 50 plus 70 plus 60 plus 20,"},{"Start":"00:57.440 ","End":"01:00.149","Text":"and that equals to 200."},{"Start":"01:00.190 ","End":"01:03.410","Text":"Now the next thing that we need to figure out,"},{"Start":"01:03.410 ","End":"01:05.660","Text":"are what are the values of X?"},{"Start":"01:05.660 ","End":"01:10.190","Text":"Now, remember we defined X as the number of cars owned by the selected family."},{"Start":"01:10.190 ","End":"01:12.455","Text":"Well, we\u0027re given that here."},{"Start":"01:12.455 ","End":"01:14.985","Text":"No car, 1 car,"},{"Start":"01:14.985 ","End":"01:16.810","Text":"2 cars, and 3 cars."},{"Start":"01:16.810 ","End":"01:20.885","Text":"That means that the values that X can have is 0,"},{"Start":"01:20.885 ","End":"01:25.695","Text":"1, 2, and 3."},{"Start":"01:25.695 ","End":"01:34.015","Text":"Now, we\u0027re asked to construct the probability function of X."},{"Start":"01:34.015 ","End":"01:36.590","Text":"That means that we want to know what\u0027s"},{"Start":"01:36.590 ","End":"01:41.015","Text":"the probability of X when it equals 0, for example."},{"Start":"01:41.015 ","End":"01:44.060","Text":"Now, in order to do that,"},{"Start":"01:44.060 ","End":"01:50.075","Text":"we have to take the proportion of families that have no car. That\u0027s right here."},{"Start":"01:50.075 ","End":"01:53.420","Text":"The proportion of families that have"},{"Start":"01:53.420 ","End":"01:56.975","Text":"no cars from the total number of families that are under community."},{"Start":"01:56.975 ","End":"01:59.540","Text":"In this case, where X equals 0,"},{"Start":"01:59.540 ","End":"02:06.670","Text":"that\u0027s 50/200 and that equals to 0.25."},{"Start":"02:07.600 ","End":"02:11.150","Text":"The probability of X equals 1."},{"Start":"02:11.150 ","End":"02:14.735","Text":"That\u0027s the probability of a family having 1 car."},{"Start":"02:14.735 ","End":"02:17.030","Text":"Well, that\u0027s the proportion of families that have"},{"Start":"02:17.030 ","End":"02:19.370","Text":"1 car from the total number of families."},{"Start":"02:19.370 ","End":"02:25.300","Text":"That\u0027s 70/200, and that\u0027s 0.35."},{"Start":"02:26.290 ","End":"02:32.625","Text":"Now, let\u0027s do that for the probability of X equals 2."},{"Start":"02:32.625 ","End":"02:38.715","Text":"That\u0027s 60/200, and that\u0027s 0.3."},{"Start":"02:38.715 ","End":"02:46.235","Text":"The probability of X equals 3, well that\u0027s 20/200."},{"Start":"02:46.235 ","End":"02:49.445","Text":"That equals to 0.1."},{"Start":"02:49.445 ","End":"02:54.750","Text":"Now, let\u0027s construct our probability function."},{"Start":"02:54.750 ","End":"02:58.305","Text":"Let\u0027s draw this in a table form."},{"Start":"02:58.305 ","End":"03:01.425","Text":"Let\u0027s do that here."},{"Start":"03:01.425 ","End":"03:05.635","Text":"X and P(X)."},{"Start":"03:05.635 ","End":"03:09.290","Text":"Now we said that the values of X are 0,"},{"Start":"03:09.290 ","End":"03:10.385","Text":"1, 2, and 3."},{"Start":"03:10.385 ","End":"03:15.210","Text":"0."},{"Start":"03:15.210 ","End":"03:19.610","Text":"We\u0027ve already figured out the probabilities of each one of the values."},{"Start":"03:19.610 ","End":"03:23.485","Text":"That\u0027s right here, that\u0027s 0.25,"},{"Start":"03:23.485 ","End":"03:28.710","Text":"0.35 for X equals 1,"},{"Start":"03:28.710 ","End":"03:32.235","Text":"for x equals 2, that\u0027s 0.3,"},{"Start":"03:32.235 ","End":"03:36.065","Text":"and for X equals 3, that\u0027s 0.1."},{"Start":"03:36.065 ","End":"03:41.075","Text":"Now, this is a probability function right here."},{"Start":"03:41.075 ","End":"03:48.425","Text":"The only thing that we have to check is that the sum of the probabilities equals 1."},{"Start":"03:48.425 ","End":"03:51.800","Text":"Well, that\u0027s 0.1 plus 0.3."},{"Start":"03:51.800 ","End":"03:56.880","Text":"That\u0027s 0.4, 0.4 plus 0.35."},{"Start":"03:56.880 ","End":"04:00.011","Text":"That\u0027s 0.75, plus 0.25."},{"Start":"04:00.011 ","End":"04:03.120","Text":"That equals 1, so we\u0027re okay."},{"Start":"04:03.120 ","End":"04:08.970","Text":"Here is our probability function for this question."}],"ID":13380},{"Watched":false,"Name":"Exercise 2","Duration":"5m ","ChapterTopicVideoID":12906,"CourseChapterTopicPlaylistID":64524,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.865","Text":"This question will be dealing with codes."},{"Start":"00:02.865 ","End":"00:07.170","Text":"Now, a 2 letter code is composed from the letters A,"},{"Start":"00:07.170 ","End":"00:11.385","Text":"B, and C. We\u0027re asked how many codes can be composed?"},{"Start":"00:11.385 ","End":"00:15.285","Text":"We are also asked to list all the possible codes."},{"Start":"00:15.285 ","End":"00:19.170","Text":"Let\u0027s get at it. Section a,"},{"Start":"00:19.170 ","End":"00:21.780","Text":"how many codes can be composed?"},{"Start":"00:21.780 ","End":"00:25.470","Text":"Well, let\u0027s define n_1 as"},{"Start":"00:25.470 ","End":"00:32.130","Text":"the possible number of letters that we can put in the first place. That\u0027s 3."},{"Start":"00:32.130 ","End":"00:33.950","Text":"We can put it in the first place,"},{"Start":"00:33.950 ","End":"00:36.020","Text":"either A or B or C,"},{"Start":"00:36.020 ","End":"00:39.270","Text":"and the same goes for the second place."},{"Start":"00:39.270 ","End":"00:43.907","Text":"Let\u0027s define n_2 as a possible number of letters that we can put in the second place."},{"Start":"00:43.907 ","End":"00:46.220","Text":"That\u0027s also 3."},{"Start":"00:46.220 ","End":"00:51.380","Text":"The possible number of letters that the code can"},{"Start":"00:51.380 ","End":"00:57.665","Text":"have is n_1 times n_2 equals 3 times 3,"},{"Start":"00:57.665 ","End":"00:59.800","Text":"that equals to 9."},{"Start":"00:59.800 ","End":"01:02.005","Text":"In section b,"},{"Start":"01:02.005 ","End":"01:04.760","Text":"we\u0027re asked to list all the possible codes."},{"Start":"01:04.760 ","End":"01:10.510","Text":"Well, in essence, what we\u0027re trying to do is to map out the sample space."},{"Start":"01:10.510 ","End":"01:14.610","Text":"Let\u0027s list out all the possible codes that we have."},{"Start":"01:14.610 ","End":"01:20.075","Text":"AA, AB, and AC."},{"Start":"01:20.075 ","End":"01:24.750","Text":"We also have BA, BB,"},{"Start":"01:24.750 ","End":"01:30.165","Text":"and BC, and we also have CA,"},{"Start":"01:30.165 ","End":"01:33.745","Text":"CB, and CC."},{"Start":"01:33.745 ","End":"01:38.495","Text":"Now, all of this is the sample space."},{"Start":"01:38.495 ","End":"01:41.630","Text":"In section c,"},{"Start":"01:41.630 ","End":"01:46.375","Text":"we define X as the number of times that B appears in the code,"},{"Start":"01:46.375 ","End":"01:50.780","Text":"and we\u0027re asked to construct the probability function of X."},{"Start":"01:50.780 ","End":"01:53.230","Text":"Before we can do that,"},{"Start":"01:53.230 ","End":"01:56.620","Text":"let\u0027s first go over the definition of X."},{"Start":"01:56.620 ","End":"02:01.243","Text":"X is the number of times that B appears in the code."},{"Start":"02:01.243 ","End":"02:02.510","Text":"Now, this is very important."},{"Start":"02:02.510 ","End":"02:06.170","Text":"We have to understand what our variable"},{"Start":"02:06.170 ","End":"02:10.340","Text":"is before we can even construct the probability function."},{"Start":"02:10.340 ","End":"02:12.575","Text":"Well, let\u0027s see,"},{"Start":"02:12.575 ","End":"02:14.660","Text":"from Question b,"},{"Start":"02:14.660 ","End":"02:16.430","Text":"of section B above,"},{"Start":"02:16.430 ","End":"02:21.150","Text":"this is our sample space and what we\u0027re asked to do."},{"Start":"02:21.150 ","End":"02:23.030","Text":"We\u0027re asked to find out, first of all,"},{"Start":"02:23.030 ","End":"02:27.110","Text":"what\u0027s the number of times that B appears in the code?"},{"Start":"02:27.110 ","End":"02:31.242","Text":"Let\u0027s see, here 0."},{"Start":"02:31.242 ","End":"02:34.620","Text":"This is the number of times that B appears in this code."},{"Start":"02:34.620 ","End":"02:36.845","Text":"Here, B appears once."},{"Start":"02:36.845 ","End":"02:38.930","Text":"Here it appears 0."},{"Start":"02:38.930 ","End":"02:41.074","Text":"Here it appears once."},{"Start":"02:41.074 ","End":"02:43.535","Text":"Here, it appears twice,"},{"Start":"02:43.535 ","End":"02:48.245","Text":"once, 0, once, and 0."},{"Start":"02:48.245 ","End":"02:55.515","Text":"This is the number of times that B appears in the code."},{"Start":"02:55.515 ","End":"03:01.380","Text":"Now we could construct the probability function of X."},{"Start":"03:01.380 ","End":"03:05.670","Text":"Now, how many values can X have?"},{"Start":"03:05.670 ","End":"03:11.790","Text":"We can see from here. It can have either 0 or 1 or 2."},{"Start":"03:12.440 ","End":"03:17.314","Text":"Let\u0027s construct our probability function."},{"Start":"03:17.314 ","End":"03:22.595","Text":"We have here the values of X, that\u0027s 0,1,"},{"Start":"03:22.595 ","End":"03:29.125","Text":"2, and probabilities of X."},{"Start":"03:29.125 ","End":"03:36.510","Text":"Now, what\u0027s the probability of X being equal to 0?"},{"Start":"03:36.510 ","End":"03:42.965","Text":"Well, all we have to do is find the relative frequency of X when it equals to 0."},{"Start":"03:42.965 ","End":"03:45.800","Text":"How many times does 0 appear here?"},{"Start":"03:45.800 ","End":"03:48.520","Text":"Well, that\u0027s once, twice,"},{"Start":"03:48.520 ","End":"03:50.775","Text":"3, and 4 times."},{"Start":"03:50.775 ","End":"03:53.130","Text":"That\u0027s 4 out of what?"},{"Start":"03:53.130 ","End":"03:58.358","Text":"We have 9 possible values for the code,"},{"Start":"03:58.358 ","End":"04:02.835","Text":"the probability of X equal 1."},{"Start":"04:02.835 ","End":"04:05.970","Text":"How many times does X equal 1?"},{"Start":"04:05.970 ","End":"04:09.390","Text":"1, 2, 3, and 4."},{"Start":"04:09.390 ","End":"04:12.195","Text":"Again, that\u0027s 4/9."},{"Start":"04:12.195 ","End":"04:16.730","Text":"What\u0027s the probability of X equaling 2?"},{"Start":"04:16.730 ","End":"04:20.185","Text":"That\u0027s only once. That\u0027s 1/9."},{"Start":"04:20.185 ","End":"04:24.965","Text":"Let\u0027s put in these probabilities back into the table."},{"Start":"04:24.965 ","End":"04:27.230","Text":"The probability of X equals 0."},{"Start":"04:27.230 ","End":"04:29.135","Text":"That\u0027s 4/9."},{"Start":"04:29.135 ","End":"04:32.220","Text":"The probability of X equals 1,"},{"Start":"04:32.220 ","End":"04:39.215","Text":"that\u0027s 4/9, and the probability of X equaling 2, that\u0027s 1/9."},{"Start":"04:39.215 ","End":"04:46.510","Text":"All we have to do now is to make sure that the sum of the probabilities equals 1."},{"Start":"04:46.510 ","End":"04:49.700","Text":"That\u0027s 4 plus 4 plus 1/9."},{"Start":"04:49.700 ","End":"04:54.320","Text":"That means it\u0027s 9/9 equals 1, so we\u0027re okay."},{"Start":"04:54.320 ","End":"05:00.510","Text":"This is our probability function."}],"ID":13381},{"Watched":false,"Name":"Exercise 3","Duration":"6m 1s","ChapterTopicVideoID":12907,"CourseChapterTopicPlaylistID":64524,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.060","Text":"In this question, we\u0027ll be talking about the chances of a student"},{"Start":"00:03.060 ","End":"00:06.465","Text":"passing in economics and statistics exam."},{"Start":"00:06.465 ","End":"00:11.580","Text":"Now, a student takes semester exams in economics and statistics."},{"Start":"00:11.580 ","End":"00:14.310","Text":"The chances of passing the economics exam are"},{"Start":"00:14.310 ","End":"00:19.230","Text":"80 percent and the chances of passing the statistics exam are 90 percent."},{"Start":"00:19.230 ","End":"00:23.595","Text":"The chances of passing both exams are 75 percent."},{"Start":"00:23.595 ","End":"00:27.765","Text":"Now let X be the number of exams that a student passes,"},{"Start":"00:27.765 ","End":"00:31.110","Text":"and we have to construct the probability function of X."},{"Start":"00:31.110 ","End":"00:33.330","Text":"In order to do that,"},{"Start":"00:33.330 ","End":"00:36.075","Text":"let\u0027s define our variables first."},{"Start":"00:36.075 ","End":"00:42.600","Text":"Let A be the event that the student passes the economics exam"},{"Start":"00:42.600 ","End":"00:53.045","Text":"and B is the event that a student passes the statistics exam."},{"Start":"00:53.045 ","End":"00:56.210","Text":"Now the probability of A is given to us,"},{"Start":"00:56.210 ","End":"01:02.090","Text":"that\u0027s 80 percent and the probability of B is 90 percent,"},{"Start":"01:02.090 ","End":"01:12.080","Text":"and the probability of passing both the economics and the statistics exam is 75 percent."},{"Start":"01:12.080 ","End":"01:13.625","Text":"Now, we see that here,"},{"Start":"01:13.625 ","End":"01:15.080","Text":"that\u0027s given to us."},{"Start":"01:15.080 ","End":"01:17.495","Text":"That\u0027s 80 percent for economics,"},{"Start":"01:17.495 ","End":"01:19.715","Text":"90 percent for statistics,"},{"Start":"01:19.715 ","End":"01:23.320","Text":"and 75 percent for both."},{"Start":"01:23.320 ","End":"01:29.405","Text":"Now, there are several ways that we can present the data."},{"Start":"01:29.405 ","End":"01:36.635","Text":"Well, we can do it through a probability matrix or we can do this through Venn diagrams."},{"Start":"01:36.635 ","End":"01:41.704","Text":"Now, I want to show you how we do this through a Venn diagram."},{"Start":"01:41.704 ","End":"01:45.470","Text":"Let\u0027s draw this up, and here it is."},{"Start":"01:45.470 ","End":"01:49.640","Text":"We can see the probabilities right on the Venn diagram."},{"Start":"01:49.640 ","End":"01:54.005","Text":"Let\u0027s start with a probability of A intersect B."},{"Start":"01:54.005 ","End":"01:55.790","Text":"We, that\u0027s 75 percent."},{"Start":"01:55.790 ","End":"02:02.595","Text":"We can see that here in the intersect between A and B."},{"Start":"02:02.595 ","End":"02:07.240","Text":"Now let\u0027s look at the probability of A, that\u0027s 80 percent."},{"Start":"02:07.240 ","End":"02:12.160","Text":"Now, 75 percent is in the intersection."},{"Start":"02:12.160 ","End":"02:14.425","Text":"In order to make it up to 80,"},{"Start":"02:14.425 ","End":"02:17.590","Text":"that means that only 5 percent is outside of"},{"Start":"02:17.590 ","End":"02:21.040","Text":"the intersect in order for the probability of all of A to be"},{"Start":"02:21.040 ","End":"02:25.160","Text":"80 percent and this 5 percent just represents the people who will"},{"Start":"02:25.160 ","End":"02:30.235","Text":"pass the economics exam but did not pass the statistics exam."},{"Start":"02:30.235 ","End":"02:34.870","Text":"Likewise, the probability of B is 90 percent."},{"Start":"02:34.870 ","End":"02:39.370","Text":"Now we know that 75 percent of the 90 percent is in the intersect."},{"Start":"02:39.370 ","End":"02:46.215","Text":"That means that 15 percent has to be outside of the intersect and all that"},{"Start":"02:46.215 ","End":"02:49.790","Text":"means is that 15 percent of the students have passed"},{"Start":"02:49.790 ","End":"02:54.485","Text":"the statistics exam but did not pass the economics exam."},{"Start":"02:54.485 ","End":"03:02.870","Text":"Now, if we look at all the probabilities within the events A,"},{"Start":"03:02.870 ","End":"03:04.384","Text":"B, and the intersect,"},{"Start":"03:04.384 ","End":"03:08.210","Text":"we see that we\u0027re missing about 5 percent, that\u0027s here."},{"Start":"03:08.210 ","End":"03:13.400","Text":"What does that mean? That means that this is a probability of"},{"Start":"03:13.400 ","End":"03:20.345","Text":"the students that did not pass either the economics or the statistics exam."},{"Start":"03:20.345 ","End":"03:22.955","Text":"Now, having said that,"},{"Start":"03:22.955 ","End":"03:26.600","Text":"let\u0027s get back to our question. What do we want to do?"},{"Start":"03:26.600 ","End":"03:31.085","Text":"We want to construct a probability function of X,"},{"Start":"03:31.085 ","End":"03:33.005","Text":"where X is what?"},{"Start":"03:33.005 ","End":"03:36.695","Text":"The number of exams that the student passes."},{"Start":"03:36.695 ","End":"03:42.140","Text":"How many exams can the student pass?"},{"Start":"03:42.140 ","End":"03:45.454","Text":"Well, he can pass either 0 exams,"},{"Start":"03:45.454 ","End":"03:49.385","Text":"or he doesn\u0027t pass either the statistics or the economics exam,"},{"Start":"03:49.385 ","End":"03:51.440","Text":"he can pass 1 exam,"},{"Start":"03:51.440 ","End":"03:53.090","Text":"and it doesn\u0027t matter which right now,"},{"Start":"03:53.090 ","End":"03:55.535","Text":"it could be the economics of the statistics exam,"},{"Start":"03:55.535 ","End":"03:58.095","Text":"or he can pass both exams."},{"Start":"03:58.095 ","End":"04:06.930","Text":"What\u0027s the probability of"},{"Start":"04:06.930 ","End":"04:10.330","Text":"X being equal to 0?"},{"Start":"04:10.330 ","End":"04:16.670","Text":"Well, we have that right here. That\u0027s 0.05."},{"Start":"04:16.670 ","End":"04:21.230","Text":"What\u0027s the probability of X being equal to 1?"},{"Start":"04:21.230 ","End":"04:24.695","Text":"That means that we\u0027re looking at the probability of"},{"Start":"04:24.695 ","End":"04:29.510","Text":"a student passing the economics and not passing the statistics,"},{"Start":"04:29.510 ","End":"04:36.235","Text":"that\u0027s 0.05 plus or"},{"Start":"04:36.235 ","End":"04:39.230","Text":"the probability that a student"},{"Start":"04:39.230 ","End":"04:46.125","Text":"passes the statistics but doesn\u0027t pass the economics, that\u0027s 0.15."},{"Start":"04:46.125 ","End":"04:49.315","Text":"That comes out to 0.2."},{"Start":"04:49.315 ","End":"04:55.810","Text":"Then what\u0027s left is the probability of X passing both exams."},{"Start":"04:55.810 ","End":"04:59.875","Text":"Well, that\u0027s right here and intersect, that\u0027s 0.75."},{"Start":"04:59.875 ","End":"05:04.855","Text":"Now we have all the information that we need to construct a probability function."},{"Start":"05:04.855 ","End":"05:11.875","Text":"Let\u0027s do that. We have X and we have the probability of X."},{"Start":"05:11.875 ","End":"05:16.410","Text":"Now, X can be either 0 or 1 or 2, we said that."},{"Start":"05:16.410 ","End":"05:23.320","Text":"We\u0027ve already calculated the probability of X where X equals 0, that\u0027s 0.05."},{"Start":"05:23.860 ","End":"05:29.530","Text":"The probability of X being equal to 1, well,"},{"Start":"05:29.530 ","End":"05:38.700","Text":"that\u0027s 0.2, and the probability of X being equal to 2, that\u0027s 75 percent."},{"Start":"05:38.700 ","End":"05:42.770","Text":"Excellent. All we have to do right now is"},{"Start":"05:42.770 ","End":"05:47.180","Text":"to make sure that the sum of our probabilities add up to 1."},{"Start":"05:47.180 ","End":"05:51.080","Text":"That\u0027s 75 percent plus 20 percent,"},{"Start":"05:51.080 ","End":"05:54.560","Text":"that\u0027s 95 percent plus 5 percent, that\u0027s 100 percent."},{"Start":"05:54.560 ","End":"06:01.650","Text":"We\u0027re good. This then is the probability function of X."}],"ID":13382},{"Watched":false,"Name":"Exercise 4","Duration":"5m 3s","ChapterTopicVideoID":12908,"CourseChapterTopicPlaylistID":64524,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.104","Text":"In this question we\u0027ll be talking about chances of a person winning a series of games."},{"Start":"00:06.104 ","End":"00:11.130","Text":"Now, the chances of winning 1 specific game, a 30 percent."},{"Start":"00:11.130 ","End":"00:14.415","Text":"A person plays the game until he wins,"},{"Start":"00:14.415 ","End":"00:15.945","Text":"when he wins he quits."},{"Start":"00:15.945 ","End":"00:19.935","Text":"But in any case, he doesn\u0027t play the game more than 4 times."},{"Start":"00:19.935 ","End":"00:23.835","Text":"We define X as the number of times he plays the game,"},{"Start":"00:23.835 ","End":"00:27.795","Text":"and we\u0027re asked to construct the probability function for X."},{"Start":"00:27.795 ","End":"00:31.035","Text":"Now, before we can do that,"},{"Start":"00:31.035 ","End":"00:34.075","Text":"we want to describe the data."},{"Start":"00:34.075 ","End":"00:38.060","Text":"Now, the best way to describe the data is by probability tree."},{"Start":"00:38.060 ","End":"00:39.710","Text":"Why is that?"},{"Start":"00:39.710 ","End":"00:45.275","Text":"Well, that\u0027s because the data or the story is presented to us in stages."},{"Start":"00:45.275 ","End":"00:47.375","Text":"He plays the first game."},{"Start":"00:47.375 ","End":"00:49.250","Text":"If he wins, he quits."},{"Start":"00:49.250 ","End":"00:51.290","Text":"If he doesn\u0027t, he goes on to the next stage,"},{"Start":"00:51.290 ","End":"00:52.640","Text":"which is the next game,"},{"Start":"00:52.640 ","End":"00:57.065","Text":"and again, he plays that if he wins he quits and so on and so forth."},{"Start":"00:57.065 ","End":"00:59.300","Text":"Now, if that\u0027s the case,"},{"Start":"00:59.300 ","End":"01:02.125","Text":"let\u0027s construct our probability tree."},{"Start":"01:02.125 ","End":"01:04.505","Text":"In here we see the first stage."},{"Start":"01:04.505 ","End":"01:08.390","Text":"If he wins, he quit and if he doesn\u0027t,"},{"Start":"01:08.390 ","End":"01:11.130","Text":"then he plays another game."},{"Start":"01:11.720 ","End":"01:14.430","Text":"Here\u0027s the other game the second game."},{"Start":"01:14.430 ","End":"01:15.570","Text":"If he wins again,"},{"Start":"01:15.570 ","End":"01:18.500","Text":"he quits and if he doesn\u0027t win,"},{"Start":"01:18.500 ","End":"01:21.310","Text":"then he goes onto play the third game."},{"Start":"01:21.310 ","End":"01:23.155","Text":"If he wins he quits,"},{"Start":"01:23.155 ","End":"01:26.525","Text":"if he loses, he goes on to play the fourth game,"},{"Start":"01:26.525 ","End":"01:28.415","Text":"and at this stage,"},{"Start":"01:28.415 ","End":"01:31.760","Text":"he either wins or loses, but that\u0027s it."},{"Start":"01:31.760 ","End":"01:36.515","Text":"He doesn\u0027t play anymore. If that\u0027s the case,"},{"Start":"01:36.515 ","End":"01:46.265","Text":"let\u0027s see what\u0027s the probability of X being 1 or equaling 1?"},{"Start":"01:46.265 ","End":"01:52.535","Text":"What\u0027s that? That\u0027s basically the probability of a person winning at the first game."},{"Start":"01:52.535 ","End":"01:56.135","Text":"Well, that\u0027s 0.3, that\u0027s right here."},{"Start":"01:56.135 ","End":"01:59.360","Text":"He wins. Now what\u0027s"},{"Start":"01:59.360 ","End":"02:04.550","Text":"the probability of a person winning on the second try or the second game?"},{"Start":"02:04.550 ","End":"02:06.770","Text":"Well, that\u0027s the probability of the person"},{"Start":"02:06.770 ","End":"02:10.940","Text":"losing in the first game and winning in the second game,"},{"Start":"02:10.940 ","End":"02:18.665","Text":"that\u0027s 0.7 losing in the first game times winning on the second, that\u0027s 0.3."},{"Start":"02:18.665 ","End":"02:22.430","Text":"The probability of X equals 3."},{"Start":"02:22.430 ","End":"02:24.740","Text":"That means he loses in the first game,"},{"Start":"02:24.740 ","End":"02:26.690","Text":"loses in the second game,"},{"Start":"02:26.690 ","End":"02:28.820","Text":"and in the third game he wins."},{"Start":"02:28.820 ","End":"02:36.484","Text":"That equals to 0.7 times 0.7 times 0.3."},{"Start":"02:36.484 ","End":"02:41.045","Text":"Loses, loses, and then he wins."},{"Start":"02:41.045 ","End":"02:45.980","Text":"Now, what\u0027s the probability of X equals 4,"},{"Start":"02:45.980 ","End":"02:48.020","Text":"that the person plays 4 games?"},{"Start":"02:48.020 ","End":"02:50.655","Text":"Well, he loses in the first,"},{"Start":"02:50.655 ","End":"02:52.305","Text":"loses on the second,"},{"Start":"02:52.305 ","End":"02:53.990","Text":"loses on the third."},{"Start":"02:53.990 ","End":"03:02.180","Text":"We have 0.7 times 0.7 times 0.7,"},{"Start":"03:02.180 ","End":"03:04.520","Text":"and then he either wins or loses."},{"Start":"03:04.520 ","End":"03:08.270","Text":"That means that you either multiply it by the probability of"},{"Start":"03:08.270 ","End":"03:14.990","Text":"winning or you multiply by the probability of him losing,"},{"Start":"03:14.990 ","End":"03:21.875","Text":"that\u0027s 0.7 times 0.7 times 0.7 times 0.7."},{"Start":"03:21.875 ","End":"03:24.470","Text":"This is the part where he wins,"},{"Start":"03:24.470 ","End":"03:26.270","Text":"the probability of him winning on"},{"Start":"03:26.270 ","End":"03:30.830","Text":"the fourth game and this is the probability of him losing in the fourth game."},{"Start":"03:30.830 ","End":"03:35.650","Text":"Now, let\u0027s calculate the probabilities."},{"Start":"03:35.650 ","End":"03:42.160","Text":"For X equals 1 obviously, that equals 0.3."},{"Start":"03:43.400 ","End":"03:47.430","Text":"For X equals 2,"},{"Start":"03:47.430 ","End":"03:51.700","Text":"well, that equals to 0.21."},{"Start":"03:52.700 ","End":"03:59.265","Text":"For X equals 3, well,"},{"Start":"03:59.265 ","End":"04:07.475","Text":"that equals to 0.147 and for X equals 4,"},{"Start":"04:07.475 ","End":"04:11.495","Text":"well, that equals to 0.343."},{"Start":"04:11.495 ","End":"04:17.440","Text":"Now, let\u0027s represent that better in a table."},{"Start":"04:17.440 ","End":"04:21.755","Text":"Again, we have X and we have the probability of x."},{"Start":"04:21.755 ","End":"04:23.840","Text":"Now, X can either be 1,"},{"Start":"04:23.840 ","End":"04:27.215","Text":"or 2, or 3, or 4."},{"Start":"04:27.215 ","End":"04:32.185","Text":"Now the probability of X being 1,"},{"Start":"04:32.185 ","End":"04:35.160","Text":"0.3, when X equals 2,"},{"Start":"04:35.160 ","End":"04:38.400","Text":"that\u0027s 0.21, when X equals 3,"},{"Start":"04:38.400 ","End":"04:42.720","Text":"that\u0027s 0.147, and when X equals 4,"},{"Start":"04:42.720 ","End":"04:47.255","Text":"that equals to 0.343."},{"Start":"04:47.255 ","End":"04:55.055","Text":"Now, all we need to do right now is to check that the probabilities equal 1,"},{"Start":"04:55.055 ","End":"04:57.125","Text":"and they do, trust me."},{"Start":"04:57.125 ","End":"05:03.480","Text":"This is our probability function for X."}],"ID":13383},{"Watched":false,"Name":"Exercise 5","Duration":"9m 31s","ChapterTopicVideoID":12909,"CourseChapterTopicPlaylistID":64524,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"This project deals with the success of projects."},{"Start":"00:03.510 ","End":"00:08.475","Text":"Now, a project management company manages 3 projects simultaneously."},{"Start":"00:08.475 ","End":"00:11.670","Text":"The chances of Project A succeeding are 0.7,"},{"Start":"00:11.670 ","End":"00:14.610","Text":"the chances of Project B succeeding are 0.8,"},{"Start":"00:14.610 ","End":"00:17.910","Text":"and the chances of Projects C succeeding are 0.9."},{"Start":"00:17.910 ","End":"00:21.870","Text":"The success of each project is independent of other projects."},{"Start":"00:21.870 ","End":"00:24.180","Text":"We define X as the number of"},{"Start":"00:24.180 ","End":"00:29.700","Text":"successful projects and we\u0027re asked to construct the probability function of X."},{"Start":"00:29.700 ","End":"00:34.815","Text":"Now, the first thing that we need to do is to describe the data."},{"Start":"00:34.815 ","End":"00:38.950","Text":"The best way that we can do this is through a probability tree."},{"Start":"00:38.950 ","End":"00:42.200","Text":"Now, we can describe the data in"},{"Start":"00:42.200 ","End":"00:46.505","Text":"many other techniques such as the probability matrix and so on and so forth."},{"Start":"00:46.505 ","End":"00:48.380","Text":"But as we said,"},{"Start":"00:48.380 ","End":"00:51.530","Text":"the best way we can do this is through a probability tree,"},{"Start":"00:51.530 ","End":"00:53.730","Text":"and let\u0027s draw this out right now,"},{"Start":"00:53.730 ","End":"00:55.520","Text":"and here it is."},{"Start":"00:55.520 ","End":"01:01.130","Text":"Now, the first thing we need to know is that since Project A, B,"},{"Start":"01:01.130 ","End":"01:03.980","Text":"and C are independent of each other,"},{"Start":"01:03.980 ","End":"01:11.735","Text":"then we can write out the various levels of the probability tree in various ways."},{"Start":"01:11.735 ","End":"01:13.180","Text":"For example, we can put here,"},{"Start":"01:13.180 ","End":"01:17.145","Text":"in the first level Project C and here Project A."},{"Start":"01:17.145 ","End":"01:18.665","Text":"It really doesn\u0027t matter,"},{"Start":"01:18.665 ","End":"01:21.215","Text":"but at least we need to be consistent."},{"Start":"01:21.215 ","End":"01:26.150","Text":"So let\u0027s make the first level of the tree Project A,"},{"Start":"01:26.150 ","End":"01:28.280","Text":"the second level Project B,"},{"Start":"01:28.280 ","End":"01:32.960","Text":"and the third level Project C. Now,"},{"Start":"01:32.960 ","End":"01:36.650","Text":"the first thing that we need to do is write the probabilities."},{"Start":"01:36.650 ","End":"01:38.545","Text":"Well, in Project A,"},{"Start":"01:38.545 ","End":"01:44.165","Text":"we see that the probability of success is 0.7."},{"Start":"01:44.165 ","End":"01:47.375","Text":"So if that\u0027s the case, let\u0027s write here."},{"Start":"01:47.375 ","End":"01:53.494","Text":"Let\u0027s write down 0.7 every time we see a plus that means that the project is successful."},{"Start":"01:53.494 ","End":"01:57.080","Text":"So this is the branch where"},{"Start":"01:57.080 ","End":"02:01.924","Text":"Project A is successful and this is the branch where Project A is not successful."},{"Start":"02:01.924 ","End":"02:03.740","Text":"That\u0027s because there\u0027s the minus here."},{"Start":"02:03.740 ","End":"02:09.300","Text":"So the probability of A being not successful is 0.3."},{"Start":"02:09.300 ","End":"02:14.830","Text":"Now, the probability of B succeeding is 0.8,"},{"Start":"02:14.830 ","End":"02:18.680","Text":"and let\u0027s write that down here."},{"Start":"02:18.680 ","End":"02:23.015","Text":"That\u0027s the branch of B being successful. That\u0027s 0.8."},{"Start":"02:23.015 ","End":"02:25.850","Text":"So every time we see a plus at this level,"},{"Start":"02:25.850 ","End":"02:27.484","Text":"we\u0027ll write down 0.8,"},{"Start":"02:27.484 ","End":"02:29.420","Text":"and here we have 0.8."},{"Start":"02:29.420 ","End":"02:32.360","Text":"That means that this is 0.2, that\u0027s the minus,"},{"Start":"02:32.360 ","End":"02:35.580","Text":"and here is again that\u0027s a minus, so that\u0027s 0.2,"},{"Start":"02:35.580 ","End":"02:42.440","Text":"and now let\u0027s go down to level C. The success of Project C,"},{"Start":"02:42.440 ","End":"02:47.030","Text":"the probability of Project C succeeding is 0.9."},{"Start":"02:47.030 ","End":"02:53.325","Text":"So every time we see a plus at this level we\u0027ll write 0.9."},{"Start":"02:53.325 ","End":"02:55.800","Text":"So that\u0027s here, that\u0027s 0.9,"},{"Start":"02:55.800 ","End":"03:01.555","Text":"that\u0027s 0.9, 0.9, and 0.9."},{"Start":"03:01.555 ","End":"03:05.855","Text":"That means that the probability of not succeeding is 0.1,"},{"Start":"03:05.855 ","End":"03:08.840","Text":"and we\u0027ll write that every time we see a minus."},{"Start":"03:08.840 ","End":"03:10.265","Text":"So here\u0027s the minus,"},{"Start":"03:10.265 ","End":"03:16.490","Text":"that\u0027s 0.1, 0.1, 0.1, and 0.1."},{"Start":"03:16.490 ","End":"03:23.715","Text":"Great. The next step is to define X."},{"Start":"03:23.715 ","End":"03:27.005","Text":"Now we defined X as what?"},{"Start":"03:27.005 ","End":"03:30.449","Text":"As the number of successful projects."},{"Start":"03:30.449 ","End":"03:32.135","Text":"What does that mean?"},{"Start":"03:32.135 ","End":"03:37.100","Text":"That means that every time we see a plus at each level,"},{"Start":"03:37.100 ","End":"03:38.675","Text":"then we have to count it."},{"Start":"03:38.675 ","End":"03:43.295","Text":"For example, if we go to this branch right here,"},{"Start":"03:43.295 ","End":"03:46.695","Text":"we see that Project A is successful,"},{"Start":"03:46.695 ","End":"03:48.705","Text":"Project B is successful,"},{"Start":"03:48.705 ","End":"03:51.239","Text":"and Project C is successful."},{"Start":"03:51.239 ","End":"03:54.510","Text":"That means that here, X equals 3."},{"Start":"03:54.510 ","End":"03:57.210","Text":"Now, if we go here,"},{"Start":"03:57.210 ","End":"03:59.085","Text":"Project A is successful,"},{"Start":"03:59.085 ","End":"04:00.720","Text":"Project B is successful,"},{"Start":"04:00.720 ","End":"04:02.760","Text":"and Project C not successful."},{"Start":"04:02.760 ","End":"04:05.310","Text":"That means that X equals 2."},{"Start":"04:05.310 ","End":"04:07.935","Text":"Here we have plus, minus,"},{"Start":"04:07.935 ","End":"04:10.540","Text":"plus, that\u0027s X equals 2."},{"Start":"04:10.540 ","End":"04:12.233","Text":"We\u0027re counting pluses,"},{"Start":"04:12.233 ","End":"04:16.130","Text":"and here we have plus, minus, minus."},{"Start":"04:16.130 ","End":"04:19.110","Text":"That means here X equals 1."},{"Start":"04:19.110 ","End":"04:22.795","Text":"Now, let\u0027s go to the other branch here."},{"Start":"04:22.795 ","End":"04:25.100","Text":"That\u0027s minus, plus, plus,"},{"Start":"04:25.100 ","End":"04:28.455","Text":"that\u0027s X equals 2, 2 pluses."},{"Start":"04:28.455 ","End":"04:30.120","Text":"Minus, plus, minus,"},{"Start":"04:30.120 ","End":"04:32.460","Text":"that\u0027s X equals 1."},{"Start":"04:32.460 ","End":"04:34.770","Text":"That\u0027s minus, minus,"},{"Start":"04:34.770 ","End":"04:36.980","Text":"plus that\u0027s X equals 1,"},{"Start":"04:36.980 ","End":"04:38.850","Text":"and that\u0027s minus, minus,"},{"Start":"04:38.850 ","End":"04:42.105","Text":"minus, that means that X equals 0."},{"Start":"04:42.105 ","End":"04:49.870","Text":"So now we have basically all the values that X can have."},{"Start":"04:49.870 ","End":"04:51.850","Text":"So let\u0027s just write it out here."},{"Start":"04:51.850 ","End":"04:55.870","Text":"X can have the values of 0,"},{"Start":"04:55.870 ","End":"05:00.295","Text":"1, 2, and 3."},{"Start":"05:00.295 ","End":"05:04.850","Text":"Excellent. So let\u0027s now try"},{"Start":"05:04.850 ","End":"05:09.005","Text":"to figure out what the probabilities are for each one of these values."},{"Start":"05:09.005 ","End":"05:15.860","Text":"So the probability of X being equal to 0, what does that equal?"},{"Start":"05:15.860 ","End":"05:19.400","Text":"That equals to this branch right here."},{"Start":"05:19.400 ","End":"05:28.820","Text":"That\u0027s 0.3, projecting not being successful times 0.2."},{"Start":"05:28.820 ","End":"05:35.570","Text":"That\u0027s the probability of B not being successful times 0.1,"},{"Start":"05:35.570 ","End":"05:43.895","Text":"that\u0027s the probability of C not being successful, and that\u0027s 0.006."},{"Start":"05:43.895 ","End":"05:49.805","Text":"What\u0027s the probability of X equaling 1?"},{"Start":"05:49.805 ","End":"05:55.880","Text":"That means, what\u0027s the probability of at least 1 project being successful?"},{"Start":"05:55.880 ","End":"05:58.100","Text":"Well, let\u0027s see what we have here."},{"Start":"05:58.100 ","End":"06:01.190","Text":"Well, we have X equals 1 here,"},{"Start":"06:01.190 ","End":"06:04.070","Text":"here, here, and here."},{"Start":"06:04.070 ","End":"06:09.515","Text":"So let\u0027s just write out the probabilities for each one of these branches."},{"Start":"06:09.515 ","End":"06:16.543","Text":"This branch right here it\u0027s 0.3 times 0.2 times 0.9."},{"Start":"06:16.543 ","End":"06:26.385","Text":"0.3 times 0.2 times 0.9 plus where else do we have X equals 1? Right here."},{"Start":"06:26.385 ","End":"06:34.775","Text":"0.3 times 0.8 times 0.1."},{"Start":"06:34.775 ","End":"06:35.900","Text":"That\u0027s this guy right here,"},{"Start":"06:35.900 ","End":"06:38.210","Text":"that\u0027s this branch right here."},{"Start":"06:38.210 ","End":"06:42.945","Text":"Where else do we have X equals 1 plus right here,"},{"Start":"06:42.945 ","End":"06:50.920","Text":"0.7 times 0.2 times 0.1 times 0.1,"},{"Start":"06:50.920 ","End":"06:58.090","Text":"and that equals to 0.092."},{"Start":"07:00.410 ","End":"07:04.685","Text":"So we have this for X equals 0."},{"Start":"07:04.685 ","End":"07:08.825","Text":"Here we have the probability for X equals 1."},{"Start":"07:08.825 ","End":"07:14.465","Text":"What\u0027s the probability of X equaling 2?"},{"Start":"07:14.465 ","End":"07:18.540","Text":"Well, again, that\u0027s where X equals 2."},{"Start":"07:18.540 ","End":"07:22.730","Text":"That\u0027s here, here, and here."},{"Start":"07:22.730 ","End":"07:25.985","Text":"So let\u0027s calculate that."},{"Start":"07:25.985 ","End":"07:29.870","Text":"That\u0027s 0.3 times 0.8 times"},{"Start":"07:29.870 ","End":"07:31.830","Text":"0.9"},{"Start":"07:36.080 ","End":"07:41.790","Text":"plus this guy right here,"},{"Start":"07:42.250 ","End":"07:52.190","Text":"0.7 times 0.2 times 0.9 plus this branch right here."},{"Start":"07:52.190 ","End":"08:00.260","Text":"That\u0027s plus 0.7 times 0.8 times 0.1,"},{"Start":"08:00.260 ","End":"08:07.783","Text":"and all these equals 0.498,"},{"Start":"08:07.783 ","End":"08:15.455","Text":"and what\u0027s the probability"},{"Start":"08:15.455 ","End":"08:18.470","Text":"of X being equal to 3?"},{"Start":"08:18.470 ","End":"08:22.955","Text":"Well, that\u0027s this guy right here."},{"Start":"08:22.955 ","End":"08:28.160","Text":"So it\u0027s 0.7 times"},{"Start":"08:28.160 ","End":"08:35.280","Text":"0.8 times 0.9 and that equals to 0.504."},{"Start":"08:35.770 ","End":"08:39.140","Text":"Now, if that\u0027s the case,"},{"Start":"08:39.140 ","End":"08:44.120","Text":"let\u0027s write out the probability function. That\u0027s X."},{"Start":"08:44.120 ","End":"08:50.195","Text":"That\u0027s the probability of X. X can be either 0,"},{"Start":"08:50.195 ","End":"08:53.340","Text":"1, 2, or 3."},{"Start":"08:53.340 ","End":"08:58.930","Text":"So if X equals 0, we have 0.006."},{"Start":"08:58.930 ","End":"09:04.060","Text":"If X equals 1, we have 0.092."},{"Start":"09:04.060 ","End":"09:05.890","Text":"If X equals 2,"},{"Start":"09:05.890 ","End":"09:14.895","Text":"we have 0.498 and if X equals 3, we have 0.504."},{"Start":"09:14.895 ","End":"09:19.610","Text":"Now, the only thing that needs to be done is to make"},{"Start":"09:19.610 ","End":"09:24.950","Text":"sure that the probabilities add up to 1, and they do."},{"Start":"09:24.950 ","End":"09:31.920","Text":"So, this is the probability function of X."}],"ID":13384},{"Watched":false,"Name":"Exercise 6","Duration":"2m 58s","ChapterTopicVideoID":12910,"CourseChapterTopicPlaylistID":64524,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.930","Text":"The following is a probability function of a given random variable."},{"Start":"00:03.930 ","End":"00:10.755","Text":"We have the probability of X equaling k equals k over A, where what?"},{"Start":"00:10.755 ","End":"00:12.750","Text":"Where k equals 1,"},{"Start":"00:12.750 ","End":"00:14.385","Text":"2, 3, or 4."},{"Start":"00:14.385 ","End":"00:17.205","Text":"We\u0027re asked to find the value of A."},{"Start":"00:17.205 ","End":"00:20.865","Text":"Now, in order to do that, first,"},{"Start":"00:20.865 ","End":"00:25.365","Text":"let\u0027s write down the probability function with the variable."},{"Start":"00:25.365 ","End":"00:28.330","Text":"We have X,"},{"Start":"00:28.720 ","End":"00:31.865","Text":"and we have the probability of x."},{"Start":"00:31.865 ","End":"00:34.040","Text":"Now, X can take on the values 1,"},{"Start":"00:34.040 ","End":"00:35.360","Text":"2, 3, and 4,"},{"Start":"00:35.360 ","End":"00:37.550","Text":"write here, 1,"},{"Start":"00:37.550 ","End":"00:39.890","Text":"2, 3, and 4."},{"Start":"00:39.890 ","End":"00:45.500","Text":"Now what\u0027s the probability of X given that X equals 1?"},{"Start":"00:45.500 ","End":"00:48.420","Text":"Well, that\u0027s 1 over A,"},{"Start":"00:48.650 ","End":"00:52.200","Text":"and where X equals 2,"},{"Start":"00:52.330 ","End":"00:55.655","Text":"that\u0027s 2 over A,"},{"Start":"00:55.655 ","End":"00:58.305","Text":"and where X equals 3,"},{"Start":"00:58.305 ","End":"01:00.300","Text":"that\u0027s 3 over A,"},{"Start":"01:00.300 ","End":"01:03.255","Text":"and 4, that will be 4 over A."},{"Start":"01:03.255 ","End":"01:10.650","Text":"Now, we know that the sum of all the probabilities have to add up to 1."},{"Start":"01:11.120 ","End":"01:16.710","Text":"Let\u0027s write this down,1 over A plus 2 over A plus"},{"Start":"01:16.710 ","End":"01:23.095","Text":"3 over A plus 4 over A, must equal to 1."},{"Start":"01:23.095 ","End":"01:26.735","Text":"Now let\u0027s multiply both sides by A."},{"Start":"01:26.735 ","End":"01:32.930","Text":"We have 1 plus 2 plus 3 plus 4 equals A."},{"Start":"01:32.930 ","End":"01:37.290","Text":"That means that A equals 10."},{"Start":"01:37.690 ","End":"01:45.110","Text":"Now, let\u0027s write the probability function of x,"},{"Start":"01:45.110 ","End":"01:49.675","Text":"now inserting A as 10."},{"Start":"01:49.675 ","End":"01:52.410","Text":"That\u0027ll be X,"},{"Start":"01:52.410 ","End":"01:56.020","Text":"that\u0027ll be the probability of x,"},{"Start":"01:56.390 ","End":"02:00.120","Text":"as 1,2,3, and 4."},{"Start":"02:00.120 ","End":"02:03.360","Text":"That\u0027ll be 1 over 10,"},{"Start":"02:03.360 ","End":"02:06.945","Text":"2 over 10, 3 over 10,"},{"Start":"02:06.945 ","End":"02:09.070","Text":"and 4 over 10."},{"Start":"02:09.070 ","End":"02:11.150","Text":"Now here we have this,"},{"Start":"02:11.150 ","End":"02:15.570","Text":"we have the probability function as a table."},{"Start":"02:15.570 ","End":"02:20.809","Text":"We can also write the probability function as an equation."},{"Start":"02:20.809 ","End":"02:25.415","Text":"Here we go, the probability of x being equal to k,"},{"Start":"02:25.415 ","End":"02:28.220","Text":"where k equals 1,"},{"Start":"02:28.220 ","End":"02:29.280","Text":"2, and 3,"},{"Start":"02:29.280 ","End":"02:34.090","Text":"and 4, that equals to k over 10."},{"Start":"02:34.090 ","End":"02:37.460","Text":"Now, as we said,"},{"Start":"02:37.460 ","End":"02:44.045","Text":"we can write the probability function as an equation or as a table."},{"Start":"02:44.045 ","End":"02:47.210","Text":"They\u0027re equivalent, they\u0027re the same."},{"Start":"02:47.210 ","End":"02:51.275","Text":"But getting back to our question,"},{"Start":"02:51.275 ","End":"02:53.630","Text":"we\u0027re asked to find the value of A,"},{"Start":"02:53.630 ","End":"02:55.160","Text":"and this is what we did."},{"Start":"02:55.160 ","End":"02:58.260","Text":"Here we go, A equals 10."}],"ID":13385},{"Watched":false,"Name":"Exercise 7","Duration":"3m 57s","ChapterTopicVideoID":12911,"CourseChapterTopicPlaylistID":64524,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"This problem deals with kindergarten and children and plays."},{"Start":"00:04.770 ","End":"00:07.845","Text":"Now, a kindergarten has 8 children;"},{"Start":"00:07.845 ","End":"00:10.035","Text":"5 boys and 3 girls."},{"Start":"00:10.035 ","End":"00:13.845","Text":"3 children are randomly selected to take part in a play."},{"Start":"00:13.845 ","End":"00:18.855","Text":"We define X as the number of boys selected to take part in the play."},{"Start":"00:18.855 ","End":"00:23.220","Text":"We\u0027re asked to construct the probability function of X."},{"Start":"00:23.220 ","End":"00:27.390","Text":"Now, there are many ways to solve this problem,"},{"Start":"00:27.390 ","End":"00:32.380","Text":"but let\u0027s use what we know from combinatorics."},{"Start":"00:33.530 ","End":"00:39.870","Text":"We know that we have X, and what\u0027s that?"},{"Start":"00:39.870 ","End":"00:42.970","Text":"That\u0027s the number of boys selected,"},{"Start":"00:42.970 ","End":"00:46.310","Text":"and we need the probability of X."},{"Start":"00:46.310 ","End":"00:48.350","Text":"Now how many boys can be selected?"},{"Start":"00:48.350 ","End":"00:53.435","Text":"Well, only 3 people or 3 children are selected for the play."},{"Start":"00:53.435 ","End":"00:57.470","Text":"The number of boys that are selected could be either 0,"},{"Start":"00:57.470 ","End":"00:59.450","Text":"excuse me, or 1,"},{"Start":"00:59.450 ","End":"01:02.555","Text":"or 2, or 3."},{"Start":"01:02.555 ","End":"01:05.225","Text":"Now, as we said,"},{"Start":"01:05.225 ","End":"01:08.190","Text":"what\u0027s our sample space?"},{"Start":"01:08.690 ","End":"01:19.900","Text":"Well, a sample space is basically the fact that we need to take 8 on 3."},{"Start":"01:19.900 ","End":"01:25.215","Text":"We have 8 children and we need to take 3 of them."},{"Start":"01:25.215 ","End":"01:29.100","Text":"That comes out to 56."},{"Start":"01:29.100 ","End":"01:36.675","Text":"Now, what\u0027s the probability of X being 0?"},{"Start":"01:36.675 ","End":"01:39.240","Text":"Well, as we said,"},{"Start":"01:39.240 ","End":"01:47.880","Text":"we have 5 boys and 0 are chosen times 3 girls,"},{"Start":"01:47.880 ","End":"01:53.015","Text":"and 3 of them are chosen divided by 56."},{"Start":"01:53.015 ","End":"01:59.580","Text":"That equals to 1/56."},{"Start":"01:59.840 ","End":"02:02.930","Text":"Now, let\u0027s do this."},{"Start":"02:02.930 ","End":"02:06.780","Text":"Probability of X equals 1,"},{"Start":"02:06.780 ","End":"02:08.040","Text":"1 boy is chosen."},{"Start":"02:08.040 ","End":"02:11.160","Text":"That means out of the 5 boys, 1 is chosen."},{"Start":"02:11.160 ","End":"02:13.305","Text":"That means that out of the 3 girls,"},{"Start":"02:13.305 ","End":"02:14.640","Text":"2 are chosen,"},{"Start":"02:14.640 ","End":"02:18.785","Text":"and we have to divide that by 56."},{"Start":"02:18.785 ","End":"02:23.975","Text":"That turns out to be 15/56."},{"Start":"02:23.975 ","End":"02:28.380","Text":"What\u0027s the probability of 2 boys chosen?"},{"Start":"02:28.380 ","End":"02:31.215","Text":"Well, that\u0027s 5 on 2."},{"Start":"02:31.215 ","End":"02:33.950","Text":"Now, how many girls are chosen then?"},{"Start":"02:33.950 ","End":"02:35.710","Text":"3 on 1."},{"Start":"02:35.710 ","End":"02:39.620","Text":"1 girl is chosen divided by 56,"},{"Start":"02:39.620 ","End":"02:45.110","Text":"and that equals to 30/56."},{"Start":"02:45.110 ","End":"02:49.675","Text":"The probability of 3 boys being chosen."},{"Start":"02:49.675 ","End":"02:58.065","Text":"Again that\u0027s 5 on 3 now times 3 on 0 divided by 56,"},{"Start":"02:58.065 ","End":"03:04.105","Text":"and that equals to 10/56."},{"Start":"03:04.105 ","End":"03:10.760","Text":"Now, we have all the probabilities that we need for each one of the values of X."},{"Start":"03:10.760 ","End":"03:13.310","Text":"So let\u0027s just put it back into the table."},{"Start":"03:13.310 ","End":"03:17.715","Text":"The probability of X where X equals 0,"},{"Start":"03:17.715 ","End":"03:21.900","Text":"that\u0027s 1/56, where X equals 1,"},{"Start":"03:21.900 ","End":"03:29.385","Text":"that\u0027s 15/56, X equals 2 is 30/56,"},{"Start":"03:29.385 ","End":"03:30.830","Text":"and when X equals 3,"},{"Start":"03:30.830 ","End":"03:33.830","Text":"the probability is 10/56."},{"Start":"03:33.830 ","End":"03:36.890","Text":"Now the only thing that we need to do is to make sure"},{"Start":"03:36.890 ","End":"03:40.865","Text":"that all the probabilities add up to 1,"},{"Start":"03:40.865 ","End":"03:43.730","Text":"so that\u0027s 10 plus 30, that\u0027s 40,"},{"Start":"03:43.730 ","End":"03:46.340","Text":"that\u0027s 55 plus 1,"},{"Start":"03:46.340 ","End":"03:49.835","Text":"that\u0027s 56 over 56, so we\u0027re good."},{"Start":"03:49.835 ","End":"03:56.850","Text":"So the probability function of X is this guy right here."}],"ID":13386},{"Watched":false,"Name":"Exercise 8","Duration":"8m 10s","ChapterTopicVideoID":12912,"CourseChapterTopicPlaylistID":64524,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.804","Text":"This question deals with TV ratings."},{"Start":"00:02.804 ","End":"00:05.520","Text":"Now, a survey examined whether people watched"},{"Start":"00:05.520 ","End":"00:08.860","Text":"the news broadcasts of channels 1, 2, and 3."},{"Start":"00:08.860 ","End":"00:11.624","Text":"The following information was discovered."},{"Start":"00:11.624 ","End":"00:15.180","Text":"20 percent of the viewers watch Channel 2,"},{"Start":"00:15.180 ","End":"00:18.340","Text":"8 percent of the viewers watched Channel 1,"},{"Start":"00:18.340 ","End":"00:21.885","Text":"and 10 percent of the viewers watch Channel 3."},{"Start":"00:21.885 ","End":"00:28.265","Text":"Now, in addition, 1 percent of the viewers watched all 3 channels together."},{"Start":"00:28.265 ","End":"00:32.690","Text":"10 percent of the viewers watched 2 of the 3 channels."},{"Start":"00:32.690 ","End":"00:40.175","Text":"Now, we define X as the number of news broadcasts watched by randomly selected viewer."},{"Start":"00:40.175 ","End":"00:45.230","Text":"We\u0027re asked to construct a probability function of X."},{"Start":"00:45.230 ","End":"00:49.760","Text":"Now if X is defined as the number of news broadcasts that are watched,"},{"Start":"00:49.760 ","End":"00:52.595","Text":"well, what are the values that X can have?"},{"Start":"00:52.595 ","End":"00:55.138","Text":"Well, X can be 0,"},{"Start":"00:55.138 ","End":"01:00.590","Text":"that means that no one watches any of the channels."},{"Start":"01:00.590 ","End":"01:02.150","Text":"X can be 1,"},{"Start":"01:02.150 ","End":"01:07.130","Text":"a person watches either Channel 1 or Channel 2 or Channel 3."},{"Start":"01:07.130 ","End":"01:09.155","Text":"X can be 2,"},{"Start":"01:09.155 ","End":"01:11.720","Text":"a person can watch 2 of the 3 channels,"},{"Start":"01:11.720 ","End":"01:13.055","Text":"Channel 1 and 2,"},{"Start":"01:13.055 ","End":"01:16.970","Text":"Channels 1 and 3 or Channel 2 and 3."},{"Start":"01:16.970 ","End":"01:18.920","Text":"X can be 3,"},{"Start":"01:18.920 ","End":"01:22.450","Text":"where a person watches all 3 of the channels."},{"Start":"01:22.450 ","End":"01:24.605","Text":"If that\u0027s the case,"},{"Start":"01:24.605 ","End":"01:30.755","Text":"let\u0027s now try to calculate or construct the probability function."},{"Start":"01:30.755 ","End":"01:32.630","Text":"Well, we have X right here,"},{"Start":"01:32.630 ","End":"01:35.930","Text":"then what are the probability of X?"},{"Start":"01:35.930 ","End":"01:39.125","Text":"Well, as we said X can take on the values of 0,"},{"Start":"01:39.125 ","End":"01:42.305","Text":"1, 2, and 3."},{"Start":"01:42.305 ","End":"01:49.315","Text":"Now, we already have some of the information that\u0027s given to us right here."},{"Start":"01:49.315 ","End":"01:51.450","Text":"Let\u0027s look at this thing."},{"Start":"01:51.450 ","End":"01:55.225","Text":"1 percent of the viewers watch all 3 channels together,"},{"Start":"01:55.225 ","End":"02:00.890","Text":"that means that the probability of X equals to 3. What\u0027s X?"},{"Start":"02:00.890 ","End":"02:02.945","Text":"That\u0027s the number of news broadcasts."},{"Start":"02:02.945 ","End":"02:05.660","Text":"All 3 of them are being watched and"},{"Start":"02:05.660 ","End":"02:08.474","Text":"the probability of all 3 of them being watched is 1 percent,"},{"Start":"02:08.474 ","End":"02:11.960","Text":"that means that that equals 0.01."},{"Start":"02:11.960 ","End":"02:16.785","Text":"Likewise, 10 percent of the viewers watch 2 of the 3 channels."},{"Start":"02:16.785 ","End":"02:23.660","Text":"That means that the probability of X being equal to 2 equals 10 percent."},{"Start":"02:23.660 ","End":"02:27.675","Text":"Now, let\u0027s plug in these guys right back in here."},{"Start":"02:27.675 ","End":"02:32.340","Text":"Where X equals 3 the probability is 0.01,"},{"Start":"02:32.340 ","End":"02:34.490","Text":"where X equals 2,"},{"Start":"02:34.490 ","End":"02:38.760","Text":"the probability is 0.1, that\u0027s 10 percent."},{"Start":"02:38.760 ","End":"02:42.260","Text":"Now, all we have to do then is to calculate"},{"Start":"02:42.260 ","End":"02:46.525","Text":"the probability of X where X equals 0 and X equals 1."},{"Start":"02:46.525 ","End":"02:52.745","Text":"Now, let\u0027s try to calculate the probability of X equaling 1."},{"Start":"02:52.745 ","End":"02:54.395","Text":"Before we can do that,"},{"Start":"02:54.395 ","End":"02:57.110","Text":"let\u0027s just define some more variables."},{"Start":"02:57.110 ","End":"03:07.620","Text":"Well, let\u0027s define the A as the event that you viewed Channel 1,"},{"Start":"03:08.720 ","End":"03:14.190","Text":"and B the event of viewing Channel 2,"},{"Start":"03:14.190 ","End":"03:22.380","Text":"and C the event of viewing Channel 3."},{"Start":"03:22.380 ","End":"03:27.150","Text":"The probability of B,"},{"Start":"03:27.150 ","End":"03:28.200","Text":"well that\u0027s given to us,"},{"Start":"03:28.200 ","End":"03:32.425","Text":"that\u0027s right here, that\u0027s 20 percent."},{"Start":"03:32.425 ","End":"03:36.420","Text":"The probability of A,"},{"Start":"03:37.180 ","End":"03:41.130","Text":"that\u0027s 8 percent, that\u0027s right here."},{"Start":"03:41.890 ","End":"03:52.620","Text":"The probability of C is 10 percent, right here."},{"Start":"03:53.180 ","End":"04:00.965","Text":"I think we\u0027re now already to start calculating the probability that X equals 1."},{"Start":"04:00.965 ","End":"04:06.060","Text":"Let\u0027s use a Venn diagram to help us out with our probabilities."},{"Start":"04:06.610 ","End":"04:10.865","Text":"Now, when X equals 3, what does that mean?"},{"Start":"04:10.865 ","End":"04:17.715","Text":"That means that people are watching all 3 broadcasts at the same time."},{"Start":"04:17.715 ","End":"04:19.695","Text":"On a Venn diagram,"},{"Start":"04:19.695 ","End":"04:25.420","Text":"that\u0027s the intersection of all 3 events, that\u0027s right here."},{"Start":"04:25.880 ","End":"04:28.580","Text":"We know what the probability of that."},{"Start":"04:28.580 ","End":"04:32.405","Text":"That\u0027s 1 percent, that\u0027s 0.01."},{"Start":"04:32.405 ","End":"04:36.650","Text":"Let\u0027s take a look at X equals 2."},{"Start":"04:36.650 ","End":"04:43.650","Text":"What does that mean? That means that people are watching 2 out of the 3 broadcasts."},{"Start":"04:43.650 ","End":"04:45.604","Text":"On the Venn diagram,"},{"Start":"04:45.604 ","End":"04:51.395","Text":"we see that here as the intersection of 2 of the 3 events."},{"Start":"04:51.395 ","End":"04:53.720","Text":"That\u0027s this intersection right here,"},{"Start":"04:53.720 ","End":"04:59.790","Text":"and this intersection right here,"},{"Start":"04:59.790 ","End":"05:05.840","Text":"and this intersection right here."},{"Start":"05:05.840 ","End":"05:08.915","Text":"What about X equals 1?"},{"Start":"05:08.915 ","End":"05:11.990","Text":"Well, what\u0027s the probability of X equal 1?"},{"Start":"05:11.990 ","End":"05:13.340","Text":"How can we see that?"},{"Start":"05:13.340 ","End":"05:16.100","Text":"How can we represent that on the Venn diagram?"},{"Start":"05:16.100 ","End":"05:19.505","Text":"Well, we can see that right here."},{"Start":"05:19.505 ","End":"05:22.610","Text":"This is this section right here."},{"Start":"05:22.610 ","End":"05:27.110","Text":"It\u0027s this section right here that\u0027s 1,"},{"Start":"05:27.110 ","End":"05:30.406","Text":"then it\u0027s this 1 right here,"},{"Start":"05:30.406 ","End":"05:34.110","Text":"and also this 1 right here."},{"Start":"05:41.360 ","End":"05:48.495","Text":"When we want to calculate the probability of X being equal to 1,"},{"Start":"05:48.495 ","End":"05:51.395","Text":"well, that\u0027s basically saying,"},{"Start":"05:51.395 ","End":"05:54.965","Text":"I want to add up all the probabilities here."},{"Start":"05:54.965 ","End":"05:57.005","Text":"That\u0027s probability of A,"},{"Start":"05:57.005 ","End":"05:59.780","Text":"plus the probability of B,"},{"Start":"05:59.780 ","End":"06:03.335","Text":"plus the probability of C. But hold on,"},{"Start":"06:03.335 ","End":"06:09.505","Text":"we\u0027ve counted some of the sections we\u0027ve counted them twice or even thrice."},{"Start":"06:09.505 ","End":"06:13.366","Text":"The blue sections we\u0027ve counted twice,"},{"Start":"06:13.366 ","End":"06:14.929","Text":"now what\u0027s the blue section?"},{"Start":"06:14.929 ","End":"06:17.030","Text":"That\u0027s X equals 2."},{"Start":"06:17.030 ","End":"06:24.680","Text":"That means that we have to take away 2 times the probability of X equaling 2."},{"Start":"06:24.680 ","End":"06:27.080","Text":"What about the red section?"},{"Start":"06:27.080 ","End":"06:30.185","Text":"Well, we\u0027ve counted that 3 times."},{"Start":"06:30.185 ","End":"06:37.120","Text":"We have to take away 3 times the probability of X equals 3."},{"Start":"06:37.130 ","End":"06:44.000","Text":"This expression equals the probability of X being equal to 1."},{"Start":"06:44.000 ","End":"06:47.519","Text":"Now, let\u0027s just plug in the numbers."},{"Start":"06:47.590 ","End":"06:53.300","Text":"We have probability of A is 0.08,"},{"Start":"06:53.300 ","End":"06:57.830","Text":"plus the probability of B that\u0027s 0.2,"},{"Start":"06:57.830 ","End":"07:01.340","Text":"plus the probability of C that\u0027s 0.1,"},{"Start":"07:01.340 ","End":"07:03.440","Text":"that\u0027s these guys right here,"},{"Start":"07:03.440 ","End":"07:07.415","Text":"minus 2 times the probability of X equals 2."},{"Start":"07:07.415 ","End":"07:16.700","Text":"That\u0027s here, 2 times 0.1 minus 3 times the probability of X equals 3."},{"Start":"07:16.700 ","End":"07:21.060","Text":"That\u0027s this guy right here, 0.01."},{"Start":"07:23.900 ","End":"07:28.810","Text":"That equals to 0.15."},{"Start":"07:29.570 ","End":"07:37.330","Text":"Okay. Great. Let\u0027s plug that in right here, that\u0027s 0.15."},{"Start":"07:38.240 ","End":"07:45.295","Text":"Now, since we know that all the probabilities here have to sum up to 1,"},{"Start":"07:45.295 ","End":"07:51.769","Text":"well, then we automatically know what the probability of X equaling 0."},{"Start":"07:51.769 ","End":"07:58.023","Text":"That turns out to be 0.74."},{"Start":"07:58.023 ","End":"08:01.145","Text":"Here we go."},{"Start":"08:01.145 ","End":"08:07.620","Text":"We\u0027ve calculated or constructed the probability function of X."},{"Start":"08:07.620 ","End":"08:10.540","Text":"It\u0027s this guy right here."}],"ID":13387}],"Thumbnail":null,"ID":64524},{"Name":"Expectation, Variance, and Standard Deviation","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"34s","ChapterTopicVideoID":12913,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.529","Text":"In this chapter, we\u0027ll be dealing with the expectation, variance,"},{"Start":"00:04.529 ","End":"00:08.475","Text":"and standard deviation of a discrete random variable."},{"Start":"00:08.475 ","End":"00:14.445","Text":"Now, these estimates, the expectation variance and standard deviation,"},{"Start":"00:14.445 ","End":"00:17.010","Text":"are very important,"},{"Start":"00:17.010 ","End":"00:21.120","Text":"and they\u0027re the main estimators that describe the central point of"},{"Start":"00:21.120 ","End":"00:26.940","Text":"the probability function and how scattered is the data around this central point."},{"Start":"00:26.940 ","End":"00:31.410","Text":"Instead of going into all the theory right now,"},{"Start":"00:31.410 ","End":"00:35.080","Text":"let\u0027s first start out with an example."}],"ID":13388},{"Watched":false,"Name":"Example 1","Duration":"10m 39s","ChapterTopicVideoID":12914,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.290","Text":"Now in our example,"},{"Start":"00:01.290 ","End":"00:02.520","Text":"we have a roulette wheel,"},{"Start":"00:02.520 ","End":"00:04.650","Text":"and it looks like this."},{"Start":"00:04.650 ","End":"00:10.140","Text":"Now a person turns a roulette wheel in order to win."},{"Start":"00:10.140 ","End":"00:13.800","Text":"Now in our specific roulette wheel,"},{"Start":"00:13.800 ","End":"00:15.390","Text":"we have 3 numbers."},{"Start":"00:15.390 ","End":"00:17.215","Text":"We have 10, 20, and 30."},{"Start":"00:17.215 ","End":"00:23.250","Text":"And that\u0027s the definition of the winnings that of the person turning the wheel."},{"Start":"00:23.250 ","End":"00:27.195","Text":"If we define the variable x,"},{"Start":"00:27.195 ","End":"00:30.430","Text":"as a person\u0027s winnings,"},{"Start":"00:31.250 ","End":"00:38.685","Text":"the values that x can have is either 10 or 20 or 30."},{"Start":"00:38.685 ","End":"00:43.590","Text":"That\u0027s right here. That\u0027s the case."},{"Start":"00:43.590 ","End":"00:46.880","Text":"What\u0027s the probability of x equals 10?"},{"Start":"00:46.880 ","End":"00:51.035","Text":"What\u0027s the probability of someone winning $10?"},{"Start":"00:51.035 ","End":"00:56.100","Text":"Well, that\u0027s 0.25, that\u0027s 25 percent."},{"Start":"00:56.100 ","End":"01:01.910","Text":"How do I know that? Because the area that\u0027s defined by 10,"},{"Start":"01:01.910 ","End":"01:03.920","Text":"that\u0027s a quarter of the wheel."},{"Start":"01:03.920 ","End":"01:05.570","Text":"That\u0027s 25 percent."},{"Start":"01:05.570 ","End":"01:08.705","Text":"The probability of x equals 20."},{"Start":"01:08.705 ","End":"01:14.075","Text":"Again, that\u0027s also 25 percent because it takes up a quarter of the wheel,"},{"Start":"01:14.075 ","End":"01:18.170","Text":"and the probability of x equals 30, well,"},{"Start":"01:18.170 ","End":"01:23.480","Text":"that equals 50 percent because it takes up a half of the wheel."},{"Start":"01:23.480 ","End":"01:28.700","Text":"Now that we have the values of x and their probabilities,"},{"Start":"01:28.700 ","End":"01:32.420","Text":"we can construct the probability function."},{"Start":"01:32.420 ","End":"01:34.173","Text":"Let\u0027s do that."},{"Start":"01:34.173 ","End":"01:38.735","Text":"That\u0027s x, that\u0027s the probability of x."},{"Start":"01:38.735 ","End":"01:42.575","Text":"Now, x can have value of 10,"},{"Start":"01:42.575 ","End":"01:44.150","Text":"or value 20,"},{"Start":"01:44.150 ","End":"01:45.750","Text":"or a value of 30,"},{"Start":"01:45.750 ","End":"01:50.580","Text":"and the probabilities for each 1 is 0.25,"},{"Start":"01:50.580 ","End":"01:54.115","Text":"0.25, and 0.5."},{"Start":"01:54.115 ","End":"01:57.975","Text":"That\u0027s these probabilities right here."},{"Start":"01:57.975 ","End":"02:03.230","Text":"Right now, it was very easy to define the variable,"},{"Start":"02:03.230 ","End":"02:06.105","Text":"see what the values of the variables are,"},{"Start":"02:06.105 ","End":"02:12.225","Text":"calculate the probabilities, and build the probability function."},{"Start":"02:12.225 ","End":"02:16.825","Text":"Now that we\u0027ve calculated the probability function of x,"},{"Start":"02:16.825 ","End":"02:19.345","Text":"let\u0027s take 1 step forward."},{"Start":"02:19.345 ","End":"02:29.940","Text":"What we want to do right now is calculate the expectation of x. E of x,"},{"Start":"02:29.940 ","End":"02:31.760","Text":"or the expectation of x,"},{"Start":"02:31.760 ","End":"02:38.270","Text":"is defined as the sum of x times its probability."},{"Start":"02:39.170 ","End":"02:47.575","Text":"By the way, we can express the expectation of x as the Greek letter Mu."},{"Start":"02:47.575 ","End":"02:49.700","Text":"Now, in our case,"},{"Start":"02:49.700 ","End":"02:51.545","Text":"what\u0027s the expectation of x?"},{"Start":"02:51.545 ","End":"02:55.145","Text":"That\u0027s the sum of x times the probability of x."},{"Start":"02:55.145 ","End":"02:59.255","Text":"That would be 10 times 0.25."},{"Start":"02:59.255 ","End":"03:02.300","Text":"That\u0027s x times its probability,"},{"Start":"03:02.300 ","End":"03:06.784","Text":"plus 20 times 0.25,"},{"Start":"03:06.784 ","End":"03:08.850","Text":"this guy right here,"},{"Start":"03:08.850 ","End":"03:11.962","Text":"plus 30 times 0.5,"},{"Start":"03:11.962 ","End":"03:14.225","Text":"that\u0027s this guy right here."},{"Start":"03:14.225 ","End":"03:22.700","Text":"That equals to 2.5 plus 5 plus 15,"},{"Start":"03:22.700 ","End":"03:26.855","Text":"and that equals to 22.5."},{"Start":"03:26.855 ","End":"03:32.360","Text":"Note, that is the calculation for the expectation of x."},{"Start":"03:32.360 ","End":"03:34.085","Text":"Now, what does that mean?"},{"Start":"03:34.085 ","End":"03:36.955","Text":"From a distribution perspective,"},{"Start":"03:36.955 ","End":"03:44.390","Text":"all that means is this is the average of x or the distribution function."},{"Start":"03:44.390 ","End":"03:47.255","Text":"This is the average of the distribution function."},{"Start":"03:47.255 ","End":"03:52.115","Text":"From the question perspective of our winnings,"},{"Start":"03:52.115 ","End":"03:55.160","Text":"this is what we expect to win when"},{"Start":"03:55.160 ","End":"03:58.955","Text":"we roll the roulette wheel over and over and over again."},{"Start":"03:58.955 ","End":"04:00.635","Text":"Over the long haul,"},{"Start":"04:00.635 ","End":"04:05.110","Text":"we\u0027ll be expected to win this amount right here."},{"Start":"04:05.110 ","End":"04:13.295","Text":"Now, obviously, we can earn on 1 turn of the wheel."},{"Start":"04:13.295 ","End":"04:17.360","Text":"You can earn $30 or $20 or $10."},{"Start":"04:17.360 ","End":"04:24.500","Text":"There\u0027s a certain amount of disbursement around this expectation, around this number."},{"Start":"04:24.500 ","End":"04:30.715","Text":"We\u0027re not always going to get a number like this number right here."},{"Start":"04:30.715 ","End":"04:34.520","Text":"That means that every time we turn the wheel,"},{"Start":"04:34.520 ","End":"04:35.690","Text":"there\u0027s a certain risk,"},{"Start":"04:35.690 ","End":"04:39.805","Text":"a certain uncertainty about our winnings."},{"Start":"04:39.805 ","End":"04:41.300","Text":"Over the long haul,"},{"Start":"04:41.300 ","End":"04:43.490","Text":"we know that this is what we\u0027re going to earn."},{"Start":"04:43.490 ","End":"04:46.310","Text":"But for each individual turning the wheel,"},{"Start":"04:46.310 ","End":"04:49.475","Text":"we don\u0027t really know what we\u0027re going to earn."},{"Start":"04:49.475 ","End":"04:54.480","Text":"In that case, we have to understand"},{"Start":"04:54.480 ","End":"05:01.205","Text":"what\u0027s the disbursement around the average, around the expectation."},{"Start":"05:01.205 ","End":"05:07.300","Text":"Now, this measure of disbursement is basically the variance of x."},{"Start":"05:07.300 ","End":"05:09.460","Text":"Now, the variance of x,"},{"Start":"05:09.460 ","End":"05:12.345","Text":"we can define it as Sigma squared."},{"Start":"05:12.345 ","End":"05:17.760","Text":"Sigma squared, that\u0027s a small Sigma,"},{"Start":"05:17.760 ","End":"05:19.650","Text":"a Greek letter squared,"},{"Start":"05:19.650 ","End":"05:21.236","Text":"and that equals to,"},{"Start":"05:21.236 ","End":"05:22.910","Text":"that\u0027s the definition here."},{"Start":"05:22.910 ","End":"05:27.530","Text":"That\u0027s the sum of x minus Mu."},{"Start":"05:27.530 ","End":"05:30.350","Text":"That\u0027s Mu is the expectation of x,"},{"Start":"05:30.350 ","End":"05:33.865","Text":"x minus the expectation x squared,"},{"Start":"05:33.865 ","End":"05:37.350","Text":"times the probability of x."},{"Start":"05:37.350 ","End":"05:44.875","Text":"Basically what we\u0027re taking is we\u0027re taking the average distances of all the x\u0027s,"},{"Start":"05:44.875 ","End":"05:49.485","Text":"from the expectation of x."},{"Start":"05:49.485 ","End":"05:54.884","Text":"We\u0027re taking all the average distances we\u0027re summing them."},{"Start":"05:54.884 ","End":"05:59.705","Text":"In our case, that\u0027ll be 10 minus"},{"Start":"05:59.705 ","End":"06:06.105","Text":"22.5 squared times 0.25."},{"Start":"06:06.105 ","End":"06:09.650","Text":"That\u0027s 10 minus the expectation,"},{"Start":"06:09.650 ","End":"06:16.200","Text":"that\u0027s 22.5, times the probability of x plus,"},{"Start":"06:16.200 ","End":"06:24.555","Text":"here we go, 20 minus 22.5 times 0.25,"},{"Start":"06:24.555 ","End":"06:31.889","Text":"plus 30 minus 22.5 squared,"},{"Start":"06:31.889 ","End":"06:38.020","Text":"excuse me, I forgot to square it right here, times 0.5."},{"Start":"06:38.020 ","End":"06:46.570","Text":"Now, that turns out to be 68.75."},{"Start":"06:46.570 ","End":"06:50.135","Text":"Now, what does this number mean?"},{"Start":"06:50.135 ","End":"07:00.425","Text":"Well, this is the average distance of x from the expectation of x."},{"Start":"07:00.425 ","End":"07:02.405","Text":"The distance squared."},{"Start":"07:02.405 ","End":"07:08.186","Text":"Now, this isn\u0027t as intuitive as it should be,"},{"Start":"07:08.186 ","End":"07:11.870","Text":"but we can think of it as the disbursement of"},{"Start":"07:11.870 ","End":"07:20.315","Text":"all the data of the probability function around the expectation."},{"Start":"07:20.315 ","End":"07:23.285","Text":"Now, this is a complicated expression."},{"Start":"07:23.285 ","End":"07:30.790","Text":"We can simplify the bit by saying that the variance of"},{"Start":"07:30.790 ","End":"07:41.280","Text":"x equals the expectation of x squared minus the expectation squared of x."},{"Start":"07:41.590 ","End":"07:49.925","Text":"Now, I won\u0027t try to prove that this expression equals this expression."},{"Start":"07:49.925 ","End":"07:52.040","Text":"That\u0027s a little bit too complicated."},{"Start":"07:52.040 ","End":"07:53.810","Text":"You\u0027re going to have to take my word for it."},{"Start":"07:53.810 ","End":"07:55.430","Text":"But what I can\u0027t tell you,"},{"Start":"07:55.430 ","End":"07:57.110","Text":"that in most cases,"},{"Start":"07:57.110 ","End":"08:01.295","Text":"this expression is much easier to work with."},{"Start":"08:01.295 ","End":"08:05.440","Text":"Let\u0027s just take a look at how this works."},{"Start":"08:05.440 ","End":"08:08.345","Text":"Now, in our case,"},{"Start":"08:08.345 ","End":"08:11.135","Text":"what\u0027s the expectation of x squared?"},{"Start":"08:11.135 ","End":"08:13.580","Text":"That equals, now we take x,"},{"Start":"08:13.580 ","End":"08:15.975","Text":"10, we square it,"},{"Start":"08:15.975 ","End":"08:17.760","Text":"that\u0027s right here,"},{"Start":"08:17.760 ","End":"08:27.408","Text":"and we multiply it by the probability 0.25 plus 20 squared,"},{"Start":"08:27.408 ","End":"08:29.030","Text":"that\u0027s this guy right here,"},{"Start":"08:29.030 ","End":"08:36.305","Text":"times 0.25, plus 30 squared times 0.5."},{"Start":"08:36.305 ","End":"08:44.165","Text":"Now, we have to take away the expectation squared of x,"},{"Start":"08:44.165 ","End":"08:47.360","Text":"and that\u0027s minus 22.5."},{"Start":"08:47.360 ","End":"08:49.996","Text":"That\u0027s this guy right here squared."},{"Start":"08:49.996 ","End":"08:56.990","Text":"You\u0027re going to have to believe me that this turns out also to be 68.75,"},{"Start":"08:56.990 ","End":"08:59.360","Text":"and as I said,"},{"Start":"08:59.360 ","End":"09:04.700","Text":"this is the measure of disbursement of the probability function."},{"Start":"09:04.700 ","End":"09:09.140","Text":"But it isn\u0027t that intuitive because it\u0027s in square units."},{"Start":"09:09.140 ","End":"09:11.500","Text":"It\u0027s the distance squared."},{"Start":"09:11.500 ","End":"09:15.635","Text":"In order to make this a little bit more understandable,"},{"Start":"09:15.635 ","End":"09:19.325","Text":"we\u0027ve defined the standard deviation."},{"Start":"09:19.325 ","End":"09:20.690","Text":"The standard deviation,"},{"Start":"09:20.690 ","End":"09:22.999","Text":"you can see that right here,"},{"Start":"09:22.999 ","End":"09:25.875","Text":"this is Sigma of x,"},{"Start":"09:25.875 ","End":"09:28.100","Text":"not Sigma squared like we have here,"},{"Start":"09:28.100 ","End":"09:29.330","Text":"this is Sigma of x,"},{"Start":"09:29.330 ","End":"09:32.898","Text":"that\u0027s the square root of the variance,"},{"Start":"09:32.898 ","End":"09:36.650","Text":"and that\u0027s the square root of 68.75,"},{"Start":"09:36.650 ","End":"09:39.890","Text":"which turns out to be 8.29."},{"Start":"09:39.890 ","End":"09:42.095","Text":"Now, what\u0027s this number?"},{"Start":"09:42.095 ","End":"09:47.130","Text":"Again, if we are expected to"},{"Start":"09:47.130 ","End":"09:53.250","Text":"earn $22.5 from this roulette wheel,"},{"Start":"09:53.250 ","End":"10:03.695","Text":"then our disbursement winning disbursement over time would be $8.29."},{"Start":"10:03.695 ","End":"10:06.980","Text":"That means we\u0027re going to be on average"},{"Start":"10:06.980 ","End":"10:15.710","Text":"$8.29 away from our average earnings, which is right here."},{"Start":"10:15.710 ","End":"10:21.935","Text":"Now, having understood what the expectation is,"},{"Start":"10:21.935 ","End":"10:24.020","Text":"what the variance is,"},{"Start":"10:24.020 ","End":"10:26.525","Text":"and what the standard deviation is,"},{"Start":"10:26.525 ","End":"10:31.280","Text":"now, your job is to start working on the problems"},{"Start":"10:31.280 ","End":"10:36.215","Text":"by yourselves and only then after you\u0027ve had time to work on them,"},{"Start":"10:36.215 ","End":"10:39.720","Text":"look at the solution videos."}],"ID":13389},{"Watched":false,"Name":"Exercise 1","Duration":"4m 54s","ChapterTopicVideoID":12915,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.415","Text":"A person plays a game of chance."},{"Start":"00:02.415 ","End":"00:05.250","Text":"We define X is the amount he wins."},{"Start":"00:05.250 ","End":"00:08.580","Text":"The probability function of X is as follows;"},{"Start":"00:08.580 ","End":"00:14.355","Text":"he wins minus 30 or he loses 30 with a probability of 40 percent."},{"Start":"00:14.355 ","End":"00:18.945","Text":"He doesn\u0027t win or he doesn\u0027t lose with probability of 0.1 percent."},{"Start":"00:18.945 ","End":"00:23.184","Text":"He wins 20 with a probability of 30 percent,"},{"Start":"00:23.184 ","End":"00:25.740","Text":"and he wins 40 with a probability of 20 percent."},{"Start":"00:25.740 ","End":"00:31.960","Text":"You were asked, what are the expectation variance and standard deviation of X?"},{"Start":"00:31.960 ","End":"00:35.535","Text":"Well, let\u0027s get started."},{"Start":"00:35.535 ","End":"00:42.345","Text":"The expectation of X is equal to"},{"Start":"00:42.345 ","End":"00:52.160","Text":"the sum of X times the probability for each X, for each value."},{"Start":"00:52.160 ","End":"01:02.329","Text":"Now that equals to 1 minus 30 times 0.4 plus 0 times"},{"Start":"01:02.329 ","End":"01:07.160","Text":"0.1 plus 20 times"},{"Start":"01:07.160 ","End":"01:14.235","Text":"0.3 plus 40 times 0.2,"},{"Start":"01:14.235 ","End":"01:18.150","Text":"and that turns out to be 2."},{"Start":"01:18.150 ","End":"01:21.335","Text":"That\u0027s the expectation."},{"Start":"01:21.335 ","End":"01:25.030","Text":"Now, let\u0027s look at the variance."},{"Start":"01:25.030 ","End":"01:34.730","Text":"The variance of X is equal to the sum over all the values of X of"},{"Start":"01:34.730 ","End":"01:44.660","Text":"the X minus the expectation squared times the probability of X_i."},{"Start":"01:44.660 ","End":"01:54.570","Text":"Now that also equals to the sum over all the values of X of X"},{"Start":"01:54.570 ","End":"01:59.210","Text":"squared times the probability of"},{"Start":"01:59.210 ","End":"02:07.020","Text":"X minus the expectation squared of X."},{"Start":"02:08.690 ","End":"02:13.835","Text":"As you said, that equals to the variance, Sigma squared."},{"Start":"02:13.835 ","End":"02:17.045","Text":"Well, what does that equal to?"},{"Start":"02:17.045 ","End":"02:19.010","Text":"Let\u0027s go back to our table."},{"Start":"02:19.010 ","End":"02:26.420","Text":"That\u0027s minus 30 squared times 0.4 plus 0 squared times"},{"Start":"02:26.420 ","End":"02:36.670","Text":"0.1 plus 20 squared times 0.3 plus 40 squared times 0.2."},{"Start":"02:36.670 ","End":"02:41.960","Text":"Let\u0027s not forget to subtract 2 squared."},{"Start":"02:41.960 ","End":"02:45.540","Text":"That\u0027s our expectation squared."},{"Start":"02:45.820 ","End":"02:53.820","Text":"That equals to 796."},{"Start":"02:53.820 ","End":"02:55.980","Text":"What\u0027s a standard deviation?"},{"Start":"02:55.980 ","End":"03:00.575","Text":"The standard deviation is the square root of the variance."},{"Start":"03:00.575 ","End":"03:08.490","Text":"That equals to square root of 796,"},{"Start":"03:08.490 ","End":"03:15.130","Text":"that equals to 28.21."},{"Start":"03:15.980 ","End":"03:23.600","Text":"Again, in this specific game with less specific probability function,"},{"Start":"03:23.600 ","End":"03:27.610","Text":"we can expect to win 2 dollars"},{"Start":"03:27.610 ","End":"03:30.320","Text":"but there\u0027s a specific risk"},{"Start":"03:30.320 ","End":"03:32.555","Text":"where there is a certain risk in playing this game because you can"},{"Start":"03:32.555 ","End":"03:36.770","Text":"also lose 30 dollars with a probability"},{"Start":"03:36.770 ","End":"03:41.875","Text":"of 0.4 or not win or lose with a probability of 0.1."},{"Start":"03:41.875 ","End":"03:46.580","Text":"This risk right here is, basically,"},{"Start":"03:46.580 ","End":"03:52.055","Text":"it\u0027s expressed as the standard deviation being 28.21."},{"Start":"03:52.055 ","End":"03:56.600","Text":"Now, most people, what they do is they add and"},{"Start":"03:56.600 ","End":"04:01.100","Text":"subtract standard deviation from the expectation."},{"Start":"04:01.100 ","End":"04:04.355","Text":"Now, that\u0027s not really the correct thing to do"},{"Start":"04:04.355 ","End":"04:08.045","Text":"because the probability function isn\u0027t symmetrical."},{"Start":"04:08.045 ","End":"04:09.620","Text":"You can see that right here,"},{"Start":"04:09.620 ","End":"04:14.825","Text":"there is a 0.4 chance of losing 30 dollars,"},{"Start":"04:14.825 ","End":"04:18.755","Text":"but there\u0027s a 0.2 chance of winning 40 dollars."},{"Start":"04:18.755 ","End":"04:21.995","Text":"We can see that this isn\u0027t symmetrical and therefore,"},{"Start":"04:21.995 ","End":"04:28.490","Text":"it\u0027s not really correct to add and subtract the standard deviation from the expectation."},{"Start":"04:28.490 ","End":"04:36.550","Text":"What we can say though is that we are expected to win 2 dollars in this game"},{"Start":"04:36.550 ","End":"04:45.443","Text":"and that we can be as far away as 2 dollars8.21 from the expectation,"},{"Start":"04:45.443 ","End":"04:48.745","Text":"again, because of the risk we\u0027re taking right here."},{"Start":"04:48.745 ","End":"04:52.100","Text":"Great. We\u0027ve solved this problem."},{"Start":"04:52.100 ","End":"04:54.570","Text":"Let\u0027s go on to the next."}],"ID":13390},{"Watched":false,"Name":"Exercise 2","Duration":"7m 32s","ChapterTopicVideoID":12916,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.135","Text":"In this question, we\u0027ll be talking about bank accounts."},{"Start":"00:03.135 ","End":"00:07.185","Text":"Now, there are 2 bank branches in a given community;"},{"Start":"00:07.185 ","End":"00:09.915","Text":"City Bank and Union Bank."},{"Start":"00:09.915 ","End":"00:13.290","Text":"50 percent of the population have an account at City Bank,"},{"Start":"00:13.290 ","End":"00:16.140","Text":"and 40 percent have an account at Union Bank."},{"Start":"00:16.140 ","End":"00:20.205","Text":"20 percent of the populations have no bank account at all."},{"Start":"00:20.205 ","End":"00:26.700","Text":"Now, let X be the number of bank branches in which a person has an account."},{"Start":"00:26.700 ","End":"00:30.930","Text":"We have to calculate the expectation of X."},{"Start":"00:30.930 ","End":"00:34.810","Text":"Now, in order to calculate the expectation of X,"},{"Start":"00:34.810 ","End":"00:38.320","Text":"we need to build a probability function for X."},{"Start":"00:38.320 ","End":"00:43.700","Text":"Before that, well, let\u0027s try to describe the data."},{"Start":"00:43.700 ","End":"00:49.835","Text":"Now, we can use either Venn diagrams or probability matrices."},{"Start":"00:49.835 ","End":"00:54.515","Text":"I prefer to use a probability matrix in this problem."},{"Start":"00:54.515 ","End":"00:58.605","Text":"So let\u0027s build 1,"},{"Start":"00:58.605 ","End":"01:00.360","Text":"and here it is."},{"Start":"01:00.360 ","End":"01:04.050","Text":"Great. Now, we have this,"},{"Start":"01:04.050 ","End":"01:09.150","Text":"but we still don\u0027t have our variables."},{"Start":"01:09.150 ","End":"01:15.470","Text":"Let\u0027s define A as a person that"},{"Start":"01:15.470 ","End":"01:22.530","Text":"has an account at the City Bank."},{"Start":"01:25.310 ","End":"01:33.230","Text":"B, we\u0027ll define that as a person having an account at the Union Bank."},{"Start":"01:35.180 ","End":"01:40.530","Text":"Now, let\u0027s describe the data."},{"Start":"01:40.530 ","End":"01:42.740","Text":"Probability of A,"},{"Start":"01:42.740 ","End":"01:46.630","Text":"that means that the probability of a person having an account at"},{"Start":"01:46.630 ","End":"01:53.125","Text":"the City Bank is 0.5 or 50 percent."},{"Start":"01:53.125 ","End":"01:56.180","Text":"That\u0027s given right here."},{"Start":"01:57.990 ","End":"02:07.430","Text":"The probability of a person having an account at Union Bank is 40 percent."},{"Start":"02:07.430 ","End":"02:12.265","Text":"The probability of a person not having an account at all,"},{"Start":"02:12.265 ","End":"02:19.205","Text":"not in the City Bank and not at the Union Bank, is 20 percent."},{"Start":"02:19.205 ","End":"02:22.275","Text":"Again, we see that right here,"},{"Start":"02:22.275 ","End":"02:27.040","Text":"40 percent and 20 percent."},{"Start":"02:27.440 ","End":"02:32.350","Text":"Now, because we wanted to use the probability matrix,"},{"Start":"02:32.350 ","End":"02:35.905","Text":"well, let\u0027s fill in the probabilities that we already have."},{"Start":"02:35.905 ","End":"02:38.225","Text":"Well, the probability of A,"},{"Start":"02:38.225 ","End":"02:40.890","Text":"as we said is 50 percent,"},{"Start":"02:40.890 ","End":"02:44.200","Text":"that\u0027s right here, 0.5."},{"Start":"02:44.200 ","End":"02:50.670","Text":"The probability of B is 0.4. Now that\u0027s right here, it\u0027s 0.4."},{"Start":"02:51.350 ","End":"02:55.740","Text":"The probability of not A and not B, well,"},{"Start":"02:55.740 ","End":"02:56.865","Text":"here\u0027s not A,"},{"Start":"02:56.865 ","End":"02:58.180","Text":"here\u0027s not B,"},{"Start":"02:58.180 ","End":"03:02.930","Text":"and this is the intersection right here, that\u0027s 0.2."},{"Start":"03:02.930 ","End":"03:08.400","Text":"Now, because we\u0027re so familiar with the probability matrix,"},{"Start":"03:08.470 ","End":"03:11.015","Text":"it\u0027s very easy to fill it that way."},{"Start":"03:11.015 ","End":"03:13.740","Text":"Here, that\u0027ll be 0.6,"},{"Start":"03:13.740 ","End":"03:16.380","Text":"0.4 plus 0.6 equals 1."},{"Start":"03:16.380 ","End":"03:18.990","Text":"This equals to 0.5."},{"Start":"03:18.990 ","End":"03:20.925","Text":"Let\u0027s fill in the rest."},{"Start":"03:20.925 ","End":"03:28.060","Text":"That\u0027ll be 0.4, 0.1, 0.3."},{"Start":"03:28.340 ","End":"03:34.150","Text":"We pretty much have all the probabilities that we need."},{"Start":"03:35.480 ","End":"03:40.340","Text":"Right now because we\u0027ve defined everything that we can,"},{"Start":"03:40.340 ","End":"03:46.010","Text":"let\u0027s go and build our probability function for X."},{"Start":"03:46.010 ","End":"03:49.065","Text":"Now again, what\u0027s X?"},{"Start":"03:49.065 ","End":"03:54.070","Text":"X is the number of bank branches."},{"Start":"03:59.300 ","End":"04:01.590","Text":"What are the values of the X?"},{"Start":"04:01.590 ","End":"04:07.040","Text":"Well, we can have a person not having an account at all,"},{"Start":"04:07.040 ","End":"04:13.190","Text":"or we can have a person having an account at 1 of the 2 branches;"},{"Start":"04:13.190 ","End":"04:15.185","Text":"City Bank or Union Bank,"},{"Start":"04:15.185 ","End":"04:21.875","Text":"or we can have a person that has an account both at the City Bank and at the Union Bank."},{"Start":"04:21.875 ","End":"04:25.500","Text":"Great. Let\u0027s see."},{"Start":"04:25.500 ","End":"04:28.775","Text":"What is our probability function?"},{"Start":"04:28.775 ","End":"04:30.990","Text":"Well, as we know,"},{"Start":"04:31.440 ","End":"04:37.815","Text":"we\u0027ll write the probability function like this here we have X,"},{"Start":"04:37.815 ","End":"04:42.720","Text":"the values of X are 0, 1, and 2."},{"Start":"04:42.720 ","End":"04:48.775","Text":"Now, the probability that X equals 0, what does that mean?"},{"Start":"04:48.775 ","End":"04:55.135","Text":"That\u0027s the probability of a person not having any bank account in any branch."},{"Start":"04:55.135 ","End":"04:58.150","Text":"Well, that\u0027s given, that\u0027s 0.2."},{"Start":"04:58.150 ","End":"05:01.065","Text":"That\u0027s right here, 0.2."},{"Start":"05:01.065 ","End":"05:05.870","Text":"Again, 20 percent of the population have no bank account."},{"Start":"05:06.430 ","End":"05:15.245","Text":"Now, what\u0027s the probability of X equals 2?"},{"Start":"05:15.245 ","End":"05:19.535","Text":"Well, X equals 2 means that a person has a bank account"},{"Start":"05:19.535 ","End":"05:24.365","Text":"both at the Union Bank and at the City Bank."},{"Start":"05:24.365 ","End":"05:27.600","Text":"Well, that\u0027s right here, that\u0027s 0.1."},{"Start":"05:30.170 ","End":"05:36.695","Text":"In order for the probability equal to 1,"},{"Start":"05:36.695 ","End":"05:40.730","Text":"a person having only 1 bank account,"},{"Start":"05:40.730 ","End":"05:45.590","Text":"be it at the Union Bank or City Bank, is 0.7."},{"Start":"05:45.590 ","End":"05:48.200","Text":"Now, again, let\u0027s look at it."},{"Start":"05:48.200 ","End":"05:56.685","Text":"This thing right here is the probability where X equals 0."},{"Start":"05:56.685 ","End":"06:03.135","Text":"He doesn\u0027t have a bank account at the City Bank or at the Union Bank."},{"Start":"06:03.135 ","End":"06:08.910","Text":"Here, that\u0027s the probability X equals"},{"Start":"06:08.910 ","End":"06:18.310","Text":"2 where a person has a bank account both at the City Bank and at the Union Bank."},{"Start":"06:18.310 ","End":"06:26.410","Text":"Right here, the sum of these probabilities is the probability of X equals 1,"},{"Start":"06:26.410 ","End":"06:32.910","Text":"where a person has a bank account at the Union Bank,"},{"Start":"06:32.910 ","End":"06:35.650","Text":"but not at the City Bank or a person"},{"Start":"06:35.650 ","End":"06:39.025","Text":"having an account at the City Bank and not at the Union Bank,"},{"Start":"06:39.025 ","End":"06:41.655","Text":"and here are the probabilities."},{"Start":"06:41.655 ","End":"06:46.385","Text":"So what we have to do right now is just to calculate the expectation."},{"Start":"06:46.385 ","End":"06:50.840","Text":"Now the expectation is the multiplication of"},{"Start":"06:50.840 ","End":"06:56.340","Text":"each value of X by its probabilities and summing that up."},{"Start":"06:58.690 ","End":"07:02.820","Text":"Let\u0027s write this in the proper color."},{"Start":"07:05.030 ","End":"07:13.700","Text":"The expectation of X is equal to 0 times 0.2,"},{"Start":"07:13.700 ","End":"07:15.409","Text":"that\u0027s these guys right here,"},{"Start":"07:15.409 ","End":"07:24.435","Text":"plus 1 times 0.7 plus 2 times 0.1,"},{"Start":"07:24.435 ","End":"07:28.270","Text":"and that equals to 0.9."},{"Start":"07:28.700 ","End":"07:32.440","Text":"That\u0027s the expectation of X."}],"ID":13391},{"Watched":false,"Name":"Exercise 3 - Part a","Duration":"7m 6s","ChapterTopicVideoID":12917,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"In this question we\u0027ll be calling families and"},{"Start":"00:02.640 ","End":"00:05.745","Text":"asking them about their satellite TV connection."},{"Start":"00:05.745 ","End":"00:08.580","Text":"Now, it\u0027s known that 20 percent of"},{"Start":"00:08.580 ","End":"00:11.745","Text":"families have a satellite TV connection in their homes."},{"Start":"00:11.745 ","End":"00:17.685","Text":"In a survey, a pollster seeks to interview a family with its satellite TV connection."},{"Start":"00:17.685 ","End":"00:20.730","Text":"He randomly telephones a family and continues"},{"Start":"00:20.730 ","End":"00:23.865","Text":"until he finds a family with a satellite TV connection."},{"Start":"00:23.865 ","End":"00:28.120","Text":"In any case, the pollster doesn\u0027t call more than 5 families."},{"Start":"00:28.120 ","End":"00:34.160","Text":"Now, we define X as the number of families called in a survey and we\u0027re asked first to"},{"Start":"00:34.160 ","End":"00:36.980","Text":"construct a probability function of X and"},{"Start":"00:36.980 ","End":"00:41.645","Text":"second to calculate the expectation and standard deviation of X."},{"Start":"00:41.645 ","End":"00:47.300","Text":"Great. Now, before we can start answering these questions,"},{"Start":"00:47.300 ","End":"00:49.120","Text":"we have to describe the data."},{"Start":"00:49.120 ","End":"00:51.470","Text":"The best way to do this is with"},{"Start":"00:51.470 ","End":"00:56.390","Text":"a probability tree and that\u0027s because the story that\u0027s given to us,"},{"Start":"00:56.390 ","End":"00:58.100","Text":"it\u0027s given to us in stages."},{"Start":"00:58.100 ","End":"01:03.260","Text":"A pollster calls the first family if he succeeds with a 20 percent chance, then he stops."},{"Start":"01:03.260 ","End":"01:09.290","Text":"If he doesn\u0027t, then he goes on to the next family and so on and so forth 5 times."},{"Start":"01:09.290 ","End":"01:11.585","Text":"If that\u0027s the case,"},{"Start":"01:11.585 ","End":"01:16.115","Text":"let\u0027s start to construct our probability tree."},{"Start":"01:16.115 ","End":"01:18.455","Text":"Now, as we said,"},{"Start":"01:18.455 ","End":"01:20.510","Text":"here\u0027s the first level."},{"Start":"01:20.510 ","End":"01:22.520","Text":"Now, if he succeeds,"},{"Start":"01:22.520 ","End":"01:25.970","Text":"he succeeds with a 20 percent chance,"},{"Start":"01:25.970 ","End":"01:33.695","Text":"and if he doesn\u0027t with an 80 percent chance and then he goes on to the next level."},{"Start":"01:33.695 ","End":"01:35.855","Text":"Again, if he succeeds,"},{"Start":"01:35.855 ","End":"01:39.920","Text":"he succeeds with a 20 percent chance and if he doesn\u0027t"},{"Start":"01:39.920 ","End":"01:44.120","Text":"succeed in catching a family with a satellite connection,"},{"Start":"01:44.120 ","End":"01:50.825","Text":"then again, that\u0027s an 80 percent chance of not succeeding and so on and so forth."},{"Start":"01:50.825 ","End":"01:54.830","Text":"You get the picture. Having said that,"},{"Start":"01:54.830 ","End":"02:00.840","Text":"let\u0027s see how the whole tree looks and then we\u0027ll take it from there."},{"Start":"02:01.430 ","End":"02:05.555","Text":"This is how our probability tree looks."},{"Start":"02:05.555 ","End":"02:12.440","Text":"The next step is to see what the definition is of our variable."},{"Start":"02:12.440 ","End":"02:22.690","Text":"Now X, our variable is defined as the number of families called in the survey."},{"Start":"02:22.690 ","End":"02:29.680","Text":"Now it\u0027s very important to understand what X is and what the values of X are so we"},{"Start":"02:29.680 ","End":"02:37.895","Text":"can construct a probability function and the expectation of standard deviations."},{"Start":"02:37.895 ","End":"02:43.840","Text":"Now, X is a number of families so let\u0027s take a look at the tree."},{"Start":"02:43.840 ","End":"02:47.420","Text":"What is the first level?"},{"Start":"02:47.420 ","End":"02:52.500","Text":"Here is X equals 1,"},{"Start":"02:53.180 ","End":"02:56.685","Text":"here X equals 2."},{"Start":"02:56.685 ","End":"02:58.980","Text":"What does that mean that X equals 1,"},{"Start":"02:58.980 ","End":"03:04.360","Text":"that the pollster called the first family and he succeeded."},{"Start":"03:05.570 ","End":"03:08.280","Text":"What does X equals 2 mean?"},{"Start":"03:08.280 ","End":"03:10.500","Text":"That he called the first family,"},{"Start":"03:10.500 ","End":"03:14.629","Text":"he didn\u0027t succeed in finding a family with a satellite connection."},{"Start":"03:14.629 ","End":"03:20.555","Text":"He called a second family and he succeeded with the second family and so on and so forth."},{"Start":"03:20.555 ","End":"03:23.930","Text":"X equals 3, X equals 4,"},{"Start":"03:23.930 ","End":"03:26.900","Text":"and X equals 5."},{"Start":"03:26.900 ","End":"03:36.167","Text":"Now, that means that X can take on the values of 1,"},{"Start":"03:36.167 ","End":"03:38.945","Text":"2, 3, 4, and 5."},{"Start":"03:38.945 ","End":"03:43.710","Text":"Now, what\u0027s the probability of X equals 1?"},{"Start":"03:43.710 ","End":"03:47.340","Text":"Again, X equals 1 means that he called the first family and he"},{"Start":"03:47.340 ","End":"03:53.265","Text":"succeeded on the first try in catching a family with a satellite connection."},{"Start":"03:53.265 ","End":"03:55.295","Text":"Well, that\u0027s 0.2."},{"Start":"03:55.295 ","End":"03:58.355","Text":"He succeeded and he stops."},{"Start":"03:58.355 ","End":"04:02.770","Text":"What does the probability of X equals 2 means?"},{"Start":"04:02.770 ","End":"04:05.050","Text":"That means that he called the first family,"},{"Start":"04:05.050 ","End":"04:09.550","Text":"he didn\u0027t catch that family as having a satellite connection."},{"Start":"04:09.550 ","End":"04:10.990","Text":"But on the second try,"},{"Start":"04:10.990 ","End":"04:14.815","Text":"he caught a family that had a satellite connection."},{"Start":"04:14.815 ","End":"04:17.795","Text":"That means it\u0027s at 0.8,"},{"Start":"04:17.795 ","End":"04:28.950","Text":"he missed out here and he succeeded here times 0.2 and that equals to 0.16."},{"Start":"04:30.050 ","End":"04:33.085","Text":"Now, when we continue on,"},{"Start":"04:33.085 ","End":"04:38.660","Text":"then these are the calculations of the probabilities for X equals 3,"},{"Start":"04:38.660 ","End":"04:41.940","Text":"where he didn\u0027t succeed on the first try,"},{"Start":"04:41.940 ","End":"04:43.785","Text":"didn\u0027t succeed on the second try,"},{"Start":"04:43.785 ","End":"04:45.660","Text":"and at third try, he did succeed."},{"Start":"04:45.660 ","End":"04:50.480","Text":"That means that\u0027s 0.8 squared, excuse me,"},{"Start":"04:50.480 ","End":"04:55.640","Text":"that\u0027s 0.8 times 0.8 times 0.2, and so on and so forth."},{"Start":"04:55.640 ","End":"04:59.660","Text":"For X equals 4, he missed out on the first try, second try,"},{"Start":"04:59.660 ","End":"05:04.865","Text":"third try, that\u0027s 0.8 cubed times 0.2."},{"Start":"05:04.865 ","End":"05:10.295","Text":"Now, what happens when X equals 5?"},{"Start":"05:10.295 ","End":"05:12.980","Text":"Well, when X equals 5,"},{"Start":"05:12.980 ","End":"05:20.615","Text":"he basically missed out on all 4 families before this family,"},{"Start":"05:20.615 ","End":"05:24.410","Text":"he didn\u0027t catch a family with a satellite connection,"},{"Start":"05:24.410 ","End":"05:25.640","Text":"and the fifth try,"},{"Start":"05:25.640 ","End":"05:30.325","Text":"he either succeeded or again, he missed out."},{"Start":"05:30.325 ","End":"05:38.280","Text":"That means that that equals to 0.8^5 that\u0027s this branch right here,"},{"Start":"05:38.280 ","End":"05:40.605","Text":"0.8 times 0.8 and so on,"},{"Start":"05:40.605 ","End":"05:47.730","Text":"5 times plus 0.8^4 that missed out on"},{"Start":"05:47.730 ","End":"05:53.480","Text":"our other 4 previous families times 0.2 he succeeded on"},{"Start":"05:53.480 ","End":"06:00.420","Text":"the fifth family and that equals to 0.4096."},{"Start":"06:07.130 ","End":"06:09.875","Text":"Now that we\u0027ve calculated"},{"Start":"06:09.875 ","End":"06:15.920","Text":"all the probabilities from the data that we\u0027ve described here in the tree,"},{"Start":"06:15.920 ","End":"06:20.155","Text":"let\u0027s build the probability function for X."},{"Start":"06:20.155 ","End":"06:27.500","Text":"Excuse me. That means that we have the X here and we have the probability of X."},{"Start":"06:27.500 ","End":"06:32.600","Text":"Now, X can equal 1,"},{"Start":"06:32.600 ","End":"06:37.040","Text":"2, 3, 4, or 5."},{"Start":"06:37.040 ","End":"06:41.140","Text":"Now, if X equals 1, that\u0027s 0.2,"},{"Start":"06:41.140 ","End":"06:47.250","Text":"if X equals 2, that\u0027s 0.16 gives me 0.128."},{"Start":"06:47.250 ","End":"06:52.380","Text":"That\u0027s 0.1024 and when X equals 5,"},{"Start":"06:52.380 ","End":"06:58.340","Text":"we have 0.4096 and here we"},{"Start":"06:58.340 ","End":"07:06.690","Text":"basically answered the first question of constructing a probability function for X."}],"ID":13392},{"Watched":false,"Name":"Exercise 3 - Part b","Duration":"2m 26s","ChapterTopicVideoID":12918,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.395","Text":"Now in section b,"},{"Start":"00:01.395 ","End":"00:05.250","Text":"we want to calculate the expectation and standard deviation of X."},{"Start":"00:05.250 ","End":"00:14.290","Text":"Now from section a, this is a probability function that we calculated on the string."},{"Start":"00:14.600 ","End":"00:19.710","Text":"Let\u0027s get started. The expectation of X is"},{"Start":"00:19.710 ","End":"00:25.935","Text":"basically summing of all of X times its probabilities,"},{"Start":"00:25.935 ","End":"00:30.089","Text":"and that equals to 1 times"},{"Start":"00:30.089 ","End":"00:36.990","Text":"0.2 plus 2 times 0.16 plus,"},{"Start":"00:36.990 ","End":"00:38.805","Text":"and so on and so forth."},{"Start":"00:38.805 ","End":"00:44.415","Text":"The so on and so forth is 3 times this plus 4 times this,"},{"Start":"00:44.415 ","End":"00:46.020","Text":"plus 5 times this."},{"Start":"00:46.020 ","End":"00:50.380","Text":"That ultimately comes out to 3.3616."},{"Start":"00:53.570 ","End":"00:57.020","Text":"That\u0027s the expectation of X."},{"Start":"00:57.020 ","End":"01:05.105","Text":"That means that the average number of families that the poster calls is 3.3616."},{"Start":"01:05.105 ","End":"01:09.695","Text":"Now, let\u0027s calculate the variance of X."},{"Start":"01:09.695 ","End":"01:15.875","Text":"The variance of X is defined as the sum of X squared times"},{"Start":"01:15.875 ","End":"01:23.620","Text":"the probability for each X minus the expectation squared."},{"Start":"01:23.630 ","End":"01:34.005","Text":"That also equals to 1 squared times 0.2 plus 2 squared times 0.16,"},{"Start":"01:34.005 ","End":"01:35.790","Text":"1 squared times this,"},{"Start":"01:35.790 ","End":"01:37.080","Text":"2 squared times this,"},{"Start":"01:37.080 ","End":"01:38.850","Text":"and again, 3 squared times this,"},{"Start":"01:38.850 ","End":"01:48.840","Text":"and so on and so forth, minus 3.3616 squared."},{"Start":"01:48.840 ","End":"01:54.830","Text":"That equals to 2.57."},{"Start":"01:54.830 ","End":"01:56.495","Text":"That\u0027s the variance."},{"Start":"01:56.495 ","End":"01:58.430","Text":"But we didn\u0027t ask for the variance."},{"Start":"01:58.430 ","End":"02:00.725","Text":"We asked for the standard deviation."},{"Start":"02:00.725 ","End":"02:07.420","Text":"Now the standard deviation of X is simply the square root of the variance,"},{"Start":"02:07.420 ","End":"02:11.825","Text":"and that equals to square root of 2.57,"},{"Start":"02:11.825 ","End":"02:16.220","Text":"and that equals to 1.603."},{"Start":"02:17.180 ","End":"02:21.255","Text":"That\u0027s the standard deviation of X,"},{"Start":"02:21.255 ","End":"02:26.440","Text":"and this is the expectation of X."}],"ID":13393},{"Watched":false,"Name":"Exercise 4 - Part a","Duration":"5m 30s","ChapterTopicVideoID":12919,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"This question deals with opening doors."},{"Start":"00:02.760 ","End":"00:05.745","Text":"A person has a key ring with 5 keys,"},{"Start":"00:05.745 ","End":"00:08.910","Text":"only 1 of which opens the door to his home."},{"Start":"00:08.910 ","End":"00:12.930","Text":"He tries the keys randomly and after trying a given key,"},{"Start":"00:12.930 ","End":"00:16.335","Text":"he takes it off the ring in order to avoid using it again."},{"Start":"00:16.335 ","End":"00:21.014","Text":"Let X denote the number of attempts until the door opens."},{"Start":"00:21.014 ","End":"00:24.480","Text":"Now we\u0027re asked to construct the probability function of"},{"Start":"00:24.480 ","End":"00:28.935","Text":"X and to calculate the expectation and variance of X."},{"Start":"00:28.935 ","End":"00:33.080","Text":"Well, the first thing that we need to do is to describe"},{"Start":"00:33.080 ","End":"00:37.025","Text":"the data and what\u0027s the best technique to describe the data?"},{"Start":"00:37.025 ","End":"00:39.595","Text":"Well, here we\u0027ll use a probability tree."},{"Start":"00:39.595 ","End":"00:41.060","Text":"Again, why is that?"},{"Start":"00:41.060 ","End":"00:44.540","Text":"Because the story is given to us in stages,"},{"Start":"00:44.540 ","End":"00:47.555","Text":"the person chooses one of the 5 rings."},{"Start":"00:47.555 ","End":"00:49.520","Text":"If it works, then great,"},{"Start":"00:49.520 ","End":"00:53.960","Text":"if it doesn\u0027t, then he takes it off the ring and then he has 4 keys left."},{"Start":"00:53.960 ","End":"00:55.565","Text":"He chooses one of the 4."},{"Start":"00:55.565 ","End":"00:56.945","Text":"He tries that out."},{"Start":"00:56.945 ","End":"01:00.635","Text":"If it works, if it opens the doors then that\u0027s great."},{"Start":"01:00.635 ","End":"01:02.769","Text":"If it doesn\u0027t, then he goes down to the third,"},{"Start":"01:02.769 ","End":"01:05.250","Text":"and fourth, and fifth try."},{"Start":"01:05.990 ","End":"01:09.950","Text":"If we\u0027re going to use a probability tree,"},{"Start":"01:09.950 ","End":"01:12.050","Text":"then let\u0027s get to it."},{"Start":"01:12.050 ","End":"01:15.390","Text":"Again, the first stage."},{"Start":"01:15.650 ","End":"01:19.905","Text":"He can try and if he succeeds on the first stage,"},{"Start":"01:19.905 ","End":"01:21.895","Text":"then what\u0027s the probability of him succeeding?"},{"Start":"01:21.895 ","End":"01:24.610","Text":"That\u0027s 1 out of the 5. Why is that?"},{"Start":"01:24.610 ","End":"01:30.960","Text":"Because only 1 opens the door and he has 5 keys in the key chain."},{"Start":"01:30.960 ","End":"01:33.180","Text":"1 out of the 5 opens the door."},{"Start":"01:33.180 ","End":"01:36.400","Text":"That means that 4 out of the 5 do not."},{"Start":"01:36.400 ","End":"01:38.155","Text":"That\u0027s on the first try."},{"Start":"01:38.155 ","End":"01:40.420","Text":"Now, on the second try,"},{"Start":"01:40.420 ","End":"01:43.315","Text":"how many keys does he have in the second try?"},{"Start":"01:43.315 ","End":"01:47.350","Text":"Well, he has 4 keys and one of the 4 will open"},{"Start":"01:47.350 ","End":"01:52.090","Text":"the door and that means that 3 of the 4 will not."},{"Start":"01:52.090 ","End":"01:54.480","Text":"Now, on the third try,"},{"Start":"01:54.480 ","End":"01:56.390","Text":"how many keys does he have left?"},{"Start":"01:56.390 ","End":"02:00.350","Text":"Well, he has 3 keys and one of the 3 will open the door,"},{"Start":"02:00.350 ","End":"02:03.395","Text":"and 2 out of 3 will not."},{"Start":"02:03.395 ","End":"02:06.545","Text":"In the fourth try how many keys does he have?"},{"Start":"02:06.545 ","End":"02:12.320","Text":"He has 2, so one of the 2 opens the door and one of the 2 will not."},{"Start":"02:12.320 ","End":"02:14.855","Text":"Now, on the last try,"},{"Start":"02:14.855 ","End":"02:20.715","Text":"he only has 1 key left so that will certainly open the door."},{"Start":"02:20.715 ","End":"02:27.215","Text":"Now that we\u0027ve constructed a probability tree right here and"},{"Start":"02:27.215 ","End":"02:33.645","Text":"we\u0027ve understood the logic behind the probabilities for each step of the way,"},{"Start":"02:33.645 ","End":"02:36.950","Text":"then first thing that we need to do is again,"},{"Start":"02:36.950 ","End":"02:42.110","Text":"look at our variable and try to understand that"},{"Start":"02:42.110 ","End":"02:48.980","Text":"the variable is X and X is the number of attempts until the door opens."},{"Start":"02:48.980 ","End":"02:52.850","Text":"That\u0027s basically given to us right here."},{"Start":"02:52.850 ","End":"02:57.115","Text":"This is the first stage where X equals 1."},{"Start":"02:57.115 ","End":"02:59.570","Text":"If he succeeds, that\u0027s great."},{"Start":"02:59.570 ","End":"03:02.000","Text":"If he doesn\u0027t, then he goes on to the next stage."},{"Start":"03:02.000 ","End":"03:07.640","Text":"That\u0027s X equals 2 and if he succeeds, not in the 2nd,"},{"Start":"03:07.640 ","End":"03:10.130","Text":"if he succeeds on the 3rd try,"},{"Start":"03:10.130 ","End":"03:11.600","Text":"then X equals 3,"},{"Start":"03:11.600 ","End":"03:14.525","Text":"the 3rd attempt and the fourth attempt,"},{"Start":"03:14.525 ","End":"03:18.130","Text":"and finally the fifth attempt."},{"Start":"03:18.130 ","End":"03:20.195","Text":"If that\u0027s the case,"},{"Start":"03:20.195 ","End":"03:25.625","Text":"what\u0027s the probability of X being equal to 1? What does that mean?"},{"Start":"03:25.625 ","End":"03:29.555","Text":"That he succeeds in opening the door on the first try."},{"Start":"03:29.555 ","End":"03:33.990","Text":"Well, that equals to 1 over 5."},{"Start":"03:34.130 ","End":"03:37.825","Text":"That\u0027s this branch right here, and then he stops."},{"Start":"03:37.825 ","End":"03:43.450","Text":"Now, what happens with a probability of X being equal to 2?"},{"Start":"03:43.450 ","End":"03:45.940","Text":"That means that he failed in"},{"Start":"03:45.940 ","End":"03:51.370","Text":"his first attempt and tried this on the 2nd attempt and he succeeded."},{"Start":"03:51.370 ","End":"03:59.540","Text":"That means that it\u0027s 4 over 5 times 1 over 4."},{"Start":"03:59.540 ","End":"04:04.365","Text":"That\u0027s this branch and he succeeded. What does that equal?"},{"Start":"04:04.365 ","End":"04:07.090","Text":"Well, 4 and 4 cancel each other out,"},{"Start":"04:07.090 ","End":"04:09.610","Text":"and that equals to 1 over 5."},{"Start":"04:09.610 ","End":"04:15.235","Text":"Now, what happens with the third trial?"},{"Start":"04:15.235 ","End":"04:19.310","Text":"Well, he didn\u0027t succeed on the first try."},{"Start":"04:19.310 ","End":"04:21.530","Text":"He didn\u0027t succeed on the 2nd try,"},{"Start":"04:21.530 ","End":"04:23.980","Text":"but he did succeed in the 3rd try."},{"Start":"04:23.980 ","End":"04:34.450","Text":"That\u0027s 4 over 5 times 3 over 4 times 1 over 3."},{"Start":"04:35.030 ","End":"04:37.260","Text":"What does that equal?"},{"Start":"04:37.260 ","End":"04:39.900","Text":"Well, again, this cancels out with this,"},{"Start":"04:39.900 ","End":"04:41.235","Text":"this cancels out with this,"},{"Start":"04:41.235 ","End":"04:43.140","Text":"and we have 1 over 5."},{"Start":"04:43.140 ","End":"04:47.415","Text":"As you can very well guess everything here,"},{"Start":"04:47.415 ","End":"04:51.750","Text":"all the probabilities are 1 over 5."},{"Start":"04:51.750 ","End":"04:58.570","Text":"We can construct our probability tree or our probability function very easily."},{"Start":"04:58.570 ","End":"05:01.000","Text":"We have X,"},{"Start":"05:01.000 ","End":"05:04.610","Text":"we have the probability of X."},{"Start":"05:04.610 ","End":"05:07.025","Text":"Now X can be either 1,"},{"Start":"05:07.025 ","End":"05:08.705","Text":"2, 3, 4, 5."},{"Start":"05:08.705 ","End":"05:12.590","Text":"So it\u0027s 1, 2, 3, 4,"},{"Start":"05:12.590 ","End":"05:20.670","Text":"and 5 and the probabilities are 1 over 5 for each of the values of X."},{"Start":"05:22.160 ","End":"05:30.000","Text":"There we go and here we\u0027ve completed section a."}],"ID":13394},{"Watched":false,"Name":"Exercise 4 - Part b","Duration":"1m 43s","ChapterTopicVideoID":12920,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.235","Text":"In Section b, we\u0027re asked to calculate the expectation and variance of x."},{"Start":"00:05.235 ","End":"00:06.540","Text":"From Section A,"},{"Start":"00:06.540 ","End":"00:08.700","Text":"we\u0027ve calculated the probability function,"},{"Start":"00:08.700 ","End":"00:10.485","Text":"that\u0027s here, right here."},{"Start":"00:10.485 ","End":"00:14.505","Text":"Let\u0027s get started with the expectation of x."},{"Start":"00:14.505 ","End":"00:22.605","Text":"The expectation of x is defined as the sum of x times its probability."},{"Start":"00:22.605 ","End":"00:33.270","Text":"That equals to 1 times 1/5 plus 2 times 1/5 plus 3 times a 1/5,"},{"Start":"00:33.270 ","End":"00:34.785","Text":"and so on and so forth,"},{"Start":"00:34.785 ","End":"00:38.090","Text":"and that equals to 3."},{"Start":"00:38.090 ","End":"00:40.715","Text":"That means, on average,"},{"Start":"00:40.715 ","End":"00:45.420","Text":"he\u0027s expected to open the door on the 3rd time."},{"Start":"00:45.770 ","End":"00:48.185","Text":"If that\u0027s the case,"},{"Start":"00:48.185 ","End":"00:55.290","Text":"let\u0027s go on and calculate the variance."},{"Start":"00:55.790 ","End":"00:58.849","Text":"What\u0027s the variance?"},{"Start":"00:58.849 ","End":"01:06.260","Text":"The variance is defined as the sum of x squared times"},{"Start":"01:06.260 ","End":"01:15.545","Text":"the probability of the x minus the expectation squared of x."},{"Start":"01:15.545 ","End":"01:19.950","Text":"That equals to 1 squared times"},{"Start":"01:19.950 ","End":"01:26.550","Text":"1/5 plus 2 squared times 1/5 plus,"},{"Start":"01:26.550 ","End":"01:27.690","Text":"and so on and so forth,"},{"Start":"01:27.690 ","End":"01:28.980","Text":"3 squared times 1/5,"},{"Start":"01:28.980 ","End":"01:30.120","Text":"4 squared times 1/5,"},{"Start":"01:30.120 ","End":"01:34.780","Text":"and so on, minus 3 squared."},{"Start":"01:34.850 ","End":"01:39.360","Text":"That comes out to 2."},{"Start":"01:39.360 ","End":"01:43.300","Text":"Here we\u0027ve solved section B."}],"ID":13395},{"Watched":false,"Name":"Exercise 5","Duration":"4m 14s","ChapterTopicVideoID":12921,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.585","Text":"In this question, we have a probability function of x,"},{"Start":"00:03.585 ","End":"00:05.010","Text":"which is a random variable."},{"Start":"00:05.010 ","End":"00:07.290","Text":"It\u0027s given to us like this."},{"Start":"00:07.290 ","End":"00:10.440","Text":"We have x with the values of 2,"},{"Start":"00:10.440 ","End":"00:11.490","Text":"4, 6,"},{"Start":"00:11.490 ","End":"00:14.280","Text":"and 8 and we have the probability of"},{"Start":"00:14.280 ","End":"00:17.745","Text":"x where we see the 2 of the probabilities are missing."},{"Start":"00:17.745 ","End":"00:23.010","Text":"Now, we\u0027re given that the expectation of x is 4.2 and"},{"Start":"00:23.010 ","End":"00:28.335","Text":"we\u0027re asked to find the missing probabilities in the table right here and here."},{"Start":"00:28.335 ","End":"00:32.105","Text":"Secondly, we want to calculate the variance of x."},{"Start":"00:32.105 ","End":"00:34.085","Text":"So let\u0027s get at it."},{"Start":"00:34.085 ","End":"00:39.300","Text":"Let\u0027s first define as P,"},{"Start":"00:39.300 ","End":"00:42.830","Text":"the probability where x equals 2."},{"Start":"00:42.830 ","End":"00:45.485","Text":"Now, that\u0027s the case."},{"Start":"00:45.485 ","End":"00:49.830","Text":"What\u0027s the probability of x being equal to 6?"},{"Start":"00:50.330 ","End":"00:58.760","Text":"Probability of x being equal to 6 is 1 minus the sum of all the other probabilities."},{"Start":"00:58.760 ","End":"01:05.585","Text":"So that\u0027s 1 minus p plus 0.3 plus 0.2."},{"Start":"01:05.585 ","End":"01:09.740","Text":"When we figure this out,"},{"Start":"01:09.740 ","End":"01:19.100","Text":"we see that the probability of x being equal to 6 is 0.5 minus p. First of all,"},{"Start":"01:19.100 ","End":"01:21.125","Text":"let\u0027s write that here."},{"Start":"01:21.125 ","End":"01:25.880","Text":"So 0.5 minus p. Now we still don\u0027t know what p is."},{"Start":"01:25.880 ","End":"01:27.740","Text":"Let\u0027s calculate that."},{"Start":"01:27.740 ","End":"01:33.520","Text":"We\u0027re given that the expectation of x is 4.2."},{"Start":"01:33.520 ","End":"01:35.510","Text":"So let\u0027s use that."},{"Start":"01:35.510 ","End":"01:39.155","Text":"First of all, what\u0027s the definition of the expectation?"},{"Start":"01:39.155 ","End":"01:44.590","Text":"That\u0027s the sum of x times the probability of x."},{"Start":"01:44.590 ","End":"01:47.975","Text":"So let\u0027s just plug the numbers in."},{"Start":"01:47.975 ","End":"01:56.040","Text":"That\u0027s 2 times p plus 4 times 0.3 plus"},{"Start":"01:56.040 ","End":"02:05.970","Text":"6 times 0.5 minus p plus 8 times 0.2."},{"Start":"02:05.970 ","End":"02:11.205","Text":"That equals to 2p plus 1.2"},{"Start":"02:11.205 ","End":"02:18.335","Text":"plus 3 minus 6p plus 1.6."},{"Start":"02:18.335 ","End":"02:25.220","Text":"Now let\u0027s just calculate the cell that\u0027s 2p minus 6p,"},{"Start":"02:25.220 ","End":"02:27.700","Text":"that\u0027s minus 4p,"},{"Start":"02:27.700 ","End":"02:29.530","Text":"1.2 plus 3,"},{"Start":"02:29.530 ","End":"02:34.820","Text":"that\u0027s 4.2, that\u0027s 5.8 plus 5.8."},{"Start":"02:34.820 ","End":"02:37.940","Text":"Now we know that the expectation of x is 4.2,"},{"Start":"02:37.940 ","End":"02:41.110","Text":"so that equals to 4.2."},{"Start":"02:41.110 ","End":"02:53.755","Text":"That\u0027s what sides 4p equals to 1.6 and therefore p equals to 0.4."},{"Start":"02:53.755 ","End":"02:56.870","Text":"Excellent. If p equals 0.4,"},{"Start":"02:56.870 ","End":"03:00.740","Text":"we can plug that right here. That\u0027s 0.4."},{"Start":"03:00.740 ","End":"03:05.780","Text":"And here the probability for x equals 6,"},{"Start":"03:05.780 ","End":"03:12.655","Text":"that\u0027s 0.5 minus 0.4, that\u0027s 0.1."},{"Start":"03:12.655 ","End":"03:17.560","Text":"The next step is to calculate the variance of x."},{"Start":"03:17.560 ","End":"03:21.850","Text":"Now, what\u0027s the definition of the variance?"},{"Start":"03:21.850 ","End":"03:26.590","Text":"Well, that equals to the sum of x squared times"},{"Start":"03:26.590 ","End":"03:32.065","Text":"the probability of x minus the expectation squared of x."},{"Start":"03:32.065 ","End":"03:33.820","Text":"That\u0027s the definition."},{"Start":"03:33.820 ","End":"03:36.280","Text":"So let\u0027s plug in the numbers here that we have in"},{"Start":"03:36.280 ","End":"03:40.450","Text":"the probability function and we can calculate the variance."},{"Start":"03:40.450 ","End":"03:45.030","Text":"Well, that\u0027s 2 squared times 0.4,"},{"Start":"03:45.030 ","End":"03:47.010","Text":"that\u0027s this guy right here."},{"Start":"03:47.010 ","End":"03:55.050","Text":"Plus 4 squared times 0.3 plus 6 squared times 0.1 plus 8"},{"Start":"03:55.050 ","End":"04:03.270","Text":"squared times 0.2 minus 4.2 squared."},{"Start":"04:03.270 ","End":"04:10.100","Text":"That comes out to 5.16."},{"Start":"04:10.100 ","End":"04:14.370","Text":"And that is the variance of x."}],"ID":13396},{"Watched":false,"Name":"Exercise 6","Duration":"3m 39s","ChapterTopicVideoID":12922,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.390","Text":"In this question, we have a discrete random variable that has the values of minus 5,"},{"Start":"00:06.390 ","End":"00:08.280","Text":"0, and plus 5."},{"Start":"00:08.280 ","End":"00:13.380","Text":"The variable\u0027s expectation is 0 and its variance is 10."},{"Start":"00:13.380 ","End":"00:17.660","Text":"Now, we have to find the probability function."},{"Start":"00:17.660 ","End":"00:21.335","Text":"Well, first of all, let\u0027s write down what we know."},{"Start":"00:21.335 ","End":"00:26.190","Text":"We know that the expectation of x equals 0,"},{"Start":"00:26.190 ","End":"00:30.820","Text":"and we know that the variance of x equals 10."},{"Start":"00:30.820 ","End":"00:35.760","Text":"Now, we want to find the probability function."},{"Start":"00:36.620 ","End":"00:38.865","Text":"Let\u0027s write this out,"},{"Start":"00:38.865 ","End":"00:41.940","Text":"that\u0027s the probability of X,"},{"Start":"00:41.940 ","End":"00:48.635","Text":"and X receives the values of minus 5,0 and plus 5."},{"Start":"00:48.635 ","End":"00:54.440","Text":"Now, in order for the expectation to equal to 0,"},{"Start":"00:54.440 ","End":"01:00.800","Text":"that means that the distribution of the probability must be symmetrical."},{"Start":"01:00.800 ","End":"01:02.045","Text":"Why is that?"},{"Start":"01:02.045 ","End":"01:08.065","Text":"Because the distance from this point to 0 and this point to 0 are equal."},{"Start":"01:08.065 ","End":"01:10.800","Text":"Let\u0027s write down here,"},{"Start":"01:10.800 ","End":"01:18.330","Text":"that\u0027s p and define that as p because they\u0027re symmetrical."},{"Start":"01:18.330 ","End":"01:21.590","Text":"What\u0027s the probability of 0?"},{"Start":"01:21.590 ","End":"01:30.420","Text":"Well, that\u0027s 1 minus 2p because the sum of other probabilities have to add up to 1."},{"Start":"01:31.690 ","End":"01:39.340","Text":"Now what we have to do is we have to calculate for p. Now,"},{"Start":"01:39.340 ","End":"01:43.805","Text":"we didn\u0027t use this data right here."},{"Start":"01:43.805 ","End":"01:47.750","Text":"Let\u0027s right now use this data to calculate what p is."},{"Start":"01:47.750 ","End":"01:51.750","Text":"Well, the variance of X is what?"},{"Start":"01:51.750 ","End":"01:55.425","Text":"That\u0027s equal to the sum of X squared"},{"Start":"01:55.425 ","End":"02:03.330","Text":"times p(X) minus the expectation square root of X."},{"Start":"02:03.500 ","End":"02:06.255","Text":"Let\u0027s plug in the numbers."},{"Start":"02:06.255 ","End":"02:10.650","Text":"That\u0027s minus 5 squared times p,"},{"Start":"02:10.650 ","End":"02:13.305","Text":"that\u0027s this guy right here,"},{"Start":"02:13.305 ","End":"02:18.120","Text":"plus 0 squared times 1 minus 2p,"},{"Start":"02:18.120 ","End":"02:20.505","Text":"that\u0027s this guy right here,"},{"Start":"02:20.505 ","End":"02:25.770","Text":"plus 5 squared times p,"},{"Start":"02:25.770 ","End":"02:29.910","Text":"that\u0027s this guy right here, minus 0 squared."},{"Start":"02:29.910 ","End":"02:33.120","Text":"That\u0027s the expectation squared where expectation equals 0,"},{"Start":"02:33.120 ","End":"02:34.485","Text":"so that\u0027s 0 squared,"},{"Start":"02:34.485 ","End":"02:38.160","Text":"and all this has to equal to what? To 10."},{"Start":"02:39.730 ","End":"02:42.830","Text":"We have minus 5 squared,"},{"Start":"02:42.830 ","End":"02:49.025","Text":"that\u0027s 25p plus, all that\u0027s 0."},{"Start":"02:49.025 ","End":"02:50.840","Text":"That\u0027s 5 squared times p,"},{"Start":"02:50.840 ","End":"02:55.290","Text":"that\u0027s 25p minus 0 squared,"},{"Start":"02:55.290 ","End":"02:58.755","Text":"that\u0027s 0, so we have that equaling 10."},{"Start":"02:58.755 ","End":"03:06.195","Text":"That means that 50p equals to 10,"},{"Start":"03:06.195 ","End":"03:10.925","Text":"or p equals to 10 over 50,"},{"Start":"03:10.925 ","End":"03:15.445","Text":"and that equals to 0.2."},{"Start":"03:15.445 ","End":"03:19.805","Text":"If that\u0027s the case, then we\u0027ll go right back up here,"},{"Start":"03:19.805 ","End":"03:25.075","Text":"p equals 0.2, here also 0.2."},{"Start":"03:25.075 ","End":"03:30.270","Text":"That means 1 minus 2 times 0.2 or 2 times 0.2 that\u0027s 0.4,"},{"Start":"03:30.270 ","End":"03:34.320","Text":"so 1 minus 0.4 is 0.6."},{"Start":"03:34.320 ","End":"03:38.610","Text":"That is a probability function."}],"ID":13397},{"Watched":false,"Name":"Exercise 7","Duration":"8m 19s","ChapterTopicVideoID":12923,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.380","Text":"In this question, the probability distribution of X,"},{"Start":"00:03.380 ","End":"00:04.635","Text":"a random variable,"},{"Start":"00:04.635 ","End":"00:08.700","Text":"is as follows: X has the values 1, 3,"},{"Start":"00:08.700 ","End":"00:13.035","Text":"and K with the probabilities of a 1/4 when X equals 1,"},{"Start":"00:13.035 ","End":"00:14.745","Text":"1/2 when X equals 3,"},{"Start":"00:14.745 ","End":"00:18.885","Text":"and 1/4 when X equals K. We\u0027re asked,"},{"Start":"00:18.885 ","End":"00:20.592","Text":"what value of K,"},{"Start":"00:20.592 ","End":"00:22.605","Text":"this guy right here,"},{"Start":"00:22.605 ","End":"00:26.195","Text":"will give the minimum value for the variance of X?"},{"Start":"00:26.195 ","End":"00:29.120","Text":"Now, in order to calculate the variance,"},{"Start":"00:29.120 ","End":"00:32.330","Text":"let\u0027s first calculate the expectation."},{"Start":"00:32.330 ","End":"00:35.255","Text":"What is the expectation of X?"},{"Start":"00:35.255 ","End":"00:40.580","Text":"Well, it\u0027s defined as the sum of X times the probability of X,"},{"Start":"00:40.580 ","End":"00:51.310","Text":"and that equals to 1 times 1/4 plus 3 times 1/2 plus K times 1/4,"},{"Start":"00:51.380 ","End":"00:55.890","Text":"and that equals to 0.25,"},{"Start":"00:55.890 ","End":"01:00.825","Text":"that\u0027s right here, and this is 1.5 right here,"},{"Start":"01:00.825 ","End":"01:05.825","Text":"and this is 0.25 times K. Now,"},{"Start":"01:05.825 ","End":"01:07.580","Text":"let\u0027s just add the things up,"},{"Start":"01:07.580 ","End":"01:14.625","Text":"so we have 1.75 plus 0.25K."},{"Start":"01:14.625 ","End":"01:18.880","Text":"Now, that\u0027s the expectation."},{"Start":"01:19.820 ","End":"01:23.625","Text":"Let\u0027s now calculate the variance."},{"Start":"01:23.625 ","End":"01:26.010","Text":"What\u0027s the variance of X?"},{"Start":"01:26.010 ","End":"01:32.540","Text":"By definition, that\u0027s the sum of X squared times the probability of"},{"Start":"01:32.540 ","End":"01:40.780","Text":"X minus the expectation squared of X."},{"Start":"01:40.780 ","End":"01:43.610","Text":"Now, let\u0027s just plug in the numbers."},{"Start":"01:43.610 ","End":"01:50.720","Text":"That\u0027s 1 squared times 1/4 plus 3 squared times"},{"Start":"01:50.720 ","End":"02:00.700","Text":"1/2 plus K squared times 1/4 minus the expectation squared."},{"Start":"02:00.700 ","End":"02:10.250","Text":"Now that\u0027s 1.75 plus 0.25K squared."},{"Start":"02:11.510 ","End":"02:13.650","Text":"What does that equal to?"},{"Start":"02:13.650 ","End":"02:20.430","Text":"That equals to 0.25 plus,"},{"Start":"02:20.430 ","End":"02:22.860","Text":"that\u0027s 3 squared 1/2 of 9,"},{"Start":"02:22.860 ","End":"02:29.100","Text":"that\u0027s 4.5 plus 0.25K"},{"Start":"02:29.100 ","End":"02:33.620","Text":"squared minus the expectation squared,"},{"Start":"02:33.620 ","End":"02:35.225","Text":"we\u0027ll leave this,"},{"Start":"02:35.225 ","End":"02:43.260","Text":"that\u0027s 1.75 plus 0.25K squared."},{"Start":"02:44.540 ","End":"02:49.085","Text":"Just to complete this calculation,"},{"Start":"02:49.085 ","End":"02:57.470","Text":"that equals to 4.75 plus 0.25K"},{"Start":"02:57.470 ","End":"03:06.930","Text":"squared minus 1.75 plus 0.25K squared."},{"Start":"03:07.820 ","End":"03:13.820","Text":"We found and we\u0027ve calculated what the variance of X was."},{"Start":"03:13.820 ","End":"03:17.755","Text":"That\u0027s this long expression right here,"},{"Start":"03:17.755 ","End":"03:21.665","Text":"but we weren\u0027t asked to find the variance of X."},{"Start":"03:21.665 ","End":"03:27.980","Text":"We were asked what value of K will give the minimum value for the variance of X."},{"Start":"03:27.980 ","End":"03:31.729","Text":"If we look at the expression for the variance,"},{"Start":"03:31.729 ","End":"03:36.590","Text":"we can look at it as a function of K."},{"Start":"03:36.590 ","End":"03:44.299","Text":"What we need to do is to find the minimum of that function."},{"Start":"03:44.299 ","End":"03:47.225","Text":"Let\u0027s write down this function."},{"Start":"03:47.225 ","End":"03:56.645","Text":"First of all, the function of K is equal to 4.75 plus"},{"Start":"03:56.645 ","End":"04:01.340","Text":"0.25K squared minus"},{"Start":"04:01.340 ","End":"04:08.549","Text":"1.75 plus 0.25K squared."},{"Start":"04:08.549 ","End":"04:10.260","Text":"This is the same function,"},{"Start":"04:10.260 ","End":"04:14.125","Text":"this is the same expression as we have here in the variance,"},{"Start":"04:14.125 ","End":"04:17.830","Text":"but now we\u0027re looking at it in terms of"},{"Start":"04:17.830 ","End":"04:23.455","Text":"K. Now in order to find a minimum or maximum of a function,"},{"Start":"04:23.455 ","End":"04:26.840","Text":"then we need to take the first derivative."},{"Start":"04:30.920 ","End":"04:34.265","Text":"Now, the first derivative of K, what we\u0027re doing is we\u0027re taking"},{"Start":"04:34.265 ","End":"04:40.430","Text":"the derivative with respect to K of this function."},{"Start":"04:42.140 ","End":"04:45.520","Text":"Once we have this derivative,"},{"Start":"04:45.520 ","End":"04:47.620","Text":"we have to equate that to 0,"},{"Start":"04:47.620 ","End":"04:49.285","Text":"we have to solve for K,"},{"Start":"04:49.285 ","End":"04:52.295","Text":"and that will give us basically"},{"Start":"04:52.295 ","End":"05:00.810","Text":"the point at which the function F of K will be either at a minimum or a maximum."},{"Start":"05:01.060 ","End":"05:09.140","Text":"Let\u0027s start. The first derivative of the function is as follows."},{"Start":"05:09.140 ","End":"05:11.690","Text":"This goes away,"},{"Start":"05:11.690 ","End":"05:20.360","Text":"and this is 0.5K minus 2 times 1.75"},{"Start":"05:20.360 ","End":"05:31.645","Text":"plus 0.25K times the internal derivative right here, that\u0027s 0.25."},{"Start":"05:31.645 ","End":"05:35.320","Text":"Now we have to equate that to 0,"},{"Start":"05:35.320 ","End":"05:38.125","Text":"but let\u0027s make the syllabus simpler first."},{"Start":"05:38.125 ","End":"05:48.355","Text":"That ultimately turns out to be 0.5K minus"},{"Start":"05:48.355 ","End":"05:54.130","Text":"0.875 minus"},{"Start":"05:54.130 ","End":"06:00.310","Text":"0.125K,"},{"Start":"06:00.310 ","End":"06:15.080","Text":"which is equal to 0.375K minus 0.875."},{"Start":"06:15.080 ","End":"06:20.610","Text":"As we said, we have to equate the first derivative to 0."},{"Start":"06:21.650 ","End":"06:27.394","Text":"So F tag of K, the first derivative,"},{"Start":"06:27.394 ","End":"06:32.520","Text":"which equals to 0.375K,"},{"Start":"06:32.520 ","End":"06:34.515","Text":"that\u0027s right here,"},{"Start":"06:34.515 ","End":"06:41.429","Text":"minus 0.875, we have to equate that to 0."},{"Start":"06:41.429 ","End":"06:53.770","Text":"That means that 0.375 times K equals to 0.875."},{"Start":"06:54.200 ","End":"06:58.740","Text":"That means that K equals to"},{"Start":"06:58.740 ","End":"07:05.850","Text":"0.875 divided by 0.375,"},{"Start":"07:05.850 ","End":"07:10.780","Text":"and that equals to 2.33."},{"Start":"07:10.850 ","End":"07:16.745","Text":"Now we figured out that the function F of K,"},{"Start":"07:16.745 ","End":"07:19.040","Text":"which is basically the variance,"},{"Start":"07:19.040 ","End":"07:25.045","Text":"gets its minimum or maximum when K equals 2.33."},{"Start":"07:25.045 ","End":"07:29.345","Text":"Now, we have to figure out whether it is a maximum or a minimum."},{"Start":"07:29.345 ","End":"07:33.480","Text":"What we need to do is take the second derivative."},{"Start":"07:33.830 ","End":"07:45.715","Text":"We take the second derivative with respect to K of the function that equals to this,"},{"Start":"07:45.715 ","End":"07:49.310","Text":"which equals 2,"},{"Start":"07:49.310 ","End":"07:51.520","Text":"and here\u0027s the function right here."},{"Start":"07:51.520 ","End":"07:55.550","Text":"We have to take the derivative of this function,"},{"Start":"07:55.550 ","End":"07:59.150","Text":"and that equals to 0.375,"},{"Start":"07:59.150 ","End":"08:01.085","Text":"it\u0027s greater than 0,"},{"Start":"08:01.085 ","End":"08:04.810","Text":"that means that we have a minimum."},{"Start":"08:04.810 ","End":"08:14.150","Text":"We can say for sure that when K equals 2.33 in the probability function,"},{"Start":"08:14.150 ","End":"08:18.870","Text":"then the variance of X will be a minimum."}],"ID":13398},{"Watched":false,"Name":"Binomial Probability Density Function","Duration":"8m 25s","ChapterTopicVideoID":13494,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.505","Text":"In this chapter, we\u0027ll be talking about special probability distributions,"},{"Start":"00:05.505 ","End":"00:08.665","Text":"specifically, binomial distribution."},{"Start":"00:08.665 ","End":"00:12.180","Text":"This will be the first of a number of chapters that will deal"},{"Start":"00:12.180 ","End":"00:16.200","Text":"with various special probability distributions."},{"Start":"00:16.200 ","End":"00:21.090","Text":"In order for us to work with these distributions,"},{"Start":"00:21.090 ","End":"00:23.625","Text":"we have to recognize them."},{"Start":"00:23.625 ","End":"00:29.250","Text":"It\u0027ll be up to you to identify them by yourselves through the context of the question,"},{"Start":"00:29.250 ","End":"00:31.905","Text":"as with the data that\u0027s provided to you."},{"Start":"00:31.905 ","End":"00:35.355","Text":"Once you\u0027ve identified these distributions,"},{"Start":"00:35.355 ","End":"00:42.224","Text":"you can use either the predefined expression or equation for the distribution,"},{"Start":"00:42.224 ","End":"00:47.429","Text":"and you can also calculate the expectation and variance."},{"Start":"00:47.429 ","End":"00:49.625","Text":"Again they\u0027re predefined equations,"},{"Start":"00:49.625 ","End":"00:51.455","Text":"so you don\u0027t have to worry about them."},{"Start":"00:51.455 ","End":"00:54.710","Text":"All you need to do is just to identify"},{"Start":"00:54.710 ","End":"01:02.195","Text":"the special distribution and know what the expectation and variance equations are."},{"Start":"01:02.195 ","End":"01:06.200","Text":"Having said that, I want to introduce to you"},{"Start":"01:06.200 ","End":"01:10.565","Text":"a very special concept that\u0027s called a Bernoulli trial."},{"Start":"01:10.565 ","End":"01:14.360","Text":"A Bernoulli trial is a trial in which"},{"Start":"01:14.360 ","End":"01:17.975","Text":"there are 2 possible results and only 2 possible results,"},{"Start":"01:17.975 ","End":"01:19.880","Text":"and we\u0027ll call them success or failure."},{"Start":"01:19.880 ","End":"01:26.015","Text":"Example of that could be either a product coming out of the production line,"},{"Start":"01:26.015 ","End":"01:28.730","Text":"which is either good or faulty,"},{"Start":"01:28.730 ","End":"01:32.015","Text":"a person that is employed or unemployed,"},{"Start":"01:32.015 ","End":"01:35.870","Text":"a coin toss that produces heads or tails,"},{"Start":"01:35.870 ","End":"01:37.340","Text":"and so on and so forth."},{"Start":"01:37.340 ","End":"01:41.870","Text":"The binomial distribution calculates"},{"Start":"01:41.870 ","End":"01:47.015","Text":"the probability of success when n independent Bernoulli trials are repeated."},{"Start":"01:47.015 ","End":"01:52.235","Text":"Let\u0027s take an example."},{"Start":"01:52.235 ","End":"01:54.650","Text":"Let\u0027s say we have a coin toss."},{"Start":"01:54.650 ","End":"01:59.315","Text":"We want to toss the coin 5 times."},{"Start":"01:59.315 ","End":"02:02.810","Text":"That means that n, in our case, equals 5."},{"Start":"02:02.810 ","End":"02:07.085","Text":"We know that each toss is independent of each other."},{"Start":"02:07.085 ","End":"02:12.020","Text":"We also know that the probability of, let\u0027s say,"},{"Start":"02:12.020 ","End":"02:22.040","Text":"getting not a tail but a head is equal to 50 percent, so P equals 0.5."},{"Start":"02:22.040 ","End":"02:25.395","Text":"In 1 Bernoulli trial,"},{"Start":"02:25.395 ","End":"02:28.635","Text":"the probability is 0.5."},{"Start":"02:28.635 ","End":"02:31.550","Text":"When we want to know,"},{"Start":"02:31.550 ","End":"02:33.725","Text":"what are the chances?"},{"Start":"02:33.725 ","End":"02:37.241","Text":"What is the,"},{"Start":"02:37.241 ","End":"02:38.873","Text":"here we go,"},{"Start":"02:38.873 ","End":"02:43.415","Text":"number of heads in 5 tosses,"},{"Start":"02:43.415 ","End":"02:46.355","Text":"that\u0027s basically the definition of our success."},{"Start":"02:46.355 ","End":"02:50.589","Text":"X would be the number of heads in 5 tosses."},{"Start":"02:50.589 ","End":"02:52.835","Text":"So when we have that,"},{"Start":"02:52.835 ","End":"03:00.480","Text":"we can define X as being distributed with a binomial distribution."},{"Start":"03:02.720 ","End":"03:07.280","Text":"As we said, the binomial distribution calculates"},{"Start":"03:07.280 ","End":"03:11.510","Text":"the probability of success when n independent Bernoulli trials are repeated."},{"Start":"03:11.510 ","End":"03:14.330","Text":"Just like in our example,"},{"Start":"03:14.330 ","End":"03:15.649","Text":"coin toss example,"},{"Start":"03:15.649 ","End":"03:17.810","Text":"let\u0027s define X as the total number of"},{"Start":"03:17.810 ","End":"03:22.465","Text":"successes when repeating n independent Bernoulli trials."},{"Start":"03:22.465 ","End":"03:25.369","Text":"Let p be the chance of success,"},{"Start":"03:25.369 ","End":"03:28.325","Text":"and q is the chance of failure in an individual trial."},{"Start":"03:28.325 ","End":"03:38.515","Text":"Let\u0027s define p as a probability of success in 1 trial."},{"Start":"03:38.515 ","End":"03:48.160","Text":"Since there are only 2 possible answers or 2 possible results in a Bernoulli trial,"},{"Start":"03:48.160 ","End":"03:57.490","Text":"then the probability of failure is equal to 1 minus p,"},{"Start":"03:57.490 ","End":"04:00.680","Text":"and we\u0027ll call that q."},{"Start":"04:02.180 ","End":"04:10.781","Text":"Again let p be the chances of success and q the chances of failure."},{"Start":"04:10.781 ","End":"04:15.100","Text":"So now we can define X as the number of"},{"Start":"04:15.100 ","End":"04:20.770","Text":"total number of successes that will be distributed."},{"Start":"04:20.770 ","End":"04:22.945","Text":"That\u0027s the sign for distributed binomial,"},{"Start":"04:22.945 ","End":"04:26.200","Text":"B for binomial with 2 parameters, n and p,"},{"Start":"04:26.200 ","End":"04:31.060","Text":"n being the number of independent trials and p being"},{"Start":"04:31.060 ","End":"04:36.855","Text":"the probability of success in a Bernoulli trial."},{"Start":"04:36.855 ","End":"04:43.900","Text":"Once we\u0027ve defined X as being binomially distributed with n and p,"},{"Start":"04:43.900 ","End":"04:48.025","Text":"we can write out the probability function."},{"Start":"04:48.025 ","End":"04:52.150","Text":"The probability function of X is this."},{"Start":"04:52.150 ","End":"04:57.580","Text":"The probability of x equaling k is equal to n over"},{"Start":"04:57.580 ","End":"05:04.870","Text":"k times p to the power of k times 1 minus p to the power of n minus k,"},{"Start":"05:04.870 ","End":"05:08.200","Text":"where k equals 0, 1, 2, 3,"},{"Start":"05:08.200 ","End":"05:12.888","Text":"and so on and so forth until n. That\u0027s n trials."},{"Start":"05:12.888 ","End":"05:18.890","Text":"Let\u0027s take a look at the components of the distribution function."},{"Start":"05:18.890 ","End":"05:22.450","Text":"Let\u0027s take a look at n over k. What does that mean?"},{"Start":"05:22.450 ","End":"05:23.740","Text":"What\u0027s that defined?"},{"Start":"05:23.740 ","End":"05:33.948","Text":"N over k is defined as n factorial over k factorial times n minus k factorial."},{"Start":"05:33.948 ","End":"05:42.230","Text":"N factorial, it just means that it\u0027s that multiplication of all the numbers from 1 till"},{"Start":"05:42.230 ","End":"05:46.430","Text":"n. That means it\u0027s 1 times 2 times 3 times n minus 2 times"},{"Start":"05:46.430 ","End":"05:50.870","Text":"n minus 1 times n. A special definition is,"},{"Start":"05:50.870 ","End":"05:54.671","Text":"where n equals 0, 0 factorial equals 1."},{"Start":"05:54.671 ","End":"06:00.613","Text":"You\u0027re not expected to calculate this thing by yourself."},{"Start":"06:00.613 ","End":"06:02.420","Text":"You have a calculator,"},{"Start":"06:02.420 ","End":"06:04.310","Text":"and you can do that."},{"Start":"06:04.310 ","End":"06:07.100","Text":"Once you know what the function of the calculator is,"},{"Start":"06:07.100 ","End":"06:08.914","Text":"you just plug in the numbers,"},{"Start":"06:08.914 ","End":"06:14.027","Text":"and it\u0027ll spew out the results."},{"Start":"06:14.027 ","End":"06:22.600","Text":"Having identified the expression for the binomial distribution,"},{"Start":"06:22.600 ","End":"06:32.240","Text":"it\u0027s given to us that the expression for the expectation of X is equals to n times p,"},{"Start":"06:32.240 ","End":"06:36.455","Text":"n being the number of times we will repeat a Bernoulli trial,"},{"Start":"06:36.455 ","End":"06:40.115","Text":"and p is the probability of success for each trial."},{"Start":"06:40.115 ","End":"06:49.219","Text":"N times p would be the expectation of getting a success over all the trials."},{"Start":"06:49.219 ","End":"06:50.870","Text":"What\u0027s the variance?"},{"Start":"06:50.870 ","End":"06:54.646","Text":"The variance of X is equal to n times p times q,"},{"Start":"06:54.646 ","End":"07:01.830","Text":"number of trials times the probability of success times the probability of failure."},{"Start":"07:01.960 ","End":"07:08.780","Text":"Let\u0027s sum up the idea of a binomial distribution."},{"Start":"07:08.780 ","End":"07:10.985","Text":"As we said previously,"},{"Start":"07:10.985 ","End":"07:14.975","Text":"that you have to identify these types of"},{"Start":"07:14.975 ","End":"07:19.760","Text":"distributions through the context of the question or to the data."},{"Start":"07:19.760 ","End":"07:24.740","Text":"That means we\u0027re looking for a set of criteria that will allow us to"},{"Start":"07:24.740 ","End":"07:32.915","Text":"identify this distribution as being the binomial distribution."},{"Start":"07:32.915 ","End":"07:34.850","Text":"This is it right here."},{"Start":"07:34.850 ","End":"07:42.350","Text":"In order for a binomial distribution to be relevant to our question,"},{"Start":"07:42.350 ","End":"07:45.095","Text":"the following conditions must be met."},{"Start":"07:45.095 ","End":"07:52.160","Text":"First of all, the same Bernoulli trial has to be repeated independently of each other,"},{"Start":"07:52.160 ","End":"07:56.585","Text":"secondly, the trial is repeated n times,"},{"Start":"07:56.585 ","End":"08:01.670","Text":"and thirdly, X is defined as the total number of successes obtained."},{"Start":"08:01.670 ","End":"08:05.120","Text":"If these 3 conditions are met,"},{"Start":"08:05.120 ","End":"08:08.060","Text":"then we have a binomial distribution,"},{"Start":"08:08.060 ","End":"08:13.945","Text":"and we can use this function right here,"},{"Start":"08:13.945 ","End":"08:20.900","Text":"and we can use these expressions to calculate the expectation and variance."},{"Start":"08:20.900 ","End":"08:25.230","Text":"Enough of theory, let\u0027s take a look at an example."}],"ID":14135},{"Watched":false,"Name":"Example 8 - Part a","Duration":"7m 35s","ChapterTopicVideoID":13495,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.415","Text":"In our example, 80 percent of the people in a given country have a driving license."},{"Start":"00:05.415 ","End":"00:09.630","Text":"Now, 10 people are randomly chosen from the country."},{"Start":"00:09.630 ","End":"00:15.720","Text":"We\u0027re asked, what\u0027s the probability that exactly 9 of them have a driving license?"},{"Start":"00:15.720 ","End":"00:21.240","Text":"Secondly, what\u0027s the probability that at least 9 of them have a driving license?"},{"Start":"00:21.240 ","End":"00:24.000","Text":"Thirdly, what is the expectation and"},{"Start":"00:24.000 ","End":"00:28.690","Text":"standard deviation of the number of people sampled having a driving license?"},{"Start":"00:28.700 ","End":"00:31.175","Text":"Let\u0027s get started."},{"Start":"00:31.175 ","End":"00:34.250","Text":"The first thing that we want to do is to"},{"Start":"00:34.250 ","End":"00:38.665","Text":"ask whether we have a binomial distribution or not."},{"Start":"00:38.665 ","End":"00:40.745","Text":"When we do that,"},{"Start":"00:40.745 ","End":"00:45.605","Text":"we have to see if the conditions for the binomial distributions are met."},{"Start":"00:45.605 ","End":"00:50.105","Text":"Let\u0027s take a look at the conditions. Here they are."},{"Start":"00:50.105 ","End":"00:58.085","Text":"The first condition talks about the same Bernoulli trial that\u0027s repeated independently."},{"Start":"00:58.085 ","End":"00:59.855","Text":"Well, first of all,"},{"Start":"00:59.855 ","End":"01:03.050","Text":"let\u0027s talk about independence."},{"Start":"01:03.050 ","End":"01:10.145","Text":"Now, when we sample a small number of people from a very large population,"},{"Start":"01:10.145 ","End":"01:15.890","Text":"we can assume that the people are independent of each other."},{"Start":"01:15.890 ","End":"01:23.180","Text":"In all cases where we have a small sample size from a large population,"},{"Start":"01:23.180 ","End":"01:26.495","Text":"we can assume independence. That\u0027s true."},{"Start":"01:26.495 ","End":"01:29.295","Text":"Now, the second part."},{"Start":"01:29.295 ","End":"01:32.075","Text":"Let\u0027s talk about Bernoulli trials."},{"Start":"01:32.075 ","End":"01:35.165","Text":"Do we have Bernoulli trials here?"},{"Start":"01:35.165 ","End":"01:38.120","Text":"Well, I believe that we do. Why is that?"},{"Start":"01:38.120 ","End":"01:39.860","Text":"Well, first of all, let\u0027s define"},{"Start":"01:39.860 ","End":"01:46.905","Text":"success as a person"},{"Start":"01:46.905 ","End":"01:50.181","Text":"having a license,"},{"Start":"01:50.181 ","End":"01:58.610","Text":"and failure as a person not having a license, no license."},{"Start":"01:59.010 ","End":"02:03.625","Text":"Now, what\u0027s the probability of a person having a license?"},{"Start":"02:03.625 ","End":"02:05.275","Text":"That\u0027s 80 percent."},{"Start":"02:05.275 ","End":"02:06.565","Text":"We have that right here."},{"Start":"02:06.565 ","End":"02:09.530","Text":"We have a driving license."},{"Start":"02:09.530 ","End":"02:17.140","Text":"That means that the probability of a success is 0.8."},{"Start":"02:17.140 ","End":"02:19.440","Text":"Now, what\u0027s the probability of failure."},{"Start":"02:19.440 ","End":"02:21.420","Text":"Well, that\u0027s 1 minus p,"},{"Start":"02:21.420 ","End":"02:23.520","Text":"and we call that q."},{"Start":"02:23.520 ","End":"02:26.375","Text":"That equals to 0.2."},{"Start":"02:26.375 ","End":"02:29.845","Text":"Now, what else do we have?"},{"Start":"02:29.845 ","End":"02:32.588","Text":"Well, how do we define?"},{"Start":"02:32.588 ","End":"02:36.345","Text":"Success here as we said,"},{"Start":"02:36.345 ","End":"02:40.890","Text":"is the number of people"},{"Start":"02:42.720 ","End":"02:54.080","Text":"having a license when we sample 10 people."},{"Start":"02:55.160 ","End":"02:57.700","Text":"This is number 3."},{"Start":"02:57.700 ","End":"02:59.495","Text":"Now, what about number 2?"},{"Start":"02:59.495 ","End":"03:01.220","Text":"The trial is repeated 10 times."},{"Start":"03:01.220 ","End":"03:03.560","Text":"Well, obviously, we take 10 people,"},{"Start":"03:03.560 ","End":"03:05.480","Text":"we sample 1 after the other."},{"Start":"03:05.480 ","End":"03:10.300","Text":"Each sample basically is a Bernoulli trial here."},{"Start":"03:10.300 ","End":"03:15.620","Text":"Having the probability of success or having a license is"},{"Start":"03:15.620 ","End":"03:22.050","Text":"0.8 and the probability of failure of person not having a license is 0.2."},{"Start":"03:22.050 ","End":"03:26.630","Text":"The conditions for a binomial distribution is met."},{"Start":"03:26.630 ","End":"03:33.715","Text":"We can say now that X is distributed binomial,"},{"Start":"03:33.715 ","End":"03:40.720","Text":"where n equals 10 and p equals 0.8."},{"Start":"03:42.950 ","End":"03:49.820","Text":"Now that we know that X is distributed binomially with n equals 10 and p equals 0.8,"},{"Start":"03:49.820 ","End":"03:53.990","Text":"we can write the expression for the binomial distribution."},{"Start":"03:53.990 ","End":"04:01.760","Text":"We say that the probability of X being equal to k equals n over"},{"Start":"04:01.760 ","End":"04:08.042","Text":"k times p to"},{"Start":"04:08.042 ","End":"04:14.790","Text":"the power of k times q to the power of n minus k. That\u0027s the probability distribution."},{"Start":"04:14.790 ","End":"04:18.585","Text":"Let\u0027s plug in the numbers for Section a."},{"Start":"04:18.585 ","End":"04:23.320","Text":"We\u0027re talking about k being equal to 9 and we\u0027re"},{"Start":"04:23.320 ","End":"04:28.900","Text":"asked what\u0027s the probability where x equals 9?"},{"Start":"04:28.940 ","End":"04:32.310","Text":"Again, let\u0027s plug in the numbers."},{"Start":"04:32.310 ","End":"04:34.335","Text":"N equals 10,"},{"Start":"04:34.335 ","End":"04:36.900","Text":"that\u0027s right here, 10 people."},{"Start":"04:36.900 ","End":"04:40.540","Text":"What\u0027s k? Well, k equals 9."},{"Start":"04:41.720 ","End":"04:44.295","Text":"It\u0027s n over 9,"},{"Start":"04:44.295 ","End":"04:51.420","Text":"or basically it\u0027s 10 over 9 times 0.8."},{"Start":"04:51.420 ","End":"04:55.270","Text":"P equals 0.8."},{"Start":"04:56.300 ","End":"05:00.375","Text":"0.8 to the power of 9 times 0.2,"},{"Start":"05:00.375 ","End":"05:02.385","Text":"that\u0027s q, that comes here,"},{"Start":"05:02.385 ","End":"05:04.470","Text":"n minus k, n minus k,"},{"Start":"05:04.470 ","End":"05:07.830","Text":"that\u0027s 10 minus 9, that\u0027s 1."},{"Start":"05:07.830 ","End":"05:11.830","Text":"Now, when we do the calculation,"},{"Start":"05:11.830 ","End":"05:16.570","Text":"that comes out to 0.268."},{"Start":"05:16.570 ","End":"05:19.569","Text":"Now, we can do this mechanically,"},{"Start":"05:19.569 ","End":"05:22.305","Text":"but let\u0027s what\u0027s going on here."},{"Start":"05:22.305 ","End":"05:29.210","Text":"Let\u0027s try to understand this outside of the framework of the expression."},{"Start":"05:29.400 ","End":"05:34.230","Text":"Well, here I\u0027ve written a series of letters,"},{"Start":"05:34.230 ","End":"05:38.065","Text":"S being success and F being a failure."},{"Start":"05:38.065 ","End":"05:39.645","Text":"Now, what was asked?"},{"Start":"05:39.645 ","End":"05:44.030","Text":"What\u0027s the probability that exactly 9 of them have a driver\u0027s license?"},{"Start":"05:44.030 ","End":"05:46.490","Text":"Well, if none of them have a driver\u0027s license,"},{"Start":"05:46.490 ","End":"05:48.770","Text":"that\u0027s means that 1 does not."},{"Start":"05:48.770 ","End":"05:50.705","Text":"We have 9 successes,"},{"Start":"05:50.705 ","End":"05:56.085","Text":"9 people having a driving license and the last 1 doesn\u0027t."},{"Start":"05:56.085 ","End":"06:00.500","Text":"That\u0027s what\u0027s the probability of a success as 0.8."},{"Start":"06:00.500 ","End":"06:03.860","Text":"That\u0027s basically 0.8 times 0.8 times 0.8,"},{"Start":"06:03.860 ","End":"06:07.015","Text":"9 times, times a failure."},{"Start":"06:07.015 ","End":"06:10.025","Text":"Well, that\u0027s 0.2 to the power of 1,"},{"Start":"06:10.025 ","End":"06:14.060","Text":"we only have 1 failure but again,"},{"Start":"06:14.060 ","End":"06:17.975","Text":"we\u0027re not asked what\u0027s the probability that"},{"Start":"06:17.975 ","End":"06:23.805","Text":"the last person doesn\u0027t have a driver license."},{"Start":"06:23.805 ","End":"06:26.010","Text":"We\u0027re asked that 1 of them doesn\u0027t have."},{"Start":"06:26.010 ","End":"06:28.550","Text":"Really, it doesn\u0027t matter where is the last person"},{"Start":"06:28.550 ","End":"06:33.715","Text":"or any person within the series of the 10 people."},{"Start":"06:33.715 ","End":"06:36.210","Text":"Here we see an example of that."},{"Start":"06:36.210 ","End":"06:38.180","Text":"It doesn\u0027t have to be the last person here,"},{"Start":"06:38.180 ","End":"06:39.650","Text":"it can read this person,"},{"Start":"06:39.650 ","End":"06:41.000","Text":"the third person here,"},{"Start":"06:41.000 ","End":"06:43.070","Text":"and so on and so forth."},{"Start":"06:43.070 ","End":"06:46.670","Text":"Now, how many options do we have here?"},{"Start":"06:46.670 ","End":"06:49.470","Text":"Well, we have 10 options."},{"Start":"06:49.570 ","End":"06:56.645","Text":"Now, this 10 is exactly this component right here."},{"Start":"06:56.645 ","End":"07:01.040","Text":"When we calculate what 10 over 9 is,"},{"Start":"07:01.040 ","End":"07:03.780","Text":"that comes out to 10."},{"Start":"07:05.090 ","End":"07:13.190","Text":"This component of the distribution talks about how many combinations do we"},{"Start":"07:13.190 ","End":"07:22.585","Text":"have of 9 people having a license and 1 person not having a license."},{"Start":"07:22.585 ","End":"07:28.055","Text":"I hope that made things a little bit clearer."},{"Start":"07:28.055 ","End":"07:34.350","Text":"If that\u0027s the case, let\u0027s just go on to the next question."}],"ID":14136},{"Watched":false,"Name":"Example 8 - Parts b-c","Duration":"4m 13s","ChapterTopicVideoID":13496,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.170","Text":"Here we have the results of what we did in Section a."},{"Start":"00:07.170 ","End":"00:13.050","Text":"We\u0027ve understood that X is distributed binomially with n equals 10 and p equals 0.8,"},{"Start":"00:13.050 ","End":"00:18.930","Text":"and we\u0027ve calculated that the probability of k being equal to 9,"},{"Start":"00:18.930 ","End":"00:21.555","Text":"well that equals to 0.2678."},{"Start":"00:21.555 ","End":"00:29.550","Text":"Again, that\u0027s the probability of having 9 people exactly who have a driver\u0027s license."},{"Start":"00:29.550 ","End":"00:31.725","Text":"In Section b, we\u0027re asked,"},{"Start":"00:31.725 ","End":"00:37.130","Text":"what\u0027s the probability that at least 9 of them have a driver\u0027s license?"},{"Start":"00:37.130 ","End":"00:46.830","Text":"That means we\u0027re looking at the probability of X being greater or equal to 9."},{"Start":"00:46.830 ","End":"00:50.884","Text":"At least 9 of them have a driver\u0027s license."},{"Start":"00:50.884 ","End":"00:57.110","Text":"Well, that means that we want to know what\u0027s the probability of X"},{"Start":"00:57.110 ","End":"01:04.700","Text":"equals 9 plus the probability of X being equal to 10."},{"Start":"01:04.700 ","End":"01:11.615","Text":"We already know what the probability of X equals 9 is."},{"Start":"01:11.615 ","End":"01:14.560","Text":"We calculated that right here,"},{"Start":"01:14.560 ","End":"01:20.540","Text":"and that equals to 0.268."},{"Start":"01:20.540 ","End":"01:26.615","Text":"All we have to do is calculate the probability of X being equal to 10."},{"Start":"01:26.615 ","End":"01:29.155","Text":"Let\u0027s plug in the numbers."},{"Start":"01:29.155 ","End":"01:32.225","Text":"The probability of X being equal to 10,"},{"Start":"01:32.225 ","End":"01:36.470","Text":"that\u0027s 10 over 10"},{"Start":"01:36.470 ","End":"01:47.420","Text":"times 0.8^10 times 0.2^0."},{"Start":"01:47.420 ","End":"01:53.825","Text":"If we take out our calculator and do the calculations, well, 10 over 10,"},{"Start":"01:53.825 ","End":"01:55.985","Text":"that equals to 1,"},{"Start":"01:55.985 ","End":"02:01.940","Text":"and 0.2^0 that also equals to 1."},{"Start":"02:01.940 ","End":"02:05.870","Text":"Basically, the probability of X being equal,"},{"Start":"02:05.870 ","End":"02:07.475","Text":"greater, or equal to 9,"},{"Start":"02:07.475 ","End":"02:15.849","Text":"that equals to 0.268 plus 0.8^10."},{"Start":"02:15.849 ","End":"02:21.060","Text":"That equals to 0.376."},{"Start":"02:25.520 ","End":"02:29.360","Text":"In Section C, we\u0027re asked what are the expectations in"},{"Start":"02:29.360 ","End":"02:33.920","Text":"standard deviation of the number of people sampled having a driving license."},{"Start":"02:33.920 ","End":"02:37.190","Text":"Well, if we recall the expectation of X,"},{"Start":"02:37.190 ","End":"02:39.560","Text":"where X is distributed binomially,"},{"Start":"02:39.560 ","End":"02:44.665","Text":"is equals to n times p. Let\u0027s plug in the numbers."},{"Start":"02:44.665 ","End":"02:48.270","Text":"That\u0027s 10 times 0.8,"},{"Start":"02:48.270 ","End":"02:50.385","Text":"that equals to 8."},{"Start":"02:50.385 ","End":"02:54.485","Text":"It makes sense, and why is that?"},{"Start":"02:54.485 ","End":"02:59.825","Text":"Well, we\u0027re told that 80 percent of the people have a driving license."},{"Start":"02:59.825 ","End":"03:03.440","Text":"On the other hand, we\u0027ve sampled 10 people,"},{"Start":"03:03.440 ","End":"03:06.920","Text":"so out of the 10 people were expected to"},{"Start":"03:06.920 ","End":"03:11.950","Text":"have 8 of the 10 people to have a driving license."},{"Start":"03:11.950 ","End":"03:15.110","Text":"Let\u0027s look at the standard deviation."},{"Start":"03:15.110 ","End":"03:18.770","Text":"Well, again, we don\u0027t have the standard deviation of the above,"},{"Start":"03:18.770 ","End":"03:21.695","Text":"but we do have the equation for the variance."},{"Start":"03:21.695 ","End":"03:27.560","Text":"The variance of X is n times p times q."},{"Start":"03:27.560 ","End":"03:30.800","Text":"That equals, let\u0027s plug in the numbers;"},{"Start":"03:30.800 ","End":"03:34.160","Text":"10 times 0.8 times q,"},{"Start":"03:34.160 ","End":"03:39.440","Text":"which is 0.2, and that turns out to be 1.6."},{"Start":"03:39.440 ","End":"03:42.520","Text":"Now again, we\u0027re asked for the standard deviation."},{"Start":"03:42.520 ","End":"03:50.000","Text":"The standard deviation of x is known to be equal to the square root of the variance 1.6,"},{"Start":"03:50.000 ","End":"03:54.900","Text":"and that turns out to be 1.26."},{"Start":"03:55.040 ","End":"03:58.295","Text":"We\u0027ve answered the question,"},{"Start":"03:58.295 ","End":"04:05.640","Text":"where the expectation of X is 8 people and the standard deviation is 1.26,"},{"Start":"04:05.640 ","End":"04:12.660","Text":"and 1.26 is the average disbursement of the number of people around the expectation."}],"ID":14137},{"Watched":false,"Name":"Exercise 8 - Parts a-c","Duration":"6m 58s","ChapterTopicVideoID":13497,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"In this question, we\u0027ll be talking about unemployment;"},{"Start":"00:04.350 ","End":"00:08.460","Text":"10 percent of a population in a country are unemployed,"},{"Start":"00:08.460 ","End":"00:12.810","Text":"5 people are randomly selected from that population."},{"Start":"00:12.810 ","End":"00:16.530","Text":"X is defined as the number of unemployed people in our sample,"},{"Start":"00:16.530 ","End":"00:20.310","Text":"and we\u0027re asked, what\u0027s the probability distribution of X?"},{"Start":"00:20.310 ","End":"00:24.630","Text":"Well, my answer is that this is a binomial distribution,"},{"Start":"00:24.630 ","End":"00:28.685","Text":"and not only because we\u0027re dealing with it here in this chapter,"},{"Start":"00:28.685 ","End":"00:36.175","Text":"but we have the definitions and we have other conditions."},{"Start":"00:36.175 ","End":"00:38.660","Text":"Let\u0027s just check it out."},{"Start":"00:38.660 ","End":"00:43.475","Text":"Let\u0027s look at the conditions for a binomial distribution that\u0027s here."},{"Start":"00:43.475 ","End":"00:46.850","Text":"Here, we\u0027re talking about first condition being"},{"Start":"00:46.850 ","End":"00:50.120","Text":"the same Bernoulli trial is repeated independently."},{"Start":"00:50.120 ","End":"00:57.740","Text":"Again, when we have a small or very small sample size from a very large population,"},{"Start":"00:57.740 ","End":"01:03.630","Text":"we\u0027re sampling people from country\u0027s population,"},{"Start":"01:03.630 ","End":"01:07.580","Text":"then we can assume that they\u0027re independent of each other."},{"Start":"01:07.580 ","End":"01:12.500","Text":"Do we have a Bernoulli trial? Yes, we do."},{"Start":"01:12.500 ","End":"01:19.975","Text":"Let\u0027s define success as a person who\u0027s"},{"Start":"01:19.975 ","End":"01:28.230","Text":"unemployed and failure as"},{"Start":"01:28.230 ","End":"01:31.210","Text":"a person who\u0027s employed."},{"Start":"01:31.940 ","End":"01:36.325","Text":"What\u0027s the probability of a person being unemployed?"},{"Start":"01:36.325 ","End":"01:38.415","Text":"That\u0027s 10 percent."},{"Start":"01:38.415 ","End":"01:46.600","Text":"We\u0027re taking the relative frequency of the unemployed people in the population."},{"Start":"01:46.600 ","End":"01:49.985","Text":"That means P equals 0.1."},{"Start":"01:49.985 ","End":"01:57.365","Text":"That means that the probability of a person being employed is 1 minus P,"},{"Start":"01:57.365 ","End":"01:59.150","Text":"and we\u0027ll call that q."},{"Start":"01:59.150 ","End":"02:02.100","Text":"That equals to 0.9."},{"Start":"02:02.270 ","End":"02:05.115","Text":"The trial is repeated n times."},{"Start":"02:05.115 ","End":"02:08.735","Text":"Obviously, we\u0027re taking 5 people 1 at a time,"},{"Start":"02:08.735 ","End":"02:11.510","Text":"and we\u0027re sampling each person."},{"Start":"02:11.510 ","End":"02:19.015","Text":"Each person has a probability of 0.01 to be unemployed and 0.9 to be employed,"},{"Start":"02:19.015 ","End":"02:24.150","Text":"so n in our case equals 5."},{"Start":"02:24.150 ","End":"02:28.850","Text":"X is defined as the total number of successes obtained,"},{"Start":"02:28.850 ","End":"02:33.920","Text":"so X equals the number of success"},{"Start":"02:33.920 ","End":"02:42.820","Text":"or the number of unemployed people."},{"Start":"02:43.480 ","End":"02:46.505","Text":"We\u0027ve met other conditions here."},{"Start":"02:46.505 ","End":"02:51.515","Text":"Now we know that X is distributed binomially,"},{"Start":"02:51.515 ","End":"02:57.815","Text":"where n equals 5 and P equals 0.1."},{"Start":"02:57.815 ","End":"03:02.615","Text":"Here we\u0027ve answered question or section a."},{"Start":"03:02.615 ","End":"03:04.835","Text":"In this section we\u0027re asked,"},{"Start":"03:04.835 ","End":"03:09.650","Text":"what\u0027s the probability that exactly 1 unemployed person was selected?"},{"Start":"03:09.650 ","End":"03:21.630","Text":"We know that X is distributed binomially with n equals to 5 and P equals to 0.1."},{"Start":"03:22.070 ","End":"03:26.570","Text":"The probability of X being equal to k,"},{"Start":"03:26.570 ","End":"03:30.065","Text":"that equals to n on k,"},{"Start":"03:30.065 ","End":"03:33.485","Text":"n over k, p^k,"},{"Start":"03:33.485 ","End":"03:38.749","Text":"q^n minus k. In our situation,"},{"Start":"03:38.749 ","End":"03:40.775","Text":"let\u0027s just plug in the numbers."},{"Start":"03:40.775 ","End":"03:45.875","Text":"The probability where X equals to 1."},{"Start":"03:45.875 ","End":"03:49.160","Text":"That\u0027s what we\u0027re asked. 1 unemployed person right here."},{"Start":"03:49.160 ","End":"03:50.825","Text":"That equals to,"},{"Start":"03:50.825 ","End":"03:54.690","Text":"n equals 5 over 1."},{"Start":"03:54.820 ","End":"04:03.485","Text":"P is 0.1^k^1 times q."},{"Start":"04:03.485 ","End":"04:05.965","Text":"Now q is 0.9."},{"Start":"04:05.965 ","End":"04:08.570","Text":"That\u0027s 1 minus 0.1,"},{"Start":"04:08.570 ","End":"04:11.840","Text":"that\u0027s 0.9^n minus k. Well,"},{"Start":"04:11.840 ","End":"04:15.600","Text":"n minus k is 5 minus 1, that\u0027s 4."},{"Start":"04:15.980 ","End":"04:19.310","Text":"We don\u0027t have to use a lot of brain power,"},{"Start":"04:19.310 ","End":"04:22.460","Text":"all we need to do right now is to plug in all these numbers into"},{"Start":"04:22.460 ","End":"04:28.385","Text":"our calculator and figure out what this expression is."},{"Start":"04:28.385 ","End":"04:32.040","Text":"That comes out to 0.32805."},{"Start":"04:38.050 ","End":"04:40.460","Text":"In this section we\u0027re asked,"},{"Start":"04:40.460 ","End":"04:44.875","Text":"what\u0027s the probability that all of the selected people work?"},{"Start":"04:44.875 ","End":"04:54.335","Text":"Let\u0027s recap, X is distributed binomially with n equals 5 and p equals 0.1,"},{"Start":"04:54.335 ","End":"04:58.249","Text":"where the probability that X equals k,"},{"Start":"04:58.249 ","End":"05:02.750","Text":"that\u0027s a general formula for the binomial distribution."},{"Start":"05:02.750 ","End":"05:07.420","Text":"That\u0027s n over k, p^k,"},{"Start":"05:07.420 ","End":"05:12.400","Text":"q^n minus k. In our case,"},{"Start":"05:12.400 ","End":"05:17.270","Text":"we\u0027re looking for the probability that all the people work."},{"Start":"05:17.270 ","End":"05:21.910","Text":"X is defined as the number of unemployed people."},{"Start":"05:21.910 ","End":"05:23.800","Text":"If that\u0027s the case,"},{"Start":"05:23.800 ","End":"05:27.895","Text":"then we\u0027re looking for k being equal to 0,"},{"Start":"05:27.895 ","End":"05:32.180","Text":"that none of the people in the sample are unemployed,"},{"Start":"05:32.180 ","End":"05:35.560","Text":"or that all of the selected people work."},{"Start":"05:35.560 ","End":"05:42.170","Text":"We\u0027re basically looking for the probability of X being equal to 0."},{"Start":"05:42.170 ","End":"05:44.775","Text":"Let\u0027s plug in the numbers."},{"Start":"05:44.775 ","End":"05:48.605","Text":"That\u0027s 5 over 0 times"},{"Start":"05:48.605 ","End":"05:56.420","Text":"0.1^0 times 0.9^n minus k,"},{"Start":"05:56.420 ","End":"05:59.270","Text":"that\u0027s 5 minus 0, that\u0027s 5."},{"Start":"05:59.270 ","End":"06:02.730","Text":"That turns out to be 0.59049."},{"Start":"06:06.590 ","End":"06:09.860","Text":"Let\u0027s just take a look at the components,"},{"Start":"06:09.860 ","End":"06:13.695","Text":"5 over 0, that turns out to be 1."},{"Start":"06:13.695 ","End":"06:17.840","Text":"Any number to the power of 0 that turns out to be 1,"},{"Start":"06:17.840 ","End":"06:22.810","Text":"so we have 1 times 1 times 0.9^5."},{"Start":"06:22.810 ","End":"06:28.400","Text":"Intuitively, it makes sense because what\u0027s the probability of the person working?"},{"Start":"06:28.400 ","End":"06:33.125","Text":"That\u0027s 0.9. How many people do we have in the sample?"},{"Start":"06:33.125 ","End":"06:35.330","Text":"We have 5 people in the sample,"},{"Start":"06:35.330 ","End":"06:38.305","Text":"and they\u0027re independent of each other."},{"Start":"06:38.305 ","End":"06:41.465","Text":"If all the people are working,"},{"Start":"06:41.465 ","End":"06:49.280","Text":"then the probability of them doing that is 0.9 times 0.9 and so on and so forth, 5 times."},{"Start":"06:49.280 ","End":"06:56.980","Text":"That\u0027s 0.9^5, and that equals 0.59049."}],"ID":14138},{"Watched":false,"Name":"Exercise 8 - Parts d-f","Duration":"5m 13s","ChapterTopicVideoID":13498,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.020 ","End":"00:02.670","Text":"In this section, we\u0027re asked what\u0027s"},{"Start":"00:02.670 ","End":"00:06.195","Text":"the probability that 3 of the people in the sample work?"},{"Start":"00:06.195 ","End":"00:09.360","Text":"Well, if 3 of the people in the sample work,"},{"Start":"00:09.360 ","End":"00:11.955","Text":"that means the 2 do not work."},{"Start":"00:11.955 ","End":"00:17.445","Text":"That means that in our case, k equals 2."},{"Start":"00:17.445 ","End":"00:24.960","Text":"Again, we have x is distributed binomially with n equals 5,"},{"Start":"00:24.960 ","End":"00:28.305","Text":"and p equals 0.1."},{"Start":"00:28.305 ","End":"00:36.030","Text":"Where the probability where x equals k equals n over k,"},{"Start":"00:36.030 ","End":"00:40.845","Text":"p^k, q^n minus k,"},{"Start":"00:40.845 ","End":"00:44.625","Text":"and we said that k equals 2."},{"Start":"00:44.625 ","End":"00:53.654","Text":"We want to find out what\u0027s the probability of 3 people working which means"},{"Start":"00:53.654 ","End":"00:58.430","Text":"that we\u0027re looking at 2 people that are not working because"},{"Start":"00:58.430 ","End":"01:03.740","Text":"x again is defined as the number of unemployed people."},{"Start":"01:03.740 ","End":"01:11.630","Text":"We\u0027re looking at the probability of x equaling 2 is plug in the numbers."},{"Start":"01:11.630 ","End":"01:17.280","Text":"That\u0027s 5 over 2, 0.1^2,"},{"Start":"01:17.280 ","End":"01:21.975","Text":"0.9^3, that\u0027s 5 minus 2,"},{"Start":"01:21.975 ","End":"01:27.910","Text":"and that comes out to 0.0729."},{"Start":"01:29.920 ","End":"01:32.195","Text":"In this section, we\u0027re asked,"},{"Start":"01:32.195 ","End":"01:36.950","Text":"what\u0027s the probability that at least 1 person in the sample is unemployed?"},{"Start":"01:36.950 ","End":"01:42.890","Text":"Well, when we\u0027re dealing with the question of at least 1 person,"},{"Start":"01:42.890 ","End":"01:48.300","Text":"we\u0027re looking at x being greater or equal to 1."},{"Start":"01:49.130 ","End":"01:53.900","Text":"Now let\u0027s take a look at the values of x. X can take on the values of"},{"Start":"01:53.900 ","End":"02:02.660","Text":"0,1,2,3,4,5 where x is the number of unemployed people."},{"Start":"02:02.660 ","End":"02:08.160","Text":"Now we\u0027re asked, what\u0027s the probability that at least 1 person in the samples is"},{"Start":"02:08.160 ","End":"02:13.955","Text":"unemployed so we\u0027re looking at these guys right here where x equals 1 or 2,"},{"Start":"02:13.955 ","End":"02:16.070","Text":"or 3, or 4 or 5."},{"Start":"02:16.070 ","End":"02:21.860","Text":"Now, instead of calculating the probabilities for each 1 of these values,"},{"Start":"02:21.860 ","End":"02:23.330","Text":"and then adding them up,"},{"Start":"02:23.330 ","End":"02:26.855","Text":"it\u0027s much easier to take the complimentary set."},{"Start":"02:26.855 ","End":"02:34.780","Text":"That would equal to 1 minus the probability of x being equal to 0."},{"Start":"02:34.970 ","End":"02:39.205","Text":"Now, if we remember,"},{"Start":"02:39.205 ","End":"02:49.620","Text":"we\u0027ve already calculated the probability of x being equal to 0 so that equals 0.59049."},{"Start":"02:53.200 ","End":"02:59.075","Text":"1 minus 0.59049."},{"Start":"02:59.075 ","End":"03:00.845","Text":"Well, that equals to"},{"Start":"03:00.845 ","End":"03:10.130","Text":"0.40954 and that\u0027s the probability"},{"Start":"03:10.130 ","End":"03:14.879","Text":"of at least 1 person in the sample being unemployed."},{"Start":"03:15.370 ","End":"03:19.070","Text":"In this section, we\u0027re asked what are the expectation and"},{"Start":"03:19.070 ","End":"03:22.370","Text":"variance of the number of unemployed people in the sample?"},{"Start":"03:22.370 ","End":"03:27.500","Text":"Well, again, let\u0027s recall x is distributed binomially,"},{"Start":"03:27.500 ","End":"03:29.788","Text":"where n equals 5,"},{"Start":"03:29.788 ","End":"03:33.425","Text":"and p equals 0.1."},{"Start":"03:33.425 ","End":"03:38.570","Text":"Now, remember that the expectation of x is"},{"Start":"03:38.570 ","End":"03:46.210","Text":"defined as n times p. Let\u0027s plug in the numbers n equals 5,"},{"Start":"03:46.210 ","End":"03:53.565","Text":"p equals 0.1, and that comes out to 0.5."},{"Start":"03:53.565 ","End":"03:56.235","Text":"In a sample of 5,"},{"Start":"03:56.235 ","End":"04:01.610","Text":"with the probability of 0.1 being unemployed,"},{"Start":"04:01.610 ","End":"04:06.275","Text":"then we are expected that half a person will be unemployed in that sample."},{"Start":"04:06.275 ","End":"04:08.990","Text":"Let\u0027s take a look at the variance."},{"Start":"04:08.990 ","End":"04:13.940","Text":"The variance is defined as n times p times q."},{"Start":"04:13.940 ","End":"04:18.710","Text":"Again, that\u0027s the definition in a binomial distribution."},{"Start":"04:18.710 ","End":"04:20.390","Text":"Let\u0027s plug in the numbers."},{"Start":"04:20.390 ","End":"04:24.860","Text":"That\u0027s 5 times 0.1 times 0.9,"},{"Start":"04:24.860 ","End":"04:29.900","Text":"and that turns out to be 0.45."},{"Start":"04:29.900 ","End":"04:35.570","Text":"Now, there\u0027s an alternative way of calculating the expectation, and variance."},{"Start":"04:35.570 ","End":"04:38.690","Text":"Let\u0027s remember how we did this previously."},{"Start":"04:38.690 ","End":"04:42.665","Text":"We had x, we had p of x,"},{"Start":"04:42.665 ","End":"04:48.650","Text":"and we calculated the probabilities for each value of x,"},{"Start":"04:48.650 ","End":"04:52.820","Text":"where x could have been 0,1,2,3,4,"},{"Start":"04:52.820 ","End":"04:54.365","Text":"and 5 in our case."},{"Start":"04:54.365 ","End":"04:57.985","Text":"Now, that\u0027s too much work."},{"Start":"04:57.985 ","End":"05:00.065","Text":"Then why should we do this?"},{"Start":"05:00.065 ","End":"05:02.540","Text":"We already have the shortcuts right here."},{"Start":"05:02.540 ","End":"05:06.155","Text":"We have the proper equations for the expectation,"},{"Start":"05:06.155 ","End":"05:09.305","Text":"and variance of the binomial distributions."},{"Start":"05:09.305 ","End":"05:13.440","Text":"There it is and good luck."}],"ID":14139},{"Watched":false,"Name":"Exercise 9","Duration":"8m 27s","ChapterTopicVideoID":13499,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"In this question, we\u0027ll be talking about people who have smartphones."},{"Start":"00:03.960 ","End":"00:08.370","Text":"According to the data provided by the Ministry of Communications,"},{"Start":"00:08.370 ","End":"00:11.670","Text":"70 percent of the population have a smartphone."},{"Start":"00:11.670 ","End":"00:14.715","Text":"Now, 10 people are randomly selected."},{"Start":"00:14.715 ","End":"00:18.990","Text":"We define X as the number of people sampled who have smartphones."},{"Start":"00:18.990 ","End":"00:23.070","Text":"We\u0027re asked, what\u0027s the probability distribution of X?"},{"Start":"00:23.070 ","End":"00:26.715","Text":"Well, X has a binomial distribution."},{"Start":"00:26.715 ","End":"00:28.050","Text":"How do I know that?"},{"Start":"00:28.050 ","End":"00:31.320","Text":"Because it meets the criteria for binomial distribution."},{"Start":"00:31.320 ","End":"00:35.460","Text":"But let\u0027s take a look at the criteria. Here it is."},{"Start":"00:35.460 ","End":"00:40.310","Text":"The first criteria right here states"},{"Start":"00:40.310 ","End":"00:45.470","Text":"that we have the same Bernoulli trial that\u0027s repeated independently."},{"Start":"00:45.470 ","End":"00:48.995","Text":"Now, are we talking about a Bernoulli trial?"},{"Start":"00:48.995 ","End":"00:51.155","Text":"Well, of course, and why is that?"},{"Start":"00:51.155 ","End":"00:54.380","Text":"Because we\u0027re talking about success and failures."},{"Start":"00:54.380 ","End":"01:03.300","Text":"The success is defined as having a smartphone. Here it is."},{"Start":"01:03.300 ","End":"01:06.620","Text":"What\u0027s the probability of having a smartphone?"},{"Start":"01:06.620 ","End":"01:10.575","Text":"That\u0027s 0.7, it\u0027s given right here."},{"Start":"01:10.575 ","End":"01:12.900","Text":"What\u0027s failure?"},{"Start":"01:13.600 ","End":"01:18.630","Text":"The failure is not having a smartphone."},{"Start":"01:19.900 ","End":"01:28.730","Text":"That means that the probability of not having a smartphone is 0.3."},{"Start":"01:28.730 ","End":"01:32.825","Text":"Now, let\u0027s talk about independence."},{"Start":"01:32.825 ","End":"01:40.295","Text":"Again, if we\u0027re sampling a very small number from a very large population,"},{"Start":"01:40.295 ","End":"01:46.355","Text":"then the assumption is that all the objects in the sample are independent."},{"Start":"01:46.355 ","End":"01:51.395","Text":"In this case, we\u0027re sampling 10 people from a population."},{"Start":"01:51.395 ","End":"01:56.170","Text":"That means that the people are independent of each other."},{"Start":"01:56.170 ","End":"01:59.090","Text":"So the first criteria is met."},{"Start":"01:59.090 ","End":"02:02.285","Text":"Now the trial is repeated n times."},{"Start":"02:02.285 ","End":"02:04.790","Text":"Here, n equals, how much?"},{"Start":"02:04.790 ","End":"02:07.445","Text":"10 people, and why is that?"},{"Start":"02:07.445 ","End":"02:12.305","Text":"Because we\u0027re sampling 1 person after the other with the same probability."},{"Start":"02:12.305 ","End":"02:14.345","Text":"If we sample 1 person,"},{"Start":"02:14.345 ","End":"02:16.970","Text":"there\u0027s a probability of 70 percent that he\u0027ll"},{"Start":"02:16.970 ","End":"02:20.485","Text":"have a smartphone and 30 percent that he won\u0027t."},{"Start":"02:20.485 ","End":"02:22.940","Text":"Now, what\u0027s the third criteria?"},{"Start":"02:22.940 ","End":"02:29.840","Text":"That X is defined as the total number of successes obtained and what do we ask for?"},{"Start":"02:29.840 ","End":"02:34.315","Text":"We define X as the number of people sampled who have smartphones."},{"Start":"02:34.315 ","End":"02:45.270","Text":"Excellent. So X is defined as number of people who have a smartphone."},{"Start":"02:49.130 ","End":"02:56.960","Text":"We have here all the conditions that point to"},{"Start":"02:56.960 ","End":"03:00.500","Text":"a binomial distribution where n equals"},{"Start":"03:00.500 ","End":"03:08.240","Text":"10 and p equals 0.7 and there we go."},{"Start":"03:08.240 ","End":"03:10.325","Text":"In this section, we\u0027re asked,"},{"Start":"03:10.325 ","End":"03:14.885","Text":"what\u0027s the probability that 8 people in the sample have a smartphone?"},{"Start":"03:14.885 ","End":"03:17.900","Text":"Well, let\u0027s write this out again."},{"Start":"03:17.900 ","End":"03:26.220","Text":"X is distributed with a binomial distribution where n equals 10 and p equals 0.7."},{"Start":"03:26.220 ","End":"03:34.485","Text":"We\u0027re asked, what\u0027s the probability where X equals 8?"},{"Start":"03:34.485 ","End":"03:40.055","Text":"Now, the probability X equaling k,"},{"Start":"03:40.055 ","End":"03:43.879","Text":"that\u0027s a general equation for the binomial distribution."},{"Start":"03:43.879 ","End":"03:46.655","Text":"That equals to n over k,"},{"Start":"03:46.655 ","End":"03:49.670","Text":"p to the power of k,"},{"Start":"03:49.670 ","End":"03:55.625","Text":"q to the power of n minus k. If we\u0027re asked,"},{"Start":"03:55.625 ","End":"03:59.300","Text":"what\u0027s the probability of X equaling 8?"},{"Start":"03:59.300 ","End":"04:01.880","Text":"Let\u0027s plug in the numbers."},{"Start":"04:01.880 ","End":"04:07.070","Text":"That equals to 10/8,"},{"Start":"04:07.070 ","End":"04:11.840","Text":"10 is n and 8 is k. Now,"},{"Start":"04:11.840 ","End":"04:14.750","Text":"p is 0.7 to the power of k,"},{"Start":"04:14.750 ","End":"04:20.660","Text":"to the power of 8 times 0.3^10 minus 8,"},{"Start":"04:20.660 ","End":"04:27.450","Text":"that\u0027s 2, and that equals to 0.2335."},{"Start":"04:30.460 ","End":"04:34.325","Text":"In this section, we\u0027re asked what\u0027s the probability"},{"Start":"04:34.325 ","End":"04:37.790","Text":"that at least 9 people in the sample have a smartphone?"},{"Start":"04:37.790 ","End":"04:40.550","Text":"Let\u0027s write out our data again."},{"Start":"04:40.550 ","End":"04:43.865","Text":"X is distributed binomially,"},{"Start":"04:43.865 ","End":"04:50.000","Text":"where n equals 10 and p equals 0.7 and we know"},{"Start":"04:50.000 ","End":"04:57.470","Text":"that the general equation of the binomial distribution is n over k,"},{"Start":"04:57.470 ","End":"05:05.645","Text":"p to the power of k times q to the power of n minus k. Now, we\u0027re asked,"},{"Start":"05:05.645 ","End":"05:10.430","Text":"what\u0027s the probability that at least 9 people in the sample,"},{"Start":"05:10.430 ","End":"05:14.420","Text":"that means that X is greater or equal to 9."},{"Start":"05:14.420 ","End":"05:19.595","Text":"Because we\u0027ve sampled only 10 people,"},{"Start":"05:19.595 ","End":"05:24.620","Text":"this probability equals to the probability of X being equal to"},{"Start":"05:24.620 ","End":"05:30.540","Text":"9 plus the probability of X being equal to 10."},{"Start":"05:30.590 ","End":"05:33.470","Text":"Let\u0027s plug in the numbers."},{"Start":"05:33.470 ","End":"05:36.320","Text":"That equals to n over k,"},{"Start":"05:36.320 ","End":"05:40.815","Text":"that\u0027s 10/9, that\u0027s for this guy right here,"},{"Start":"05:40.815 ","End":"05:48.840","Text":"times 0.7^9 times 0.3^1,"},{"Start":"05:48.840 ","End":"05:53.010","Text":"k is 9 and this 1 is 10 minus 9,"},{"Start":"05:53.010 ","End":"05:55.575","Text":"n minus k, plus,"},{"Start":"05:55.575 ","End":"05:57.765","Text":"now let\u0027s figure this 1 out,"},{"Start":"05:57.765 ","End":"06:00.935","Text":"that\u0027s 10/10,"},{"Start":"06:00.935 ","End":"06:10.555","Text":"0.7^10 times 0.3^0."},{"Start":"06:10.555 ","End":"06:13.360","Text":"When you calculate this out,"},{"Start":"06:13.360 ","End":"06:19.760","Text":"this comes out to 0.1493."},{"Start":"06:19.980 ","End":"06:22.540","Text":"In this section, we\u0027re asked,"},{"Start":"06:22.540 ","End":"06:24.040","Text":"what are the expectations and"},{"Start":"06:24.040 ","End":"06:28.150","Text":"standard deviation of the number of people sampled who have a smartphone?"},{"Start":"06:28.150 ","End":"06:38.920","Text":"Again, X is distributed binomially with n equaling 10 and p equaling 0.7."},{"Start":"06:38.920 ","End":"06:43.290","Text":"Now, we know what the expectation of X is,"},{"Start":"06:43.290 ","End":"06:49.055","Text":"that equals to n times p. Let\u0027s just plug in the numbers,"},{"Start":"06:49.055 ","End":"06:54.470","Text":"n equals 10 times p which is 0.7."},{"Start":"06:54.470 ","End":"06:57.740","Text":"That means that we\u0027re talking about 7."},{"Start":"06:57.740 ","End":"07:05.340","Text":"The expectation for people who have smartphones in our sample are 7 people."},{"Start":"07:05.500 ","End":"07:09.515","Text":"Now, let\u0027s take a look at the variance."},{"Start":"07:09.515 ","End":"07:15.745","Text":"The variance of X is defined as n times p times q,"},{"Start":"07:15.745 ","End":"07:17.775","Text":"and let\u0027s plug in the numbers,"},{"Start":"07:17.775 ","End":"07:20.120","Text":"and is 10 times p,"},{"Start":"07:20.120 ","End":"07:23.150","Text":"that\u0027s 0.7 times q,"},{"Start":"07:23.150 ","End":"07:29.045","Text":"that\u0027s 0.3, and that equals to 2.1."},{"Start":"07:29.045 ","End":"07:30.530","Text":"But what are the units?"},{"Start":"07:30.530 ","End":"07:35.690","Text":"The units are people squared."},{"Start":"07:35.690 ","End":"07:38.090","Text":"Now this isn\u0027t very intuitive."},{"Start":"07:38.090 ","End":"07:40.336","Text":"Let\u0027s take the standard deviation,"},{"Start":"07:40.336 ","End":"07:41.840","Text":"that\u0027s what we\u0027re asked for."},{"Start":"07:41.840 ","End":"07:48.845","Text":"The standard deviation of X is equal to the square root of the variance,"},{"Start":"07:48.845 ","End":"07:56.090","Text":"square root of 2.1, and that equals to 1.449."},{"Start":"07:56.090 ","End":"07:58.415","Text":"The units here are people."},{"Start":"07:58.415 ","End":"08:01.460","Text":"Much more intuitive, but what does this mean?"},{"Start":"08:01.460 ","End":"08:04.040","Text":"It means that the expectation of people who have"},{"Start":"08:04.040 ","End":"08:07.309","Text":"a smartphone in the sample are 7 people,"},{"Start":"08:07.309 ","End":"08:09.965","Text":"and what we have here,"},{"Start":"08:09.965 ","End":"08:13.460","Text":"the standard deviation is the average dispersement of people"},{"Start":"08:13.460 ","End":"08:18.740","Text":"around the expectation, that\u0027s 1.449 people."},{"Start":"08:18.740 ","End":"08:21.110","Text":"Here we have it. We\u0027ve answered"},{"Start":"08:21.110 ","End":"08:28.170","Text":"both the questions for the expectation and the variance of X."}],"ID":14140},{"Watched":false,"Name":"Exercise 10 - Parts a-b","Duration":"5m 17s","ChapterTopicVideoID":13500,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.140 ","End":"00:04.260","Text":"In this question, we\u0027ll be talking about slot machines."},{"Start":"00:04.260 ","End":"00:08.970","Text":"In a casino, there\u0027s 1 row of 6 identical slot machines."},{"Start":"00:08.970 ","End":"00:12.135","Text":"Playing a game on 1 of the machine costs 5 dollars."},{"Start":"00:12.135 ","End":"00:16.920","Text":"The probability of winning $20 on each machine is 10 percent."},{"Start":"00:16.920 ","End":"00:19.875","Text":"The probability of losing is 90 percent."},{"Start":"00:19.875 ","End":"00:22.410","Text":"Now a gambler enters the casino,"},{"Start":"00:22.410 ","End":"00:25.245","Text":"and places $5 in each of the 6 machines."},{"Start":"00:25.245 ","End":"00:31.305","Text":"We\u0027re asked, what\u0027s the probability that the gambler will lose on all the machines?"},{"Start":"00:31.305 ","End":"00:35.830","Text":"Well, here we see that we\u0027re dealing with a binomial distribution."},{"Start":"00:35.830 ","End":"00:39.275","Text":"Now, in order to just prove that,"},{"Start":"00:39.275 ","End":"00:45.495","Text":"we have to make sure that it meets all the criteria that we talked about previously."},{"Start":"00:45.495 ","End":"00:47.604","Text":"Let\u0027s look at the criteria,"},{"Start":"00:47.604 ","End":"00:49.785","Text":"or the conditions, and here they are."},{"Start":"00:49.785 ","End":"00:53.540","Text":"The first 1 states that we have Bernoulli trials,"},{"Start":"00:53.540 ","End":"00:56.105","Text":"and they\u0027re independent of each other."},{"Start":"00:56.105 ","End":"00:59.150","Text":"Well, when we\u0027re talking about a Bernoulli trial,"},{"Start":"00:59.150 ","End":"01:04.410","Text":"we\u0027re talking about success and failure."},{"Start":"01:04.610 ","End":"01:09.620","Text":"Now success and failure of y in this game,"},{"Start":"01:09.620 ","End":"01:11.990","Text":"in our situation, we\u0027re talking about winning or"},{"Start":"01:11.990 ","End":"01:15.500","Text":"losing when playing the slot machine. That\u0027s a win."},{"Start":"01:15.500 ","End":"01:16.880","Text":"Success is a win,"},{"Start":"01:16.880 ","End":"01:19.685","Text":"and failure is a lose."},{"Start":"01:19.685 ","End":"01:21.980","Text":"What\u0027s the probability of winning?"},{"Start":"01:21.980 ","End":"01:27.165","Text":"Well, probability is 0.1 to win,"},{"Start":"01:27.165 ","End":"01:31.400","Text":"and q, which is the probability of losing,"},{"Start":"01:31.400 ","End":"01:35.165","Text":"that\u0027s 1 minus p, that\u0027s 0.9."},{"Start":"01:35.165 ","End":"01:37.730","Text":"Now, are they independent?"},{"Start":"01:37.730 ","End":"01:41.675","Text":"Well, the slot machines do work independent of each other,"},{"Start":"01:41.675 ","End":"01:45.065","Text":"and we can infer that they are independent."},{"Start":"01:45.065 ","End":"01:47.660","Text":"Now, the trial is repeated n times."},{"Start":"01:47.660 ","End":"01:52.744","Text":"In here, we\u0027re playing the 6 slot machines."},{"Start":"01:52.744 ","End":"01:57.685","Text":"That means that n equals 6 right here."},{"Start":"01:57.685 ","End":"02:02.480","Text":"The last criteria, x is defined as the total number of successes."},{"Start":"02:02.480 ","End":"02:09.405","Text":"I want to know how much I want when I play the 6th slot machines."},{"Start":"02:09.405 ","End":"02:15.540","Text":"That means that x is defined as the number of wins."},{"Start":"02:17.510 ","End":"02:21.980","Text":"Because I\u0027ve met all the criteria,"},{"Start":"02:21.980 ","End":"02:26.450","Text":"I know that x is distributed"},{"Start":"02:26.450 ","End":"02:32.680","Text":"binomially with n equals 6 and p equals what?"},{"Start":"02:32.680 ","End":"02:36.690","Text":"P equals 0.1."},{"Start":"02:38.660 ","End":"02:41.185","Text":"What do we ask in section a?"},{"Start":"02:41.185 ","End":"02:47.352","Text":"We\u0027re asked, what\u0027s the probability that the gambler will lose on all the machines?"},{"Start":"02:47.352 ","End":"02:51.885","Text":"We\u0027re defining x as the number of wins,"},{"Start":"02:51.885 ","End":"02:57.309","Text":"and we want to know what\u0027s the probability that the gambler will lose on all machines?"},{"Start":"02:57.309 ","End":"03:06.070","Text":"That means that we\u0027re looking at the probability where x equals 0. Why is that?"},{"Start":"03:06.070 ","End":"03:08.185","Text":"Because we don\u0027t want any wins."},{"Start":"03:08.185 ","End":"03:10.480","Text":"We want all the machines to lose."},{"Start":"03:10.480 ","End":"03:17.920","Text":"Now, do we remember what the equation is for the binomial probability?"},{"Start":"03:17.920 ","End":"03:21.665","Text":"Well, that equals to n over k,"},{"Start":"03:21.665 ","End":"03:27.500","Text":"p^k, q^n minus k. Well,"},{"Start":"03:27.500 ","End":"03:30.565","Text":"in our case, k equals 0."},{"Start":"03:30.565 ","End":"03:34.070","Text":"So the probability of X equaling 0,"},{"Start":"03:34.070 ","End":"03:35.660","Text":"that\u0027s n over k,"},{"Start":"03:35.660 ","End":"03:40.190","Text":"n is 6 and k is 0."},{"Start":"03:40.190 ","End":"03:46.490","Text":"The probability of win is 0.1^k, that\u0027s 0,"},{"Start":"03:46.490 ","End":"03:51.260","Text":"times 0.9^n minus k,"},{"Start":"03:51.260 ","End":"03:53.450","Text":"that\u0027s 6 minus 0,"},{"Start":"03:53.450 ","End":"03:59.400","Text":"that 6, and that equals to 0.5314."},{"Start":"04:05.320 ","End":"04:09.140","Text":"In this section, we\u0027re asked what\u0027s the probability that"},{"Start":"04:09.140 ","End":"04:13.025","Text":"the gambler will win on exactly 2 machines?"},{"Start":"04:13.025 ","End":"04:15.264","Text":"Well, let\u0027s recall,"},{"Start":"04:15.264 ","End":"04:19.388","Text":"X is distributed binomially with n equals 6,"},{"Start":"04:19.388 ","End":"04:24.040","Text":"and the probability of winning on a machine is 0.1."},{"Start":"04:24.040 ","End":"04:27.590","Text":"We know that the probability of x equaling k,"},{"Start":"04:27.590 ","End":"04:31.130","Text":"that\u0027s a general equation for the binomial distribution."},{"Start":"04:31.130 ","End":"04:34.970","Text":"That\u0027s n on k, p^k,"},{"Start":"04:34.970 ","End":"04:39.215","Text":"q^n minus k. Now, what are we asked?"},{"Start":"04:39.215 ","End":"04:46.310","Text":"We\u0027re asked, what\u0027s the probability that the gambler will win on exactly 2 machines?"},{"Start":"04:46.310 ","End":"04:47.945","Text":"That means that here,"},{"Start":"04:47.945 ","End":"04:50.420","Text":"k equals to 2."},{"Start":"04:50.420 ","End":"04:52.970","Text":"X has to equal to 2."},{"Start":"04:52.970 ","End":"04:55.760","Text":"Now that equals, let\u0027s plug in the numbers,"},{"Start":"04:55.760 ","End":"05:00.738","Text":"n equals 6 onto 0.1,"},{"Start":"05:00.738 ","End":"05:07.860","Text":"that\u0027s p^2 times 0.9^6 minus 2,"},{"Start":"05:07.860 ","End":"05:13.660","Text":"that\u0027s 4, and that equals to 0.0984."}],"ID":14141},{"Watched":false,"Name":"Exercise 10 - Parts c-d","Duration":"12m 24s","ChapterTopicVideoID":13501,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.110 ","End":"00:03.660","Text":"In this section, we\u0027re asked what\u0027s the probability"},{"Start":"00:03.660 ","End":"00:06.675","Text":"that the gambler will win more money than he gambles?"},{"Start":"00:06.675 ","End":"00:09.330","Text":"Well, before we can answer this question,"},{"Start":"00:09.330 ","End":"00:12.480","Text":"let\u0027s just recall some of the data that we were given."},{"Start":"00:12.480 ","End":"00:18.870","Text":"The number of machines equals 6."},{"Start":"00:18.870 ","End":"00:21.750","Text":"We have 6 machines that the gambler plays simultaneously,"},{"Start":"00:21.750 ","End":"00:28.360","Text":"and the cost of playing 1 machine,"},{"Start":"00:32.150 ","End":"00:35.040","Text":"that equals to 5 dollars."},{"Start":"00:35.040 ","End":"00:37.965","Text":"If you recall, he just puts in 5 dollars in"},{"Start":"00:37.965 ","End":"00:41.460","Text":"each of the machines and he plays the machines simultaneously."},{"Start":"00:41.460 ","End":"00:44.910","Text":"What about his winnings?"},{"Start":"00:44.910 ","End":"00:51.250","Text":"The wins for 1 machine"},{"Start":"00:52.610 ","End":"00:55.934","Text":"is 20 dollars,"},{"Start":"00:55.934 ","End":"01:00.980","Text":"and the probability of winning this 20 dollars is 0.1 for 1 machine."},{"Start":"01:00.980 ","End":"01:07.620","Text":"If the gambler plays"},{"Start":"01:07.620 ","End":"01:14.390","Text":"6 machines simultaneously and he pays 5 dollars per machine,"},{"Start":"01:14.390 ","End":"01:18.829","Text":"then the cost of playing"},{"Start":"01:18.829 ","End":"01:27.245","Text":"6 machines equals 30 dollars."},{"Start":"01:27.245 ","End":"01:30.530","Text":"We\u0027re asked again, what\u0027s"},{"Start":"01:30.530 ","End":"01:34.550","Text":"the probability that the gambler will win more money than he gambles?"},{"Start":"01:34.550 ","End":"01:38.924","Text":"If x is the number of wins,"},{"Start":"01:38.924 ","End":"01:47.200","Text":"how many wins does the gambler need to have in order to get his investment back?"},{"Start":"01:47.200 ","End":"01:51.115","Text":"Well, if he wins 20 dollars on 1 machine,"},{"Start":"01:51.115 ","End":"01:55.690","Text":"then he needs to win at least 2 machines in order to get 40 dollars,"},{"Start":"01:55.690 ","End":"01:57.895","Text":"where 40 is greater than 30."},{"Start":"01:57.895 ","End":"02:03.250","Text":"Basically, we\u0027re looking for the probability of x,"},{"Start":"02:03.250 ","End":"02:08.215","Text":"the number of wins being greater or equal to 2. Again, why?"},{"Start":"02:08.215 ","End":"02:10.960","Text":"Because if he wins 2 games or more,"},{"Start":"02:10.960 ","End":"02:14.010","Text":"then he\u0027ll win 40 dollars,"},{"Start":"02:14.010 ","End":"02:16.080","Text":"2 times 20 dollars,"},{"Start":"02:16.080 ","End":"02:17.740","Text":"that\u0027s 40 dollars or more,"},{"Start":"02:17.740 ","End":"02:21.265","Text":"and that\u0027s greater than his investment of 30 dollars."},{"Start":"02:21.265 ","End":"02:23.665","Text":"This is what we\u0027re looking for."},{"Start":"02:23.665 ","End":"02:29.495","Text":"As we said, x can have the values of 0,"},{"Start":"02:29.495 ","End":"02:35.195","Text":"1, 2, 3, 4, 5, and 6."},{"Start":"02:35.195 ","End":"02:40.680","Text":"We\u0027re looking at this set right here,"},{"Start":"02:40.680 ","End":"02:43.530","Text":"where x is either 2 or 3,"},{"Start":"02:43.530 ","End":"02:45.490","Text":"or 4, or 5, or 6."},{"Start":"02:45.490 ","End":"02:48.830","Text":"In order to calculate that,"},{"Start":"02:48.830 ","End":"02:50.945","Text":"that takes a lot of effort and energy,"},{"Start":"02:50.945 ","End":"02:53.480","Text":"I prefer taking the complimentary set,"},{"Start":"02:53.480 ","End":"03:01.020","Text":"which is 1 minus the probability of x being less than or equal to 1."},{"Start":"03:03.250 ","End":"03:05.825","Text":"What does that equal to?"},{"Start":"03:05.825 ","End":"03:10.645","Text":"Well, that\u0027s 1 minus the probability of x"},{"Start":"03:10.645 ","End":"03:16.920","Text":"equaling 0 plus the probability of x equaling 1."},{"Start":"03:17.710 ","End":"03:28.200","Text":"We know that the probability of x equaling k equals n over k,"},{"Start":"03:28.480 ","End":"03:32.240","Text":"p^k, q^n minus k;"},{"Start":"03:32.240 ","End":"03:35.730","Text":"that\u0027s the binomial distribution."},{"Start":"03:36.290 ","End":"03:39.635","Text":"We need to remember that. Why is that?"},{"Start":"03:39.635 ","End":"03:42.410","Text":"Because when we plug in the numbers here,"},{"Start":"03:42.410 ","End":"03:44.525","Text":"we\u0027ll see how we can calculate things out."},{"Start":"03:44.525 ","End":"03:49.024","Text":"What\u0027s the probability of x equaling 0?"},{"Start":"03:49.024 ","End":"03:53.015","Text":"Well, if you recall, we\u0027ve calculated that in section a,"},{"Start":"03:53.015 ","End":"03:59.045","Text":"that equals to 0.5314."},{"Start":"03:59.045 ","End":"04:02.300","Text":"Let\u0027s plug in this number that equals right here,"},{"Start":"04:02.300 ","End":"04:07.820","Text":"1 minus the probability of x equaling 0,"},{"Start":"04:07.820 ","End":"04:14.540","Text":"that\u0027s 0.5314, plus the probability of x equaling 1."},{"Start":"04:14.540 ","End":"04:18.685","Text":"Well, let\u0027s plug in the numbers right here."},{"Start":"04:18.685 ","End":"04:22.335","Text":"n is 6, k is 1,"},{"Start":"04:22.335 ","End":"04:24.100","Text":"that\u0027s 6 over 1 there,"},{"Start":"04:24.100 ","End":"04:27.605","Text":"0.1, that\u0027s the probability of a win,"},{"Start":"04:27.605 ","End":"04:30.510","Text":"to the power of 1 times 0.9,"},{"Start":"04:30.510 ","End":"04:34.470","Text":"that\u0027s our q, to the power of n minus k,"},{"Start":"04:34.470 ","End":"04:37.600","Text":"that 6 minus 1, that\u0027s 5."},{"Start":"04:38.240 ","End":"04:40.970","Text":"Having done the calculation,"},{"Start":"04:40.970 ","End":"04:44.160","Text":"this comes out to 0.1143."},{"Start":"04:46.300 ","End":"04:49.910","Text":"This is the probability of"},{"Start":"04:49.910 ","End":"04:55.750","Text":"the gambler winning more money than he invests in playing the 6 games."},{"Start":"04:56.510 ","End":"04:58.810","Text":"In this section, we\u0027re asked,"},{"Start":"04:58.810 ","End":"05:02.855","Text":"what is the expectation and standard deviation of the gambler\u0027s net profits,"},{"Start":"05:02.855 ","End":"05:05.490","Text":"prizes minus the investment?"},{"Start":"05:05.870 ","End":"05:09.395","Text":"First, let\u0027s go over some data."},{"Start":"05:09.395 ","End":"05:11.630","Text":"Well, we have 6 machines,"},{"Start":"05:11.630 ","End":"05:14.630","Text":"that\u0027s n. We have x,"},{"Start":"05:14.630 ","End":"05:17.509","Text":"which is the number of wins."},{"Start":"05:17.509 ","End":"05:23.390","Text":"The probability of winning is 0.1,"},{"Start":"05:23.390 ","End":"05:34.090","Text":"and we know that x is binomially distributed with n equals 6 and p equals 0.1."},{"Start":"05:34.090 ","End":"05:42.364","Text":"We know that the cost of playing"},{"Start":"05:42.364 ","End":"05:51.890","Text":"1 game equals 5 dollars per game or per machine,"},{"Start":"05:51.890 ","End":"05:58.740","Text":"so the cost of playing 6 games,"},{"Start":"06:02.030 ","End":"06:05.085","Text":"that equals to 30 dollars."},{"Start":"06:05.085 ","End":"06:09.840","Text":"Why is that? So 5 dollars per game per machine times 6 machines,"},{"Start":"06:09.840 ","End":"06:10.890","Text":"that\u0027s 6 times 5,"},{"Start":"06:10.890 ","End":"06:12.330","Text":"so that\u0027s 30 dollars."},{"Start":"06:12.330 ","End":"06:14.865","Text":"How much do we win?"},{"Start":"06:14.865 ","End":"06:21.190","Text":"What we win is 20 dollars per game."},{"Start":"06:22.820 ","End":"06:25.820","Text":"What else do we know?"},{"Start":"06:25.820 ","End":"06:32.415","Text":"We know that the expectation of x is n times p,"},{"Start":"06:32.415 ","End":"06:36.664","Text":"because x is binomially distributed,"},{"Start":"06:36.664 ","End":"06:42.260","Text":"so that\u0027s 6 times 0.1 and that equals to 0.6."},{"Start":"06:42.260 ","End":"06:47.880","Text":"The variance of x equals n times p times q,"},{"Start":"06:47.880 ","End":"06:56.220","Text":"and that equals to 6 times 0.1 times 0.9 and that equals to 0.54."},{"Start":"06:56.260 ","End":"07:02.780","Text":"This is basically all the information that we have to date."},{"Start":"07:02.780 ","End":"07:04.520","Text":"What are we asked?"},{"Start":"07:04.520 ","End":"07:08.270","Text":"We\u0027re asked about the gambler\u0027s net profit,"},{"Start":"07:08.270 ","End":"07:10.085","Text":"not about his wins."},{"Start":"07:10.085 ","End":"07:16.100","Text":"That means that I have some sort of transformation;"},{"Start":"07:16.100 ","End":"07:20.930","Text":"that I have to take the number of wins and I have to transform them into"},{"Start":"07:20.930 ","End":"07:26.590","Text":"profits and that reminds me of a linear transformation."},{"Start":"07:26.590 ","End":"07:32.515","Text":"Let\u0027s just take a look at the steps required for linear transformation."},{"Start":"07:32.515 ","End":"07:34.760","Text":"Here are the steps."},{"Start":"07:34.760 ","End":"07:36.545","Text":"Well, as we said,"},{"Start":"07:36.545 ","End":"07:38.660","Text":"the first step is to recognize that we\u0027re"},{"Start":"07:38.660 ","End":"07:41.315","Text":"dealing with a linear transformation and I think we are."},{"Start":"07:41.315 ","End":"07:47.675","Text":"We have to take our winnings and we have to transform them into net profits."},{"Start":"07:47.675 ","End":"07:50.480","Text":"What are the transformation?"},{"Start":"07:50.480 ","End":"07:52.595","Text":"What is the transformation rule?"},{"Start":"07:52.595 ","End":"07:56.465","Text":"Well, if y is the transformation,"},{"Start":"07:56.465 ","End":"08:04.365","Text":"we have to take the number of wins and we have to transform that into money."},{"Start":"08:04.365 ","End":"08:08.970","Text":"How much did I win according to the number of wins?"},{"Start":"08:08.970 ","End":"08:12.515","Text":"Well, I know that I win 20 dollars per game."},{"Start":"08:12.515 ","End":"08:16.200","Text":"So it has to be 20 times x,"},{"Start":"08:16.200 ","End":"08:18.435","Text":"where x is the number of wins."},{"Start":"08:18.435 ","End":"08:21.559","Text":"But because I\u0027m dealing with profits,"},{"Start":"08:21.559 ","End":"08:25.340","Text":"I have to take away from my winnings my expenses."},{"Start":"08:25.340 ","End":"08:26.885","Text":"What are my expenses?"},{"Start":"08:26.885 ","End":"08:29.520","Text":"Well, that\u0027s 30 dollars."},{"Start":"08:30.020 ","End":"08:37.725","Text":"Why is that? Because I\u0027m playing 6 machines at 5 dollars again."},{"Start":"08:37.725 ","End":"08:42.560","Text":"This looks like a good linear transformation."},{"Start":"08:42.560 ","End":"08:47.990","Text":"But we have to compare that to our general form of"},{"Start":"08:47.990 ","End":"08:57.125","Text":"a linear transformation and that looks like y equals ax plus b."},{"Start":"08:57.125 ","End":"09:01.475","Text":"Is this in the form of this?"},{"Start":"09:01.475 ","End":"09:02.810","Text":"Well, yes, obviously."},{"Start":"09:02.810 ","End":"09:07.070","Text":"We can easily identify that a equals 20,"},{"Start":"09:07.070 ","End":"09:09.450","Text":"a is a multiplier of x,"},{"Start":"09:09.450 ","End":"09:14.135","Text":"so a equals 20, and b equals minus 30."},{"Start":"09:14.135 ","End":"09:21.650","Text":"Excellent. You\u0027ve recognized that we have a linear transformation."},{"Start":"09:21.650 ","End":"09:23.660","Text":"We wrote the rules,"},{"Start":"09:23.660 ","End":"09:28.585","Text":"we simplified everything although there wasn\u0027t that much to simplify,"},{"Start":"09:28.585 ","End":"09:33.420","Text":"but we did identify a and b."},{"Start":"09:33.420 ","End":"09:36.560","Text":"Now, let\u0027s get back to our question."},{"Start":"09:36.560 ","End":"09:42.050","Text":"We\u0027re asked, what are or what is the expectation of y?"},{"Start":"09:42.050 ","End":"09:44.360","Text":"Well, if we recall,"},{"Start":"09:44.360 ","End":"09:47.920","Text":"the expectation of y looks like this."},{"Start":"09:47.920 ","End":"09:53.240","Text":"That\u0027s a times the expectation of x plus b;"},{"Start":"09:53.240 ","End":"09:57.275","Text":"that\u0027s the expectation of the linear transformation,"},{"Start":"09:57.275 ","End":"10:02.575","Text":"that equals, let\u0027s plug in the numbers, a equals 20."},{"Start":"10:02.575 ","End":"10:04.530","Text":"What\u0027s the expectation of x?"},{"Start":"10:04.530 ","End":"10:08.070","Text":"Well, that\u0027s 0.6, that\u0027s right here."},{"Start":"10:08.070 ","End":"10:11.310","Text":"What\u0027s b? B is minus 30."},{"Start":"10:11.310 ","End":"10:15.080","Text":"It\u0027s 20 times 0.6 minus 30,"},{"Start":"10:15.080 ","End":"10:19.100","Text":"and that equals to minus 18."},{"Start":"10:19.100 ","End":"10:22.410","Text":"What\u0027s this minus 18?"},{"Start":"10:22.410 ","End":"10:24.930","Text":"That\u0027s minus 18 dollars,"},{"Start":"10:24.930 ","End":"10:32.860","Text":"which basically means that if I have 6 machines and I have to pay 5 dollars"},{"Start":"10:32.860 ","End":"10:36.690","Text":"a game or 5 dollars per machine and I\u0027m"},{"Start":"10:36.690 ","End":"10:41.855","Text":"winning 20 dollars per game but at probability of 0.1,"},{"Start":"10:41.855 ","End":"10:51.145","Text":"then I\u0027m expected to lose minus 18 dollars or I\u0027m expecting to lose 18 dollars."},{"Start":"10:51.145 ","End":"10:54.815","Text":"Now, let\u0027s take a look at the variance of y."},{"Start":"10:54.815 ","End":"10:57.320","Text":"Well, the variance of y or the variance of"},{"Start":"10:57.320 ","End":"11:03.030","Text":"a linear transformation is a squared times the variance of x."},{"Start":"11:03.140 ","End":"11:07.920","Text":"What\u0027s a? a is 20,"},{"Start":"11:07.920 ","End":"11:13.640","Text":"that\u0027s 20 squared times the variance of x times 0.54,"},{"Start":"11:13.640 ","End":"11:20.300","Text":"that\u0027s right here, and that equals to 215."},{"Start":"11:20.300 ","End":"11:22.085","Text":"What are the units here?"},{"Start":"11:22.085 ","End":"11:27.805","Text":"The units here are in dollars squared and that\u0027s not intuitive at all."},{"Start":"11:27.805 ","End":"11:33.050","Text":"Also, we\u0027re asked for the standard deviation, not the variance."},{"Start":"11:33.050 ","End":"11:42.900","Text":"The standard deviation of y is the square root of the variance,"},{"Start":"11:42.900 ","End":"11:50.190","Text":"215, and that equals to 14.697, and that\u0027s okay."},{"Start":"11:50.190 ","End":"11:53.760","Text":"Right here, we\u0027re talking about this being the units,"},{"Start":"11:53.760 ","End":"11:57.125","Text":"dollars instead of dollar square right here."},{"Start":"11:57.125 ","End":"11:59.475","Text":"This is dollars."},{"Start":"11:59.475 ","End":"12:02.075","Text":"Again, we\u0027ve answered the question."},{"Start":"12:02.075 ","End":"12:05.090","Text":"We\u0027ve calculated the expectation of y,"},{"Start":"12:05.090 ","End":"12:09.800","Text":"the linear transformation or the profits of the gambler,"},{"Start":"12:09.800 ","End":"12:11.640","Text":"that\u0027s minus 18 dollars,"},{"Start":"12:11.640 ","End":"12:14.280","Text":"he\u0027s expected to lose 18 dollars."},{"Start":"12:14.280 ","End":"12:18.920","Text":"We\u0027ve calculated the standard deviation of the linear transformation or"},{"Start":"12:18.920 ","End":"12:24.540","Text":"the profits and that equals to 14.697 dollars."}],"ID":14142},{"Watched":false,"Name":"Exercise 11","Duration":"7m 12s","ChapterTopicVideoID":13502,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"In this question, we\u0027ll be talking about levels of education."},{"Start":"00:03.540 ","End":"00:05.460","Text":"Now in a given country,"},{"Start":"00:05.460 ","End":"00:09.450","Text":"the distribution of the education of people over the age of 30 is"},{"Start":"00:09.450 ","End":"00:14.400","Text":"as follows: 10 percent of the population have a low level of education,"},{"Start":"00:14.400 ","End":"00:17.175","Text":"60 percent have a high school education,"},{"Start":"00:17.175 ","End":"00:21.345","Text":"20 percent are first-degree graduates from university,"},{"Start":"00:21.345 ","End":"00:25.025","Text":"and 10 percent are second-degree or higher graduates."},{"Start":"00:25.025 ","End":"00:30.890","Text":"Now, 20 people over the age of 30 are randomly selected and were asked,"},{"Start":"00:30.890 ","End":"00:35.870","Text":"what\u0027s the probability that 5 of them are university graduates?"},{"Start":"00:37.010 ","End":"00:41.035","Text":"In order to answer this question,"},{"Start":"00:41.035 ","End":"00:44.305","Text":"let\u0027s first do a little bit of definitions."},{"Start":"00:44.305 ","End":"00:48.050","Text":"Let\u0027s see about our Bernoulli trials."},{"Start":"00:48.050 ","End":"00:49.880","Text":"When we talk about Bernoulli trials,"},{"Start":"00:49.880 ","End":"00:52.460","Text":"we\u0027re talking about success and failure."},{"Start":"00:52.460 ","End":"00:55.255","Text":"In our case, what\u0027s success?"},{"Start":"00:55.255 ","End":"00:59.680","Text":"What success is being a graduate"},{"Start":"01:00.260 ","End":"01:07.809","Text":"and failure is not being a graduate."},{"Start":"01:09.530 ","End":"01:16.505","Text":"Graduate. Now, what do we know the probability"},{"Start":"01:16.505 ","End":"01:24.615","Text":"of being a graduate that\u0027s 0.2 plus 0.1 that equals to 0.3 and why is that?"},{"Start":"01:24.615 ","End":"01:29.130","Text":"Well, 0.2 are people with first degrees and 0.1"},{"Start":"01:29.130 ","End":"01:34.275","Text":"are people with second and third degrees, that\u0027s the probabilities."},{"Start":"01:34.275 ","End":"01:37.360","Text":"Now, the probability of failure,"},{"Start":"01:37.360 ","End":"01:40.885","Text":"that\u0027s q equals 1 minus p,"},{"Start":"01:40.885 ","End":"01:42.700","Text":"that equals 1 minus 0,"},{"Start":"01:42.700 ","End":"01:47.510","Text":"3, that equals to 0.7."},{"Start":"01:48.380 ","End":"01:55.910","Text":"We have here our success and failure where we define success,"},{"Start":"01:55.910 ","End":"01:57.210","Text":"we\u0027ve defined failure,"},{"Start":"01:57.210 ","End":"02:01.165","Text":"we\u0027ve define the probability of success and failure."},{"Start":"02:01.165 ","End":"02:03.140","Text":"What do we want to know?"},{"Start":"02:03.140 ","End":"02:07.250","Text":"Well, the next thing that we want to know is how many people we have."},{"Start":"02:07.250 ","End":"02:10.305","Text":"We have 20 in our sample,"},{"Start":"02:10.305 ","End":"02:16.415","Text":"where each 1 of the people that are sampled is considered a Bernoulli trial with"},{"Start":"02:16.415 ","End":"02:23.300","Text":"these probabilities 0.3 of having probability of 0.3 of having a degree of some sort,"},{"Start":"02:23.300 ","End":"02:25.120","Text":"and 0.7 of not."},{"Start":"02:25.120 ","End":"02:30.800","Text":"Now, each 1 of the people are independent of each other and again, why is that?"},{"Start":"02:30.800 ","End":"02:37.105","Text":"Because we have a small number of people from a whole population of the country."},{"Start":"02:37.105 ","End":"02:40.940","Text":"Again, that\u0027s, the assumption is that they\u0027re independent."},{"Start":"02:40.940 ","End":"02:44.190","Text":"The last thing to define is X,"},{"Start":"02:44.190 ","End":"02:49.630","Text":"where X would be the number of graduates."},{"Start":"02:50.620 ","End":"02:56.435","Text":"The number of graduates in our sample and what are we looking for?"},{"Start":"02:56.435 ","End":"03:04.545","Text":"We\u0027re looking for the probability of x being equal to here it is."},{"Start":"03:04.545 ","End":"03:07.130","Text":"That\u0027s 5 of them are university graduates."},{"Start":"03:07.130 ","End":"03:10.200","Text":"That means that x equals 5."},{"Start":"03:10.340 ","End":"03:18.035","Text":"Now, x because it met all the criteria here,"},{"Start":"03:18.035 ","End":"03:23.420","Text":"we know that x is distributed binomially."},{"Start":"03:23.420 ","End":"03:26.100","Text":"Let\u0027s just write this out,"},{"Start":"03:26.290 ","End":"03:32.240","Text":"x is distributed binomially with n"},{"Start":"03:32.240 ","End":"03:37.870","Text":"equals 20 and p that equals to 0.3."},{"Start":"03:37.870 ","End":"03:43.535","Text":"We know that the probability where x equals k,"},{"Start":"03:43.535 ","End":"03:46.075","Text":"we know that equals n over k,"},{"Start":"03:46.075 ","End":"03:52.460","Text":"p^k, q^n minus k. In our case,"},{"Start":"03:52.460 ","End":"03:54.320","Text":"let\u0027s just plug in the numbers."},{"Start":"03:54.320 ","End":"03:59.135","Text":"The probability of X being equal to 5,"},{"Start":"03:59.135 ","End":"04:04.400","Text":"that equals to n is 20, k is 5."},{"Start":"04:04.400 ","End":"04:07.470","Text":"20 here, 20 here,"},{"Start":"04:07.470 ","End":"04:08.865","Text":"and 5 of them."},{"Start":"04:08.865 ","End":"04:11.385","Text":"We\u0027re looking at 5 successes,"},{"Start":"04:11.385 ","End":"04:13.680","Text":"so that\u0027s n over k,"},{"Start":"04:13.680 ","End":"04:17.470","Text":"20 over 5 times p. What\u0027s p?"},{"Start":"04:19.040 ","End":"04:21.900","Text":"0.3^k^5 times 0.7."},{"Start":"04:21.900 ","End":"04:25.020","Text":"That\u0027s our q^n minus k,"},{"Start":"04:25.020 ","End":"04:27.780","Text":"that\u0027s 20 minus 5, that\u0027s 15,"},{"Start":"04:27.780 ","End":"04:33.940","Text":"and that comes out to 0.1789."},{"Start":"04:34.900 ","End":"04:38.270","Text":"In this section, we\u0027re asked what\u0027s"},{"Start":"04:38.270 ","End":"04:42.485","Text":"the expectation of the number of poorly educated people in the sample?"},{"Start":"04:42.485 ","End":"04:45.065","Text":"As we said in section a,"},{"Start":"04:45.065 ","End":"04:50.855","Text":"each person in the sample is like a Bernoulli trial."},{"Start":"04:50.855 ","End":"04:58.350","Text":"We have to define our success and our failure."},{"Start":"04:58.520 ","End":"05:01.925","Text":"Now, what\u0027s our success here?"},{"Start":"05:01.925 ","End":"05:06.140","Text":"That success is a person who is poorly educated."},{"Start":"05:06.140 ","End":"05:08.660","Text":"Let\u0027s define that."},{"Start":"05:08.660 ","End":"05:13.130","Text":"Educated as opposed to a person"},{"Start":"05:13.130 ","End":"05:18.005","Text":"having a higher education or university graduate in section 8."},{"Start":"05:18.005 ","End":"05:23.810","Text":"Right now we define our success as a person being poorly educated and failure"},{"Start":"05:23.810 ","End":"05:32.050","Text":"means that he has a high school education or more."},{"Start":"05:32.050 ","End":"05:34.490","Text":"Now, what else do we know?"},{"Start":"05:34.490 ","End":"05:39.020","Text":"We know that n equals 20 because that\u0027s the sample size,"},{"Start":"05:39.020 ","End":"05:41.239","Text":"that\u0027s the number of people that we\u0027ve sampled."},{"Start":"05:41.239 ","End":"05:46.160","Text":"Now, what\u0027s the probability of success of being poorly educated?"},{"Start":"05:46.160 ","End":"05:48.575","Text":"That equals 10 percent,"},{"Start":"05:48.575 ","End":"05:51.185","Text":"this is what we got from the data."},{"Start":"05:51.185 ","End":"05:54.000","Text":"Now, what else do we know?"},{"Start":"05:54.000 ","End":"05:58.955","Text":"Let\u0027s define our w as a random variable that"},{"Start":"05:58.955 ","End":"06:05.190","Text":"counts the number of people that have a poor education."},{"Start":"06:10.760 ","End":"06:14.490","Text":"Education, there we go."},{"Start":"06:14.490 ","End":"06:16.835","Text":"What do we want to know?"},{"Start":"06:16.835 ","End":"06:20.885","Text":"Well, first of all, we know that w is distributed"},{"Start":"06:20.885 ","End":"06:28.605","Text":"binomially with n equaling 20 and p equaling 0.1."},{"Start":"06:28.605 ","End":"06:32.190","Text":"That\u0027s what\u0027s given to us right here, p equals 1,"},{"Start":"06:32.190 ","End":"06:39.110","Text":"that\u0027s the probability of success where success is a person that is poorly educated."},{"Start":"06:39.110 ","End":"06:43.669","Text":"That means that the expectation of w,"},{"Start":"06:43.669 ","End":"06:47.870","Text":"because w is distributed binomially,"},{"Start":"06:47.870 ","End":"06:51.230","Text":"that\u0027s n times p and that equals,"},{"Start":"06:51.230 ","End":"06:52.355","Text":"let\u0027s plug in the numbers,"},{"Start":"06:52.355 ","End":"06:55.835","Text":"n is 20 and p is 0.1,"},{"Start":"06:55.835 ","End":"06:58.650","Text":"that equals to 2 people."},{"Start":"07:00.020 ","End":"07:06.365","Text":"All this means is that in the sample of 20 people,"},{"Start":"07:06.365 ","End":"07:12.480","Text":"we can expect that 2 people will be poorly educated."}],"ID":14143},{"Watched":false,"Name":"Exercise 12","Duration":"8m 25s","ChapterTopicVideoID":13503,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.645","Text":"In this question, we\u0027ll be talking about students and cars."},{"Start":"00:03.645 ","End":"00:08.190","Text":"20 percent of the students attending NYU actually live in New York."},{"Start":"00:08.190 ","End":"00:10.125","Text":"Of the students living in New York,"},{"Start":"00:10.125 ","End":"00:12.180","Text":"30 percent arrive by car."},{"Start":"00:12.180 ","End":"00:14.955","Text":"Of the students who do not live in New York,"},{"Start":"00:14.955 ","End":"00:16.830","Text":"50 percent come by car."},{"Start":"00:16.830 ","End":"00:22.710","Text":"We were asked, a guard checks the bags of each student entering the campus."},{"Start":"00:22.710 ","End":"00:27.840","Text":"What\u0027s the probability that among 5 students checked by the guard,"},{"Start":"00:27.840 ","End":"00:30.915","Text":"only 1 came by car?"},{"Start":"00:30.915 ","End":"00:34.500","Text":"Well, the first thing that we have to do is"},{"Start":"00:34.500 ","End":"00:40.230","Text":"to calculate the probability of a student coming by car."},{"Start":"00:40.230 ","End":"00:42.175","Text":"Now, if we look at the question,"},{"Start":"00:42.175 ","End":"00:45.935","Text":"we see that we can solve this by probability tree."},{"Start":"00:45.935 ","End":"00:49.955","Text":"Now, let\u0027s just build the probability tree,"},{"Start":"00:49.955 ","End":"00:52.510","Text":"and here it is."},{"Start":"00:52.510 ","End":"00:57.440","Text":"As we can see, we have here the 2 levels."},{"Start":"00:57.440 ","End":"01:00.680","Text":"The first level is the level of the students,"},{"Start":"01:00.680 ","End":"01:03.170","Text":"whether they live in New York or not."},{"Start":"01:03.170 ","End":"01:10.025","Text":"Now we said that 20 percent of the students attending NYU actually live in New York."},{"Start":"01:10.025 ","End":"01:12.485","Text":"Well, here it is, that\u0027s the 20 percent."},{"Start":"01:12.485 ","End":"01:16.535","Text":"That means that 80 percent do not live in New York."},{"Start":"01:16.535 ","End":"01:20.040","Text":"Now what else? Of the students living in New York,"},{"Start":"01:20.040 ","End":"01:21.795","Text":"30 percent arrive by car."},{"Start":"01:21.795 ","End":"01:23.870","Text":"Well, that\u0027s the second level right here."},{"Start":"01:23.870 ","End":"01:27.895","Text":"Who has a car or who arrives by car and who doesn\u0027t?"},{"Start":"01:27.895 ","End":"01:30.080","Text":"Out of the people living in New York,"},{"Start":"01:30.080 ","End":"01:32.150","Text":"30 percent arrive by car,"},{"Start":"01:32.150 ","End":"01:35.030","Text":"that means 70 percent do not."},{"Start":"01:35.030 ","End":"01:38.255","Text":"Of the people who do not live in New York,"},{"Start":"01:38.255 ","End":"01:41.180","Text":"well, 50 percent of them come by car."},{"Start":"01:41.180 ","End":"01:44.085","Text":"That means the 50 percent don\u0027t."},{"Start":"01:44.085 ","End":"01:50.900","Text":"Now if we remember how to calculate the probabilities in the probability tree,"},{"Start":"01:50.900 ","End":"01:55.935","Text":"within a tree we multiply the probabilities,"},{"Start":"01:55.935 ","End":"02:01.410","Text":"and amongst the branches, we add."},{"Start":"02:01.410 ","End":"02:08.880","Text":"What do we have to do? We have to see what are the branches of students coming by car."},{"Start":"02:09.620 ","End":"02:13.385","Text":"Well, of the students who are not living in New York City,"},{"Start":"02:13.385 ","End":"02:16.670","Text":"this is the branch that we\u0027re interested in,"},{"Start":"02:16.670 ","End":"02:21.205","Text":"and of the people that are living in New York City,"},{"Start":"02:21.205 ","End":"02:25.990","Text":"this is the branch that we\u0027re interested in."},{"Start":"02:26.660 ","End":"02:34.979","Text":"The probability of a student coming by car"},{"Start":"02:34.979 ","End":"02:41.490","Text":"is 0.8 times 0.5"},{"Start":"02:41.490 ","End":"02:48.010","Text":"plus 0.2 times 0.3."},{"Start":"02:48.560 ","End":"02:55.150","Text":"That turns out to be 0.46."},{"Start":"02:56.660 ","End":"02:59.985","Text":"Let\u0027s get back to our question."},{"Start":"02:59.985 ","End":"03:05.870","Text":"Now we\u0027re asked, what\u0027s the probability that among 5 students checked by the guard,"},{"Start":"03:05.870 ","End":"03:08.410","Text":"only 1 came by car?"},{"Start":"03:08.410 ","End":"03:13.880","Text":"We\u0027re interested in 5 students,"},{"Start":"03:13.880 ","End":"03:17.740","Text":"so our sample is 5."},{"Start":"03:17.740 ","End":"03:28.320","Text":"We\u0027re interested to find out the probability where a random variable x equals 1."},{"Start":"03:28.320 ","End":"03:29.900","Text":"What\u0027s a random variable?"},{"Start":"03:29.900 ","End":"03:39.885","Text":"A random variable here we\u0027ll define that as the number of students who came by car."},{"Start":"03:39.885 ","End":"03:42.390","Text":"That\u0027s what we want to know."},{"Start":"03:42.390 ","End":"03:45.545","Text":"Now, are the students independent of each other?"},{"Start":"03:45.545 ","End":"03:53.210","Text":"Obviously. Do we have a Bernoulli trial here?"},{"Start":"03:53.210 ","End":"03:57.020","Text":"Well yes, we define success as"},{"Start":"03:57.020 ","End":"04:02.220","Text":"a person coming by car and we have a probability for that."},{"Start":"04:02.830 ","End":"04:10.760","Text":"We know that x is distributed binomially"},{"Start":"04:10.760 ","End":"04:18.750","Text":"with n equals 5 and p being equal to 0.46."},{"Start":"04:20.630 ","End":"04:23.220","Text":"If that\u0027s the case,"},{"Start":"04:23.220 ","End":"04:25.620","Text":"let\u0027s just calculate,"},{"Start":"04:25.620 ","End":"04:30.090","Text":"what\u0027s the probability of x being equal to 1?"},{"Start":"04:30.090 ","End":"04:36.521","Text":"Well, if we have a binomial distribution and the probability of x being equal to k,"},{"Start":"04:36.521 ","End":"04:38.870","Text":"that\u0027s easy for us right now."},{"Start":"04:38.870 ","End":"04:48.720","Text":"That\u0027s n/k times p^k times q^n minus k. Now,"},{"Start":"04:48.720 ","End":"04:50.955","Text":"let\u0027s just plug in the numbers."},{"Start":"04:50.955 ","End":"04:53.685","Text":"That\u0027s n, that\u0027s 5,"},{"Start":"04:53.685 ","End":"04:58.005","Text":"and k is 1. What\u0027s our p?"},{"Start":"04:58.005 ","End":"05:06.645","Text":"Our p is 0.46^1 times what\u0027s our q?"},{"Start":"05:06.645 ","End":"05:09.270","Text":"Q is 1 minus p,"},{"Start":"05:09.270 ","End":"05:13.335","Text":"which is 1 minus 0.46, that\u0027s 0.54^4,"},{"Start":"05:13.335 ","End":"05:18.810","Text":"4 is n minus k,"},{"Start":"05:18.810 ","End":"05:21.610","Text":"that\u0027s 5 minus 1, that\u0027s 4."},{"Start":"05:21.790 ","End":"05:26.910","Text":"That comes out to 0.1956."},{"Start":"05:27.460 ","End":"05:35.410","Text":"That\u0027s the probability of 1 student coming by car out of 5."},{"Start":"05:35.410 ","End":"05:37.580","Text":"In this section we\u0027re asked,"},{"Start":"05:37.580 ","End":"05:43.630","Text":"what\u0027s the probability that the majority of the 5 students came to NYU by car?"},{"Start":"05:43.630 ","End":"05:46.445","Text":"Well, basically what we\u0027re asking is,"},{"Start":"05:46.445 ","End":"05:49.235","Text":"what is the probability that x,"},{"Start":"05:49.235 ","End":"05:53.120","Text":"the number of students come by car, is the majority?"},{"Start":"05:53.120 ","End":"05:57.355","Text":"That means it has to be greater or equal to 3."},{"Start":"05:57.355 ","End":"06:00.350","Text":"2 out of the 5 students isn\u0027t a majority,"},{"Start":"06:00.350 ","End":"06:03.430","Text":"but 3 out of the 5 is."},{"Start":"06:03.560 ","End":"06:05.840","Text":"Well, if that\u0027s the case,"},{"Start":"06:05.840 ","End":"06:09.590","Text":"this equals to the probability of x being equal to"},{"Start":"06:09.590 ","End":"06:15.150","Text":"3 plus the probability of x being equal to 4,"},{"Start":"06:15.150 ","End":"06:20.040","Text":"plus the probability of x being equal to 5."},{"Start":"06:20.040 ","End":"06:26.690","Text":"Now, we know that x is binomially distributed with n"},{"Start":"06:26.690 ","End":"06:33.185","Text":"being equal to 5 and p being equal to 0.46."},{"Start":"06:33.185 ","End":"06:37.130","Text":"We know that the definition of"},{"Start":"06:37.130 ","End":"06:41.840","Text":"the binomial distribution is the probability of x being equal to k,"},{"Start":"06:41.840 ","End":"06:47.870","Text":"is n/k, p^k,"},{"Start":"06:47.870 ","End":"06:51.350","Text":"q^n minus k. If that\u0027s the case,"},{"Start":"06:51.350 ","End":"06:55.380","Text":"let\u0027s just plug in the numbers and calculate things out."},{"Start":"06:56.800 ","End":"07:00.425","Text":"Where the probability of x equals 3, well,"},{"Start":"07:00.425 ","End":"07:05.910","Text":"that\u0027s 5/3, k equals 3 here."},{"Start":"07:08.710 ","End":"07:18.335","Text":"0.46^3 times 0.54^2 plus, now here,"},{"Start":"07:18.335 ","End":"07:20.965","Text":"we\u0027re calculating where k equals 4,"},{"Start":"07:20.965 ","End":"07:32.060","Text":"so that\u0027s 5/4, 0.46^4 times 0.54^1."},{"Start":"07:33.350 ","End":"07:38.435","Text":"N minus k, 5 minus 4 plus,"},{"Start":"07:38.435 ","End":"07:43.150","Text":"what\u0027s the probability of x when x equals 5?"},{"Start":"07:43.150 ","End":"07:49.660","Text":"Well, that\u0027s 5/5"},{"Start":"07:49.660 ","End":"07:55.120","Text":", 0.46^5 times 0.54^0."},{"Start":"07:55.790 ","End":"08:03.770","Text":"This section calculates the probability when x equals 3."},{"Start":"08:03.770 ","End":"08:08.360","Text":"This section right here calculates the probability when x equals 4,"},{"Start":"08:08.360 ","End":"08:13.680","Text":"and this section calculates the probability when x equals 5."},{"Start":"08:13.850 ","End":"08:16.385","Text":"Now, when we calculate everything,"},{"Start":"08:16.385 ","End":"08:23.220","Text":"this comes out to 0.4253."}],"ID":14144},{"Watched":false,"Name":"Exercise 13 - Parts a-b","Duration":"5m 8s","ChapterTopicVideoID":13505,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.125","Text":"In this question, we\u0027ll be talking about getting the right answer on an exam."},{"Start":"00:04.125 ","End":"00:09.090","Text":"Now, John takes a multiple choice exam having 20 questions."},{"Start":"00:09.090 ","End":"00:12.870","Text":"The chances of his knowing the answer to a question are 80 percent."},{"Start":"00:12.870 ","End":"00:15.900","Text":"If he doesn\u0027t know the answer, he guesses."},{"Start":"00:15.900 ","End":"00:18.555","Text":"Each question has 4 possible answers,"},{"Start":"00:18.555 ","End":"00:20.446","Text":"only one of which is correct."},{"Start":"00:20.446 ","End":"00:26.015","Text":"We\u0027re asked, what is the chances of answering a given question correctly?"},{"Start":"00:26.015 ","End":"00:31.095","Text":"Now, this question has nothing to do with the binomial distribution."},{"Start":"00:31.095 ","End":"00:36.020","Text":"We\u0027re asked to find the probability of answering a given question"},{"Start":"00:36.020 ","End":"00:42.815","Text":"correctly when we\u0027re given the procedure of how he answers the questions."},{"Start":"00:42.815 ","End":"00:46.730","Text":"Now, this basically looks like a probability tree."},{"Start":"00:46.730 ","End":"00:52.640","Text":"Where the first level of whether John knows or doesn\u0027t know the correct answer."},{"Start":"00:52.640 ","End":"00:57.415","Text":"The second level is whether he guesses correctly or not."},{"Start":"00:57.415 ","End":"01:02.040","Text":"Let\u0027s build our probability tree. Here it is."},{"Start":"01:02.040 ","End":"01:07.220","Text":"As we said, the first level is whether John"},{"Start":"01:07.220 ","End":"01:12.890","Text":"knows or doesn\u0027t know and he knows what the probability of 80 percent. That\u0027s right here."},{"Start":"01:12.890 ","End":"01:17.630","Text":"That means that he doesn\u0027t know the probability of 20 percent."},{"Start":"01:17.630 ","End":"01:22.325","Text":"Now if we go to the second level where John guesses, now,"},{"Start":"01:22.325 ","End":"01:25.970","Text":"there are 4 possible answers to each question,"},{"Start":"01:25.970 ","End":"01:28.355","Text":"only one of which is correct."},{"Start":"01:28.355 ","End":"01:34.385","Text":"If he guesses, he can guess with a probability of 25 percent,"},{"Start":"01:34.385 ","End":"01:36.880","Text":"1/4 of guessing right."},{"Start":"01:36.880 ","End":"01:42.335","Text":"That means that he can guess wrong with a probability of 75 percent."},{"Start":"01:42.335 ","End":"01:44.930","Text":"Now, what are we asked?"},{"Start":"01:44.930 ","End":"01:48.800","Text":"We\u0027re asked, what are the chances of answering the given questions correctly,"},{"Start":"01:48.800 ","End":"01:55.110","Text":"the probability of getting the question right."},{"Start":"01:56.030 ","End":"02:01.480","Text":"Now, when we take a look at our probability tree,"},{"Start":"02:01.480 ","End":"02:08.510","Text":"we can get the question right on this branch or right here."},{"Start":"02:10.070 ","End":"02:15.520","Text":"That means that we can write right now, calculate the probability."},{"Start":"02:15.520 ","End":"02:20.200","Text":"Well, that\u0027s 0.8, that\u0027s this branch right here,"},{"Start":"02:20.200 ","End":"02:25.120","Text":"plus this branch right here,"},{"Start":"02:25.120 ","End":"02:35.240","Text":"which is 0.2 times 0.25 and that equals to 0.85."},{"Start":"02:36.620 ","End":"02:38.945","Text":"In this section, we\u0027re asked,"},{"Start":"02:38.945 ","End":"02:43.129","Text":"what are the chances of answering exactly 16 questions correctly?"},{"Start":"02:43.129 ","End":"02:52.270","Text":"Well, let\u0027s define X as the number of correct answers."},{"Start":"02:53.300 ","End":"02:56.285","Text":"Now, how many questions do we have?"},{"Start":"02:56.285 ","End":"02:58.159","Text":"We have 20 questions,"},{"Start":"02:58.159 ","End":"03:00.080","Text":"so n equals 20."},{"Start":"03:00.080 ","End":"03:04.940","Text":"What\u0027s the probability of answering a question correctly?"},{"Start":"03:04.940 ","End":"03:08.965","Text":"Well, that equals to 0.85."},{"Start":"03:08.965 ","End":"03:11.635","Text":"We\u0027ve answered that in section A."},{"Start":"03:11.635 ","End":"03:16.100","Text":"Now because a question can be answered either correctly or incorrectly,"},{"Start":"03:16.100 ","End":"03:21.760","Text":"and all questions have the same probability of being answered correctly,"},{"Start":"03:21.760 ","End":"03:25.930","Text":"then we do have a Bernoulli trial."},{"Start":"03:25.940 ","End":"03:30.440","Text":"The questions are independent and obviously answering"},{"Start":"03:30.440 ","End":"03:35.870","Text":"the questions when one question is independent of answering a different question."},{"Start":"03:35.870 ","End":"03:38.765","Text":"We have 20 of these trials,"},{"Start":"03:38.765 ","End":"03:44.990","Text":"and we want to know what is the total number of correct answers."},{"Start":"03:44.990 ","End":"03:49.535","Text":"We can honestly say that X has"},{"Start":"03:49.535 ","End":"03:58.535","Text":"a binomial distribution where n equals 20 and p equals 0.85."},{"Start":"03:58.535 ","End":"04:06.830","Text":"Now, we know that the general equation for a binomial probability is this,"},{"Start":"04:06.830 ","End":"04:10.100","Text":"where the probability of X being equal to k,"},{"Start":"04:10.100 ","End":"04:14.660","Text":"that\u0027s n/k, p^k,"},{"Start":"04:14.660 ","End":"04:23.255","Text":"q^n minus k. Where q is 1 minus p. Let\u0027s answer the question."},{"Start":"04:23.255 ","End":"04:28.310","Text":"What\u0027s the probability of X being equal to 16?"},{"Start":"04:28.310 ","End":"04:33.470","Text":"We want to know what are the chances of answering exactly 16 questions."},{"Start":"04:33.470 ","End":"04:36.860","Text":"Well, let\u0027s plug in the numbers, n equals 20."},{"Start":"04:36.860 ","End":"04:42.605","Text":"That\u0027s 20 over what, 16. What\u0027s p?"},{"Start":"04:42.605 ","End":"04:49.890","Text":"P is 0.85^16 times q,"},{"Start":"04:49.890 ","End":"04:54.290","Text":"where q is 1 minus p. That\u0027s 1 minus 0.85,"},{"Start":"04:54.290 ","End":"04:59.240","Text":"that\u0027s 0.15^n minus k. N"},{"Start":"04:59.240 ","End":"05:06.020","Text":"minus k is 4 and that turns out to be 0.1821."}],"ID":14145},{"Watched":false,"Name":"Exercise 13 - Part c","Duration":"6m 10s","ChapterTopicVideoID":13504,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In this section, we\u0027re given that John receives"},{"Start":"00:02.940 ","End":"00:06.105","Text":"5 points for each question he answers correctly,"},{"Start":"00:06.105 ","End":"00:10.995","Text":"and 1 point is deducted from his mark for each question he answers incorrectly."},{"Start":"00:10.995 ","End":"00:13.440","Text":"We\u0027re asked, what are the expectation,"},{"Start":"00:13.440 ","End":"00:16.300","Text":"and variance of the student\u0027s mark?"},{"Start":"00:16.760 ","End":"00:20.460","Text":"First of all, I believes that we\u0027re"},{"Start":"00:20.460 ","End":"00:23.490","Text":"dealing with a linear transformation here. Why is that?"},{"Start":"00:23.490 ","End":"00:25.515","Text":"Because we are taking x,"},{"Start":"00:25.515 ","End":"00:27.615","Text":"the number of correct answers,"},{"Start":"00:27.615 ","End":"00:30.900","Text":"and we\u0027re transforming them into y,"},{"Start":"00:30.900 ","End":"00:34.620","Text":"the final mark that John receives."},{"Start":"00:34.620 ","End":"00:36.255","Text":"If that\u0027s the case,"},{"Start":"00:36.255 ","End":"00:44.190","Text":"then let\u0027s take a look at the 4 stages of a linear transformation. Here they are."},{"Start":"00:44.190 ","End":"00:46.365","Text":"Well, first of all,"},{"Start":"00:46.365 ","End":"00:49.545","Text":"we\u0027ve answered the first step."},{"Start":"00:49.545 ","End":"00:53.710","Text":"We recognized that we\u0027re dealing with a linear transformation."},{"Start":"00:53.710 ","End":"00:56.285","Text":"Let\u0027s go to the second step."},{"Start":"00:56.285 ","End":"01:02.255","Text":"We have to write the transformation according to what\u0027s given to us."},{"Start":"01:02.255 ","End":"01:04.720","Text":"If that\u0027s the case, y,"},{"Start":"01:04.720 ","End":"01:08.280","Text":"the final mark is equal to what?"},{"Start":"01:08.280 ","End":"01:09.990","Text":"What\u0027s given to us?"},{"Start":"01:09.990 ","End":"01:15.740","Text":"We are given that John receives 5 points for each question he answers correctly,"},{"Start":"01:15.740 ","End":"01:23.735","Text":"and 1 point is deducted for each question he answers incorrectly."},{"Start":"01:23.735 ","End":"01:30.170","Text":"What does that mean? That means that if x is a number of questions the answers correctly,"},{"Start":"01:30.170 ","End":"01:33.590","Text":"then I have to multiply that by 5 to"},{"Start":"01:33.590 ","End":"01:42.555","Text":"transform the number of questions to points."},{"Start":"01:42.555 ","End":"01:44.900","Text":"But I have to take away what?"},{"Start":"01:44.900 ","End":"01:46.460","Text":"I have to take away 1,"},{"Start":"01:46.460 ","End":"01:47.960","Text":"that\u0027s 1 point,"},{"Start":"01:47.960 ","End":"01:52.815","Text":"times the number of questions he answers incorrectly."},{"Start":"01:52.815 ","End":"01:54.975","Text":"We have 20 questions."},{"Start":"01:54.975 ","End":"01:57.900","Text":"We know that n equals 20,"},{"Start":"01:57.900 ","End":"02:02.520","Text":"so if he answers x questions correctly then he must"},{"Start":"02:02.520 ","End":"02:08.220","Text":"answer 20 minus x questions incorrectly."},{"Start":"02:08.220 ","End":"02:16.085","Text":"This would be the linear transformation according to this question."},{"Start":"02:16.085 ","End":"02:19.130","Text":"What\u0027s the general form of a linear transformation?"},{"Start":"02:19.130 ","End":"02:23.670","Text":"That\u0027s y equals ax plus b."},{"Start":"02:23.670 ","End":"02:31.635","Text":"All we have to do is bring this transformation into this form. Let\u0027s do that."},{"Start":"02:31.635 ","End":"02:36.405","Text":"That equals to 5x minus 20,"},{"Start":"02:36.405 ","End":"02:38.205","Text":"minus 1 times 20,"},{"Start":"02:38.205 ","End":"02:42.000","Text":"minus 1 times minus 6 that\u0027s plus x,"},{"Start":"02:42.000 ","End":"02:48.615","Text":"and that equals to 6x minus 20."},{"Start":"02:48.615 ","End":"02:54.380","Text":"Here we\u0027ve brought the linear transformation into the general form."},{"Start":"02:54.380 ","End":"02:58.965","Text":"We can identify a and b where a equals what?"},{"Start":"02:58.965 ","End":"03:00.810","Text":"That is the multiply here,"},{"Start":"03:00.810 ","End":"03:06.790","Text":"that\u0027s 6, and b equals minus 20."},{"Start":"03:07.190 ","End":"03:11.560","Text":"Excellent, we\u0027ve done that."},{"Start":"03:11.840 ","End":"03:15.320","Text":"We can start and answer the question."},{"Start":"03:15.320 ","End":"03:18.935","Text":"The question being, what are the expectations and variance of the student\u0027s mark?"},{"Start":"03:18.935 ","End":"03:21.395","Text":"We\u0027re asked what are the expectation,"},{"Start":"03:21.395 ","End":"03:23.300","Text":"and variance of y."},{"Start":"03:23.300 ","End":"03:26.030","Text":"But before we do that,"},{"Start":"03:26.030 ","End":"03:33.290","Text":"let\u0027s first see about x and the expectation and variance of x."},{"Start":"03:33.290 ","End":"03:44.285","Text":"If we recall x was distributed binomially with n equaling 20,"},{"Start":"03:44.285 ","End":"03:49.180","Text":"and p being equal to 0.85."},{"Start":"03:49.180 ","End":"03:54.830","Text":"If that\u0027s the case, then the expectation of x is n times p,"},{"Start":"03:54.830 ","End":"03:59.885","Text":"which is 20 times 0.85,"},{"Start":"03:59.885 ","End":"04:02.230","Text":"and that equals to 17."},{"Start":"04:02.230 ","End":"04:08.360","Text":"The variance of x equals n times p times q,"},{"Start":"04:08.360 ","End":"04:15.480","Text":"that equals to 20 times 0.85 times 0.15,"},{"Start":"04:16.360 ","End":"04:22.950","Text":"and that equals to 2.55."},{"Start":"04:23.110 ","End":"04:27.560","Text":"Excellent. Having done that,"},{"Start":"04:27.560 ","End":"04:35.970","Text":"we can go ahead and answer the question about the expectation and variance of y."},{"Start":"04:37.520 ","End":"04:40.410","Text":"What is the expectation of y?"},{"Start":"04:40.410 ","End":"04:43.480","Text":"Y as we said is a linear transformation of x."},{"Start":"04:43.480 ","End":"04:49.385","Text":"If we recall how we get the expectation of y that"},{"Start":"04:49.385 ","End":"04:56.570","Text":"equals to a times the expectation of x plus b."},{"Start":"04:56.570 ","End":"05:01.300","Text":"In our case, a is equal to 6."},{"Start":"05:01.300 ","End":"05:03.585","Text":"What\u0027s the expectation of x?"},{"Start":"05:03.585 ","End":"05:07.305","Text":"We just figured it out right here that\u0027s 17,"},{"Start":"05:07.305 ","End":"05:09.960","Text":"and we have to add b."},{"Start":"05:09.960 ","End":"05:11.190","Text":"B is minus 20,"},{"Start":"05:11.190 ","End":"05:19.280","Text":"so we subtract 20 from that and we get 82."},{"Start":"05:19.280 ","End":"05:22.880","Text":"That\u0027s his expected final grade."},{"Start":"05:22.880 ","End":"05:27.120","Text":"Let\u0027s take a look at the variance of y. Variance of"},{"Start":"05:27.120 ","End":"05:32.165","Text":"y is defined as a squared times the variance of x,"},{"Start":"05:32.165 ","End":"05:35.120","Text":"and again, let\u0027s plug in the numbers."},{"Start":"05:35.120 ","End":"05:41.175","Text":"A squared is 6 squared times what the variance of x."},{"Start":"05:41.175 ","End":"05:45.675","Text":"The variance of x is 2.55,"},{"Start":"05:45.675 ","End":"05:56.740","Text":"and that equals to 91.8 points squared."},{"Start":"05:56.750 ","End":"06:07.295","Text":"Here we\u0027ve answered the question where we calculated the expected value of y,"},{"Start":"06:07.295 ","End":"06:09.510","Text":"and the variance of y."}],"ID":14146},{"Watched":false,"Name":"Exercise 14","Duration":"7m 15s","ChapterTopicVideoID":13506,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.070","Text":"In this question, we\u0027ll be talking about faulty products being packaged."},{"Start":"00:05.070 ","End":"00:10.425","Text":"Now, we\u0027re given that 5 percent of products coming out of a production line are faulty,"},{"Start":"00:10.425 ","End":"00:14.159","Text":"the products are packaged in a cardboard box."},{"Start":"00:14.159 ","End":"00:19.800","Text":"Each box has 10 different products that the boxes are packaged in a container."},{"Start":"00:19.800 ","End":"00:24.165","Text":"Each container has 20 boxes, and we\u0027re asked,"},{"Start":"00:24.165 ","End":"00:31.179","Text":"what is the probability of at least 1 faulty product in a randomly selected box?"},{"Start":"00:31.340 ","End":"00:37.965","Text":"Now, here we\u0027re talking about a faulty product within a box."},{"Start":"00:37.965 ","End":"00:48.760","Text":"That means that we define our success as being a faulty product."},{"Start":"00:53.540 ","End":"01:00.545","Text":"The probability of a product being faulty is 0.05."},{"Start":"01:00.545 ","End":"01:03.360","Text":"That\u0027s 5 percent right here."},{"Start":"01:03.590 ","End":"01:07.220","Text":"Now, how many products are in a box?"},{"Start":"01:07.220 ","End":"01:11.035","Text":"Well, there are 10 different products,"},{"Start":"01:11.035 ","End":"01:13.850","Text":"so n equals 10."},{"Start":"01:13.850 ","End":"01:17.970","Text":"Now the products are independent of each other, obviously."},{"Start":"01:18.050 ","End":"01:23.185","Text":"Each box is basically a Bernoulli trial."},{"Start":"01:23.185 ","End":"01:30.445","Text":"Whether it\u0027s a faulty product or not a faulty product with the same probability."},{"Start":"01:30.445 ","End":"01:39.140","Text":"We\u0027re asked, what\u0027s the number of faulty products?"},{"Start":"01:43.130 ","End":"01:49.820","Text":"Excellent. That means that we have x and it has"},{"Start":"01:49.820 ","End":"02:00.715","Text":"a binomial distribution where n equals 10 and p equals 0.05."},{"Start":"02:00.715 ","End":"02:04.070","Text":"Now let\u0027s get back to our question."},{"Start":"02:04.070 ","End":"02:08.630","Text":"We\u0027re asked, what\u0027s the probability of at least 1 faulty product?"},{"Start":"02:08.630 ","End":"02:11.750","Text":"That means that we want to know"},{"Start":"02:11.750 ","End":"02:20.340","Text":"whether probability of x being greater or equal to 1."},{"Start":"02:20.340 ","End":"02:24.350","Text":"Because x is a number of faulty products and we"},{"Start":"02:24.350 ","End":"02:28.265","Text":"want a box that has at least 1 faulty product."},{"Start":"02:28.265 ","End":"02:32.540","Text":"Well, we know that x can have the values of 0,"},{"Start":"02:32.540 ","End":"02:36.200","Text":"1, 2, and so on and so forth until 10."},{"Start":"02:36.200 ","End":"02:39.055","Text":"We\u0027re looking at this set."},{"Start":"02:39.055 ","End":"02:42.843","Text":"But again, it\u0027s very complicated to figure this out,"},{"Start":"02:42.843 ","End":"02:44.800","Text":"so let\u0027s take the complimentary set."},{"Start":"02:44.800 ","End":"02:51.980","Text":"Well, that equals to 1 minus the probability of x being equal to 0."},{"Start":"02:54.080 ","End":"02:56.275","Text":"It\u0027s 1 minus."},{"Start":"02:56.275 ","End":"03:00.025","Text":"Now what\u0027s the probability of x being equal to 0?"},{"Start":"03:00.025 ","End":"03:03.370","Text":"Well, x is binomially distributed,"},{"Start":"03:03.370 ","End":"03:08.295","Text":"so that\u0027s 10 over 0, n equals 10,"},{"Start":"03:08.295 ","End":"03:19.130","Text":"p is 0.05 to the power of 0 times 0.95 to the power of 10."},{"Start":"03:19.970 ","End":"03:25.900","Text":"That equals to 0.401."},{"Start":"03:26.680 ","End":"03:31.940","Text":"In this section, we\u0027re asked what are the expectations and standard deviation of"},{"Start":"03:31.940 ","End":"03:37.509","Text":"the number of boxes in the container with at least 1 damaged product?"},{"Start":"03:37.509 ","End":"03:39.800","Text":"Now remember in section A,"},{"Start":"03:39.800 ","End":"03:43.670","Text":"we\u0027re talking about products within a box."},{"Start":"03:43.670 ","End":"03:48.515","Text":"Here we\u0027re talking about boxes within a container."},{"Start":"03:48.515 ","End":"03:56.870","Text":"We have a binomial distribution within a binomial distribution. Let\u0027s get to work."},{"Start":"03:56.870 ","End":"04:00.540","Text":"Let\u0027s define success."},{"Start":"04:01.100 ","End":"04:03.750","Text":"Success as what?"},{"Start":"04:03.750 ","End":"04:14.530","Text":"As a box with at least 1 faulty product."},{"Start":"04:18.470 ","End":"04:23.710","Text":"Now, how many of these boxes do we have in a container?"},{"Start":"04:23.710 ","End":"04:26.240","Text":"Well, we have 20."},{"Start":"04:26.570 ","End":"04:34.610","Text":"Again, now we have a Bernoulli trial where we\u0027re looking at boxes instead of products."},{"Start":"04:34.610 ","End":"04:43.515","Text":"Now, that means that we can right now define a new random variable w, let\u0027s call it."},{"Start":"04:43.515 ","End":"04:53.370","Text":"It\u0027ll be the number of boxes with at least 1 faulty product."},{"Start":"04:53.370 ","End":"04:56.035","Text":"It\u0027s the same thing, just like here."},{"Start":"04:56.035 ","End":"04:58.175","Text":"Now, because of that,"},{"Start":"04:58.175 ","End":"05:04.370","Text":"we know that w is distributed binomially where n equals 20."},{"Start":"05:04.370 ","End":"05:07.415","Text":"Now, we\u0027re looking at boxes within a container."},{"Start":"05:07.415 ","End":"05:09.170","Text":"But with the same probability,"},{"Start":"05:09.170 ","End":"05:13.140","Text":"the probability is 0.401."},{"Start":"05:14.840 ","End":"05:21.410","Text":"We know that the probability of w being equal to k,"},{"Start":"05:21.410 ","End":"05:24.260","Text":"That equals to n over k,"},{"Start":"05:24.260 ","End":"05:27.215","Text":"p to the power of k,"},{"Start":"05:27.215 ","End":"05:33.410","Text":"q to the power of n minus k. That\u0027s the equation for a binomial distribution."},{"Start":"05:33.410 ","End":"05:37.400","Text":"Well, if that\u0027s the case, let\u0027s just put in the numbers right here."},{"Start":"05:37.400 ","End":"05:43.670","Text":"What we\u0027re looking for is what are the expectations,"},{"Start":"05:43.670 ","End":"05:47.150","Text":"standard deviation of the number of boxes in a container?"},{"Start":"05:47.150 ","End":"05:55.490","Text":"Well, we know that n equals 20 and we know that p equals 0.41."},{"Start":"05:55.490 ","End":"05:58.755","Text":"The expectation of w, well,"},{"Start":"05:58.755 ","End":"06:02.978","Text":"because w is distributed binomially,"},{"Start":"06:02.978 ","End":"06:08.040","Text":"that equals to n times p. That equals n equals 20."},{"Start":"06:08.040 ","End":"06:10.725","Text":"As we said, p is times"},{"Start":"06:10.725 ","End":"06:19.600","Text":"0.401 and that equals to 8.025 boxes."},{"Start":"06:20.750 ","End":"06:24.030","Text":"What\u0027s the variance of w?"},{"Start":"06:24.030 ","End":"06:28.580","Text":"The variance of w is n times p times q."},{"Start":"06:28.580 ","End":"06:38.590","Text":"That equals to 20 times 0.401 times 1 minus 0.401."},{"Start":"06:39.310 ","End":"06:50.135","Text":"That equals to 4.867 boxes squared."},{"Start":"06:50.135 ","End":"06:54.290","Text":"Now, what\u0027s the standard deviation of w?"},{"Start":"06:54.290 ","End":"06:59.225","Text":"Well, that\u0027s the square root of 4.867 and that"},{"Start":"06:59.225 ","End":"07:05.970","Text":"equals to 2.193 boxes."},{"Start":"07:05.970 ","End":"07:14.910","Text":"Here we have it, the standard deviation of w and the expectation of w."}],"ID":14147},{"Watched":false,"Name":"Exercise 15","Duration":"2m 30s","ChapterTopicVideoID":13507,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.875","Text":"In this question we\u0027re given that a balanced coin is tossed 5 times."},{"Start":"00:04.875 ","End":"00:08.490","Text":"We define X as the number of times that heads appear."},{"Start":"00:08.490 ","End":"00:13.240","Text":"We\u0027re asked to calculate the expectation of X squared."},{"Start":"00:13.280 ","End":"00:18.840","Text":"Let\u0027s write this out, X is the number of times that heads appear."},{"Start":"00:18.840 ","End":"00:20.950","Text":"Number of heads."},{"Start":"00:22.490 ","End":"00:25.830","Text":"We\u0027re tossing a coin 5 times,"},{"Start":"00:25.830 ","End":"00:28.275","Text":"that means n equals 5."},{"Start":"00:28.275 ","End":"00:30.965","Text":"Because it\u0027s a balanced coin,"},{"Start":"00:30.965 ","End":"00:36.095","Text":"the probability of getting a heads is 0.5."},{"Start":"00:36.095 ","End":"00:41.545","Text":"Now, this is a classic Bernoulli trial,"},{"Start":"00:41.545 ","End":"00:47.195","Text":"where each task has a probability of 0.5 of getting heads."},{"Start":"00:47.195 ","End":"00:51.590","Text":"We\u0027re asked how many heads do we have after 5 tosses?"},{"Start":"00:51.590 ","End":"00:53.105","Text":"Well, if that\u0027s the case,"},{"Start":"00:53.105 ","End":"00:57.935","Text":"then X has a binomial distribution,"},{"Start":"00:57.935 ","End":"01:03.330","Text":"where n equals 5 and p equals 0.5."},{"Start":"01:03.980 ","End":"01:07.890","Text":"Let\u0027s see what the expectation of X is."},{"Start":"01:07.890 ","End":"01:12.840","Text":"That equals to n times p. Now if we plug in the numbers,"},{"Start":"01:12.840 ","End":"01:18.205","Text":"that\u0027s 5 times 0.5 and that equals to 2.5."},{"Start":"01:18.205 ","End":"01:21.620","Text":"What\u0027s the variance of X?"},{"Start":"01:21.620 ","End":"01:26.725","Text":"Well, that\u0027s defined as n times p times q."},{"Start":"01:26.725 ","End":"01:33.060","Text":"That is 5 times 0.5 times 0.5."},{"Start":"01:33.060 ","End":"01:36.830","Text":"That equals to 1.25."},{"Start":"01:36.830 ","End":"01:40.220","Text":"Now, what else do we know about the variance?"},{"Start":"01:40.220 ","End":"01:44.705","Text":"We know that the variance of X equals"},{"Start":"01:44.705 ","End":"01:53.215","Text":"the expectation of X squared minus the expectation squared of X."},{"Start":"01:53.215 ","End":"01:56.715","Text":"Now, if we plug in these numbers,"},{"Start":"01:56.715 ","End":"01:59.670","Text":"the variance of X, that\u0027s 1.25,"},{"Start":"01:59.670 ","End":"02:02.535","Text":"We calculated that, that\u0027s 1.25."},{"Start":"02:02.535 ","End":"02:06.470","Text":"That equals to the expectation of X squared."},{"Start":"02:06.470 ","End":"02:07.910","Text":"That\u0027s what we\u0027re trying to find out,"},{"Start":"02:07.910 ","End":"02:13.220","Text":"calculate the expectation of X squared minus the expectation squared of X."},{"Start":"02:13.220 ","End":"02:15.485","Text":"Well, the expectation of X is 2.5."},{"Start":"02:15.485 ","End":"02:19.470","Text":"That\u0027s 2.5 squared."},{"Start":"02:19.970 ","End":"02:29.670","Text":"That means that the expectation of X squared equals 7.5."}],"ID":14148},{"Watched":false,"Name":"Uniform Probability Density Function","Duration":"4m 44s","ChapterTopicVideoID":13508,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.060 ","End":"00:04.990","Text":"In this chapter, we\u0027ll be discussing special discrete probabilities,"},{"Start":"00:04.990 ","End":"00:08.065","Text":"specifically the uniform distribution."},{"Start":"00:08.065 ","End":"00:11.095","Text":"Now, in this distribution,"},{"Start":"00:11.095 ","End":"00:13.634","Text":"each result has the same probability,"},{"Start":"00:13.634 ","End":"00:18.635","Text":"and the values obtained in this distribution are between a and b,"},{"Start":"00:18.635 ","End":"00:21.270","Text":"with increasing increments of 1."},{"Start":"00:21.270 ","End":"00:27.085","Text":"All that means is that the distribution has a lower limit and an upper limit."},{"Start":"00:27.085 ","End":"00:28.930","Text":"The lower limit being a,"},{"Start":"00:28.930 ","End":"00:30.940","Text":"the upper limit being b,"},{"Start":"00:30.940 ","End":"00:36.370","Text":"and X takes on the values between a and b with increments of 1."},{"Start":"00:36.370 ","End":"00:38.245","Text":"That means that it starts with a,"},{"Start":"00:38.245 ","End":"00:40.360","Text":"then a plus 1, a plus 2,"},{"Start":"00:40.360 ","End":"00:42.785","Text":"and so on and so forth until we get to b."},{"Start":"00:42.785 ","End":"00:45.725","Text":"Now, whenever we have this scenario,"},{"Start":"00:45.725 ","End":"00:51.240","Text":"then we say that X is distributed with a uniform distribution."},{"Start":"00:51.240 ","End":"00:54.950","Text":"That\u0027s U for uniform with 2 parameters, a and b,"},{"Start":"00:54.950 ","End":"00:56.480","Text":"where a is the lower limit of"},{"Start":"00:56.480 ","End":"01:00.365","Text":"the distribution and b is the upper limit of the distribution."},{"Start":"01:00.365 ","End":"01:08.190","Text":"Now, the probability function of X. Probability of X where X equals k,"},{"Start":"01:08.190 ","End":"01:12.860","Text":"what that would equal 1/b minus a plus 1."},{"Start":"01:12.860 ","End":"01:17.610","Text":"That\u0027s what makes this a uniform distribution for every"},{"Start":"01:17.610 ","End":"01:22.380","Text":"k. Where k gets a value of a, a plus 1,"},{"Start":"01:22.380 ","End":"01:27.295","Text":"a plus 2 until b for every k. Then"},{"Start":"01:27.295 ","End":"01:32.445","Text":"the probability of X equaling k is a constant."},{"Start":"01:32.445 ","End":"01:36.060","Text":"It\u0027s 1/b minus a plus 1."},{"Start":"01:36.060 ","End":"01:41.070","Text":"Now, let\u0027s go to the expectation and variance."},{"Start":"01:41.290 ","End":"01:45.900","Text":"The equation for the expectation is as follows."},{"Start":"01:45.900 ","End":"01:50.780","Text":"The expectation of X equals a plus b/2."},{"Start":"01:50.780 ","End":"01:57.725","Text":"The variance of X is b minus a plus 1 squared minus 1/12."},{"Start":"01:57.725 ","End":"02:03.380","Text":"Now, these are the equations that we have to deal with."},{"Start":"02:03.380 ","End":"02:06.845","Text":"These are the equations that we\u0027re given when X is"},{"Start":"02:06.845 ","End":"02:12.200","Text":"distributed with a uniform distribution with parameters a and b."},{"Start":"02:12.200 ","End":"02:17.725","Text":"Now, let\u0027s take a look at an example and see how we can work with these things."},{"Start":"02:17.725 ","End":"02:20.470","Text":"So in our example,"},{"Start":"02:20.470 ","End":"02:25.640","Text":"a person randomly selects a number between 1 and 100 inclusively."},{"Start":"02:25.640 ","End":"02:31.905","Text":"We\u0027re asked what\u0027s the probability function of the number and what\u0027s its expectation?"},{"Start":"02:31.905 ","End":"02:38.915","Text":"Well, here, we obviously have a uniform distribution and why is that?"},{"Start":"02:38.915 ","End":"02:46.160","Text":"Because for every number that\u0027s selected randomly between 1 and 100,"},{"Start":"02:46.160 ","End":"02:49.355","Text":"the probability would be the same probability."},{"Start":"02:49.355 ","End":"02:55.250","Text":"So let\u0027s just use our equations for the uniform probability."},{"Start":"02:55.250 ","End":"03:02.820","Text":"The probability of X being equal to k. That would equal to what?"},{"Start":"03:02.820 ","End":"03:08.025","Text":"1/b minus a plus 1."},{"Start":"03:08.025 ","End":"03:10.425","Text":"That\u0027s the probability."},{"Start":"03:10.425 ","End":"03:17.285","Text":"So again, we have a uniform probability and what\u0027s our lower limit and upper limit?"},{"Start":"03:17.285 ","End":"03:20.725","Text":"We have the lower limit being 1."},{"Start":"03:20.725 ","End":"03:22.160","Text":"It\u0027s given right here,"},{"Start":"03:22.160 ","End":"03:25.595","Text":"and the upper limit being 100."},{"Start":"03:25.595 ","End":"03:31.675","Text":"This is our X. X is a randomly selected number."},{"Start":"03:31.675 ","End":"03:37.290","Text":"So this is our probability function."},{"Start":"03:37.290 ","End":"03:39.410","Text":"When we just plug in the numbers,"},{"Start":"03:39.410 ","End":"03:40.670","Text":"we have 1 over,"},{"Start":"03:40.670 ","End":"03:44.730","Text":"now b is 100 minus a,"},{"Start":"03:44.730 ","End":"03:46.215","Text":"that\u0027s 1 plus 1,"},{"Start":"03:46.215 ","End":"03:52.775","Text":"and that equals to 1/100."},{"Start":"03:52.775 ","End":"03:59.405","Text":"So that\u0027s the probability where X equals any number between 1 and 100."},{"Start":"03:59.405 ","End":"04:04.080","Text":"Now, we\u0027re also asked about the expectation."},{"Start":"04:04.310 ","End":"04:10.295","Text":"Now, what\u0027s the equation for the expectation in a uniform distribution?"},{"Start":"04:10.295 ","End":"04:14.360","Text":"That\u0027s b plus a/2."},{"Start":"04:14.360 ","End":"04:16.460","Text":"Again, let\u0027s plug in the numbers."},{"Start":"04:16.460 ","End":"04:19.740","Text":"B is 100 plus a,"},{"Start":"04:19.740 ","End":"04:23.130","Text":"a is 1, divided by 2,"},{"Start":"04:23.130 ","End":"04:26.940","Text":"and that equals 50.5."},{"Start":"04:26.940 ","End":"04:31.450","Text":"So this is basically the middle of the distribution."},{"Start":"04:31.450 ","End":"04:33.030","Text":"Between 1 and 100,"},{"Start":"04:33.030 ","End":"04:41.840","Text":"and this is the probability of landing on any random number between a and b,"},{"Start":"04:41.840 ","End":"04:44.250","Text":"between 1 and 100."}],"ID":14149},{"Watched":false,"Name":"Exercise 16","Duration":"5m 32s","ChapterTopicVideoID":13509,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.415","Text":"This question deals with a game of chance."},{"Start":"00:02.415 ","End":"00:03.780","Text":"Now, in a lot of game,"},{"Start":"00:03.780 ","End":"00:05.580","Text":"there are 45 balls numbered from"},{"Start":"00:05.580 ","End":"00:11.055","Text":"1-45 and X is the number of the first ball drawn by the machine."},{"Start":"00:11.055 ","End":"00:18.255","Text":"Now, we can see that X is distributed with a uniform distribution. Why is that?"},{"Start":"00:18.255 ","End":"00:23.505","Text":"Because it really doesn\u0027t matter what number was chosen."},{"Start":"00:23.505 ","End":"00:28.100","Text":"The probability of getting that number is a constant."},{"Start":"00:28.100 ","End":"00:34.805","Text":"It doesn\u0027t matter if it was Number 1 or Number 10 or Number 15 and so on and so forth."},{"Start":"00:34.805 ","End":"00:37.190","Text":"For each value of X,"},{"Start":"00:37.190 ","End":"00:39.955","Text":"we have the same probability."},{"Start":"00:39.955 ","End":"00:43.400","Text":"This points to a uniform distribution."},{"Start":"00:43.400 ","End":"00:47.795","Text":"Now, what\u0027s the parameters of a uniform distribution?"},{"Start":"00:47.795 ","End":"00:50.500","Text":"Well, we have a,"},{"Start":"00:50.500 ","End":"00:53.510","Text":"which is the lower bound and that equals to 1,"},{"Start":"00:53.510 ","End":"00:56.480","Text":"that\u0027s given right here and the upper bound,"},{"Start":"00:56.480 ","End":"01:00.390","Text":"that\u0027s b, but that equals to 45."},{"Start":"01:01.070 ","End":"01:06.950","Text":"The probability of X being equal to k,"},{"Start":"01:06.950 ","End":"01:11.795","Text":"well that equals to 1/b minus a plus 1,"},{"Start":"01:11.795 ","End":"01:14.000","Text":"where k equals 1,"},{"Start":"01:14.000 ","End":"01:17.105","Text":"2 and so on and so forth until 45."},{"Start":"01:17.105 ","End":"01:19.580","Text":"Now, let\u0027s just plug in the numbers."},{"Start":"01:19.580 ","End":"01:27.555","Text":"1/b is 45, 1/45 minus 1 plus 1."},{"Start":"01:27.555 ","End":"01:33.640","Text":"That equals to 1/45."},{"Start":"01:34.520 ","End":"01:43.090","Text":"Having said that, let\u0027s calculate the first section right here, a."},{"Start":"01:43.090 ","End":"01:50.645","Text":"Now a asks us to calculate the probability where X equals 2 and as we said,"},{"Start":"01:50.645 ","End":"01:54.305","Text":"it really doesn\u0027t matter what value X has,"},{"Start":"01:54.305 ","End":"01:59.795","Text":"because X is distributed with the uniform distribution, therefore,"},{"Start":"01:59.795 ","End":"02:07.200","Text":"the value of the probability of any value of X would be 1/45."},{"Start":"02:07.910 ","End":"02:14.735","Text":"Section b asks us to calculate the probability of X being less than or equal to 30."},{"Start":"02:14.735 ","End":"02:20.195","Text":"What\u0027s the probability of X being less than or equal to 30?"},{"Start":"02:20.195 ","End":"02:23.195","Text":"Well, let\u0027s take a look."},{"Start":"02:23.195 ","End":"02:25.790","Text":"The probability of X being equal to 1,"},{"Start":"02:25.790 ","End":"02:31.505","Text":"that equals 1/45 and the probability of X equaling 2,"},{"Start":"02:31.505 ","End":"02:38.555","Text":"that\u0027s also 1/45 and so on and so forth until we get to X being equal to 30."},{"Start":"02:38.555 ","End":"02:40.970","Text":"That\u0027s also 1/45."},{"Start":"02:40.970 ","End":"02:46.490","Text":"Now, how many values of X are there between 1 and 30?"},{"Start":"02:46.490 ","End":"02:55.160","Text":"Well, that\u0027s 30 minus 1 plus 1 times the probability of each one of these values."},{"Start":"02:55.160 ","End":"03:00.275","Text":"Well, each one of these values have a probability of 1/45,"},{"Start":"03:00.275 ","End":"03:05.190","Text":"that equals to 30/45."},{"Start":"03:05.720 ","End":"03:10.690","Text":"In section c, we\u0027re asked to calculate the probability of X being"},{"Start":"03:10.690 ","End":"03:15.545","Text":"greater than 4 given that X is less than or equal to 10."},{"Start":"03:15.545 ","End":"03:19.570","Text":"That\u0027s again the probability of X being greater than"},{"Start":"03:19.570 ","End":"03:23.950","Text":"4 given that X is less than or equal to 10."},{"Start":"03:23.950 ","End":"03:28.505","Text":"Well, that\u0027s a conditional probability and we do know how to solve that."},{"Start":"03:28.505 ","End":"03:33.490","Text":"In the denominator, we\u0027ll have the probability of what\u0027s given,"},{"Start":"03:33.490 ","End":"03:36.220","Text":"X being less than or equal to 10."},{"Start":"03:36.220 ","End":"03:38.620","Text":"Now, what\u0027s the numerator?"},{"Start":"03:38.620 ","End":"03:43.390","Text":"The numerator is the probability of the intersection of these 2 guys."},{"Start":"03:43.390 ","End":"03:48.800","Text":"So that\u0027s the probability where X is greater than 4,"},{"Start":"03:48.800 ","End":"03:51.285","Text":"not equaling 4, but greater than 4,"},{"Start":"03:51.285 ","End":"03:54.670","Text":"but less than or equal to 10."},{"Start":"03:55.580 ","End":"03:59.030","Text":"Let\u0027s just see what\u0027s going on here."},{"Start":"03:59.030 ","End":"04:04.055","Text":"The probability of X being less than or equal to 10, well,"},{"Start":"04:04.055 ","End":"04:11.293","Text":"that\u0027s 10/45 and what\u0027s the probability of X being greater than 4,"},{"Start":"04:11.293 ","End":"04:16.070","Text":"not equaling 4, but greater than 4 and less than or equal to 10."},{"Start":"04:16.070 ","End":"04:20.240","Text":"Well, what are the values of X which are greater than 4 and less than or equal to 10?"},{"Start":"04:20.240 ","End":"04:22.385","Text":"That\u0027s 5, 6, 7,"},{"Start":"04:22.385 ","End":"04:24.815","Text":"8, 9 and 10, that\u0027s 6."},{"Start":"04:24.815 ","End":"04:28.940","Text":"We have 6 values of X in this range."},{"Start":"04:28.940 ","End":"04:38.350","Text":"That\u0027s 6/45 and that comes out to 0.6."},{"Start":"04:38.350 ","End":"04:42.980","Text":"In section d, we\u0027re asked to calculate the probability where X equals"},{"Start":"04:42.980 ","End":"04:49.100","Text":"k. The probability of X being equal to k, well,"},{"Start":"04:49.100 ","End":"04:52.557","Text":"in our situation here in this question,"},{"Start":"04:52.557 ","End":"04:56.635","Text":"X is distributed uniformly"},{"Start":"04:56.635 ","End":"05:03.805","Text":"with the ranges where a equals 1 and b equals 45."},{"Start":"05:03.805 ","End":"05:09.380","Text":"That\u0027s easy. That equals to 1/45 because what does this say?"},{"Start":"05:09.380 ","End":"05:11.210","Text":"This basically asks us,"},{"Start":"05:11.210 ","End":"05:18.760","Text":"what\u0027s the probability of X being equal to any value within the range between 1 and 45?"},{"Start":"05:18.760 ","End":"05:22.080","Text":"Now, we know since X is a distributor formula,"},{"Start":"05:22.080 ","End":"05:25.820","Text":"it\u0027s a constant probability of 1/45."},{"Start":"05:25.820 ","End":"05:27.770","Text":"We calculated that right here,"},{"Start":"05:27.770 ","End":"05:32.100","Text":"so this is our answer for d."}],"ID":14150},{"Watched":false,"Name":"Exercise 17","Duration":"12m 14s","ChapterTopicVideoID":13510,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.580","Text":"In this question, we\u0027ll be talking about magic."},{"Start":"00:02.580 ","End":"00:07.455","Text":"A magician wishes to select a random number between 1 and 100."},{"Start":"00:07.455 ","End":"00:11.580","Text":"Assume that there\u0027s no information about the magician and we\u0027re asked,"},{"Start":"00:11.580 ","End":"00:15.555","Text":"what\u0027s the expectation and standard deviation of the number selected?"},{"Start":"00:15.555 ","End":"00:19.605","Text":"Well, we know that we have a uniform distribution."},{"Start":"00:19.605 ","End":"00:22.965","Text":"That is that X is being distributed"},{"Start":"00:22.965 ","End":"00:27.780","Text":"uniformly and we need to know what the parameters are."},{"Start":"00:27.780 ","End":"00:30.970","Text":"Now, why is it a uniform distribution?"},{"Start":"00:30.970 ","End":"00:34.939","Text":"Because no matter what number a person selects,"},{"Start":"00:34.939 ","End":"00:40.175","Text":"the probability of getting that number is uniform, it\u0027s a constant."},{"Start":"00:40.175 ","End":"00:46.175","Text":"Now, let\u0027s look at the parameters of this uniform distribution."},{"Start":"00:46.175 ","End":"00:48.215","Text":"Well, the lower bound, that\u0027s a,"},{"Start":"00:48.215 ","End":"00:49.610","Text":"that equals to 1,"},{"Start":"00:49.610 ","End":"00:51.110","Text":"that\u0027s given right here."},{"Start":"00:51.110 ","End":"00:52.490","Text":"Now, what\u0027s the upper bound?"},{"Start":"00:52.490 ","End":"00:55.105","Text":"That\u0027s b, well, that\u0027s 100."},{"Start":"00:55.105 ","End":"01:02.420","Text":"We know that the probability of X being equal to k,"},{"Start":"01:02.420 ","End":"01:09.275","Text":"well, that equals to 1 over b minus a plus 1."},{"Start":"01:09.275 ","End":"01:11.530","Text":"Now, in our case,"},{"Start":"01:11.530 ","End":"01:19.485","Text":"probability of X being equal to k,"},{"Start":"01:19.485 ","End":"01:22.730","Text":"that equals to 1 over what\u0027s b?"},{"Start":"01:22.730 ","End":"01:26.220","Text":"b is 100 minus 1 plus 1,"},{"Start":"01:27.430 ","End":"01:32.490","Text":"and that equals to 1 over 100."},{"Start":"01:33.200 ","End":"01:35.595","Text":"If that\u0027s the case,"},{"Start":"01:35.595 ","End":"01:37.370","Text":"what are we asked?"},{"Start":"01:37.370 ","End":"01:40.400","Text":"Well, we\u0027re asked to figure out what\u0027s"},{"Start":"01:40.400 ","End":"01:43.955","Text":"the expectation and standard deviation of the number."},{"Start":"01:43.955 ","End":"01:48.750","Text":"Fine, let\u0027s get to it."},{"Start":"01:51.340 ","End":"01:56.990","Text":"Now, the expectation of X,"},{"Start":"01:56.990 ","End":"02:02.320","Text":"well, that\u0027s given to us as b plus a over 2."},{"Start":"02:02.320 ","End":"02:05.420","Text":"Let\u0027s plug in the numbers, b is 100,"},{"Start":"02:05.420 ","End":"02:12.525","Text":"plus a, that\u0027s 1 over 2 and that equals to 50.5."},{"Start":"02:12.525 ","End":"02:15.555","Text":"What\u0027s the variance of X?"},{"Start":"02:15.555 ","End":"02:25.230","Text":"Well, that\u0027s b minus a plus 1 squared minus 1 over 12."},{"Start":"02:25.230 ","End":"02:27.360","Text":"There\u0027s nothing we can do about it."},{"Start":"02:27.360 ","End":"02:29.595","Text":"This is the equation for the variance,"},{"Start":"02:29.595 ","End":"02:31.540","Text":"so let\u0027s just use it."},{"Start":"02:31.540 ","End":"02:33.475","Text":"Let\u0027s plug in the numbers right now."},{"Start":"02:33.475 ","End":"02:35.195","Text":"B is 100,"},{"Start":"02:35.195 ","End":"02:38.275","Text":"minus a, that\u0027s minus 1,"},{"Start":"02:38.275 ","End":"02:44.525","Text":"plus 1 squared minus 1 over 12,"},{"Start":"02:44.525 ","End":"02:51.750","Text":"and that equals to 833.25."},{"Start":"02:51.750 ","End":"02:54.170","Text":"But hold on, we\u0027re not asked about the variance,"},{"Start":"02:54.170 ","End":"02:57.050","Text":"we\u0027re asked about the standard deviation."},{"Start":"02:57.050 ","End":"03:00.190","Text":"The standard deviation of X,"},{"Start":"03:00.190 ","End":"03:03.109","Text":"that equals to the square root of the variance,"},{"Start":"03:03.109 ","End":"03:10.560","Text":"which is 833.25 and that equals to 28.87."},{"Start":"03:13.900 ","End":"03:17.660","Text":"In this section, we\u0027re given that the magician asks"},{"Start":"03:17.660 ","End":"03:20.720","Text":"6 people to select a number and we\u0027re asked,"},{"Start":"03:20.720 ","End":"03:24.665","Text":"what\u0027s the probability that 3 of them will select a number larger than 80?"},{"Start":"03:24.665 ","End":"03:28.370","Text":"Now, here, this reminds me more of"},{"Start":"03:28.370 ","End":"03:32.270","Text":"a binomial distribution more than a uniform distribution."},{"Start":"03:32.270 ","End":"03:36.125","Text":"But let\u0027s see how everything just gels together."},{"Start":"03:36.125 ","End":"03:42.170","Text":"First of all, let\u0027s remember that X was defined as"},{"Start":"03:42.170 ","End":"03:48.225","Text":"any number chosen randomly between 1 and 100,"},{"Start":"03:48.225 ","End":"03:54.855","Text":"so the probability was 1 over 100 to select any number."},{"Start":"03:54.855 ","End":"03:58.730","Text":"Now, what are we looking for right now?"},{"Start":"03:58.730 ","End":"04:03.845","Text":"We\u0027re looking at a new variable, Y."},{"Start":"04:03.845 ","End":"04:08.515","Text":"We\u0027ll define that is the number that\u0027s chosen,"},{"Start":"04:08.515 ","End":"04:11.715","Text":"which is over 80."},{"Start":"04:11.715 ","End":"04:14.010","Text":"That would be our success,"},{"Start":"04:14.010 ","End":"04:20.960","Text":"that would be the random variable that we\u0027ll have to deal with."},{"Start":"04:20.960 ","End":"04:27.080","Text":"Now, this random variable is binomially distributed. Now, why is that?"},{"Start":"04:27.080 ","End":"04:30.560","Text":"Well, because we\u0027re taking 3 out of"},{"Start":"04:30.560 ","End":"04:33.410","Text":"the 6 people and we want to know what\u0027s"},{"Start":"04:33.410 ","End":"04:38.150","Text":"the probability that 3 of them will select a number larger than 80."},{"Start":"04:38.150 ","End":"04:43.290","Text":"First of all, n is equal to 6,"},{"Start":"04:43.290 ","End":"04:49.230","Text":"the magician asks 6 people and k will be equal to 3,"},{"Start":"04:49.230 ","End":"04:53.045","Text":"3 of them will select a number larger than 80."},{"Start":"04:53.045 ","End":"04:57.410","Text":"Now, we need to figure out what\u0027s the probability"},{"Start":"04:57.410 ","End":"05:02.960","Text":"of a number being chosen that\u0027s greater than 80?"},{"Start":"05:02.960 ","End":"05:05.510","Text":"Well, let\u0027s go back here."},{"Start":"05:05.510 ","End":"05:09.710","Text":"X is any number chosen with a probability of 1 over 100,"},{"Start":"05:09.710 ","End":"05:15.310","Text":"so what\u0027s the probability of X being greater than 80?"},{"Start":"05:15.310 ","End":"05:17.795","Text":"Well, that equals to,"},{"Start":"05:17.795 ","End":"05:20.165","Text":"first of all, something over 100."},{"Start":"05:20.165 ","End":"05:25.925","Text":"Now, how many values of X do we have which are over 80?"},{"Start":"05:25.925 ","End":"05:29.710","Text":"Well, we have 81, 82, 83, until 100."},{"Start":"05:29.710 ","End":"05:31.070","Text":"Now, if we count them up,"},{"Start":"05:31.070 ","End":"05:33.305","Text":"we\u0027ll see that we have 20 of them,"},{"Start":"05:33.305 ","End":"05:36.035","Text":"that equals to 0.2."},{"Start":"05:36.035 ","End":"05:43.985","Text":"The probability of a number that\u0027s chosen to be over 80,"},{"Start":"05:43.985 ","End":"05:46.890","Text":"well, that would be 0.2."},{"Start":"05:47.090 ","End":"05:52.175","Text":"If that\u0027s the case and Y is now a random variable,"},{"Start":"05:52.175 ","End":"05:58.790","Text":"then we know that Y is being distributed binomially,"},{"Start":"05:58.790 ","End":"06:04.550","Text":"where n equals 6 and p equals 0.2."},{"Start":"06:04.550 ","End":"06:09.770","Text":"Now if we remember the probability of Y being equal to k,"},{"Start":"06:09.770 ","End":"06:14.840","Text":"well, that equals to n over k, p^k,"},{"Start":"06:14.840 ","End":"06:18.740","Text":"q^n minus k,"},{"Start":"06:18.740 ","End":"06:24.304","Text":"where we remember q is 1 minus p. Let\u0027s just plug in the numbers,"},{"Start":"06:24.304 ","End":"06:26.270","Text":"k here is equal to 3."},{"Start":"06:26.270 ","End":"06:31.700","Text":"The probability of Y being equal to 3 and that\u0027s what we\u0027re looking for."},{"Start":"06:31.700 ","End":"06:36.560","Text":"We\u0027re looking for 3 people who selected a number larger than 80."},{"Start":"06:36.560 ","End":"06:40.415","Text":"Well, that\u0027s the probability of Y being equal to 3."},{"Start":"06:40.415 ","End":"06:42.440","Text":"That equals to n over k,"},{"Start":"06:42.440 ","End":"06:45.875","Text":"n is 6, k is 3,"},{"Start":"06:45.875 ","End":"06:49.910","Text":"p is equal to 0.2,"},{"Start":"06:49.910 ","End":"06:56.060","Text":"0.2^3, to the power of k, times q."},{"Start":"06:56.060 ","End":"06:58.850","Text":"That\u0027s 0.8^n minus k,"},{"Start":"06:58.850 ","End":"07:00.215","Text":"6 minus 3,"},{"Start":"07:00.215 ","End":"07:02.140","Text":"that also equals 3,"},{"Start":"07:02.140 ","End":"07:05.620","Text":"and this turns out to be 0.08192."},{"Start":"07:11.680 ","End":"07:13.700","Text":"In this next section,"},{"Start":"07:13.700 ","End":"07:17.690","Text":"we\u0027re asked what are the expectation and standard deviation of the sum of"},{"Start":"07:17.690 ","End":"07:22.080","Text":"the numbers selected by the people? Let\u0027s get started."},{"Start":"07:22.080 ","End":"07:29.030","Text":"We know from the previous section that X was distributed with a uniform distribution."},{"Start":"07:29.030 ","End":"07:33.740","Text":"X was distributed with a uniform distribution,"},{"Start":"07:33.740 ","End":"07:38.255","Text":"where a equals 1 and b equals 100."},{"Start":"07:38.255 ","End":"07:43.935","Text":"Now, let\u0027s define X_i. What\u0027s that?"},{"Start":"07:43.935 ","End":"07:47.970","Text":"That would be the number that was"},{"Start":"07:47.970 ","End":"07:55.550","Text":"selected by the ith person."},{"Start":"07:55.550 ","End":"08:02.455","Text":"Now, this has the same distribution as we figured out previously."},{"Start":"08:02.455 ","End":"08:07.570","Text":"This X_i is distributed also uniformly,"},{"Start":"08:07.570 ","End":"08:12.620","Text":"where a equals 1 and b equals 100."},{"Start":"08:12.660 ","End":"08:15.625","Text":"What do we need to do right now?"},{"Start":"08:15.625 ","End":"08:22.120","Text":"Well, first of all, let\u0027s also recall that the expectation of X was"},{"Start":"08:22.120 ","End":"08:31.425","Text":"50.5 and the variance of X was 833.25."},{"Start":"08:31.425 ","End":"08:40.175","Text":"Now, that means that the expectation of the number of the ith person"},{"Start":"08:40.175 ","End":"08:51.180","Text":"is also 50.5 and the variance of X of the ith person is also 833.25."},{"Start":"08:51.180 ","End":"08:54.115","Text":"Let\u0027s see how we can continue."},{"Start":"08:54.115 ","End":"08:56.485","Text":"Now, what are we asked?"},{"Start":"08:56.485 ","End":"09:03.730","Text":"We\u0027re asked to find the expectation and standard deviation of the sum of the numbers."},{"Start":"09:03.730 ","End":"09:06.115","Text":"The sum of the numbers, that\u0027s the total."},{"Start":"09:06.115 ","End":"09:08.583","Text":"Let\u0027s just write this out."},{"Start":"09:08.583 ","End":"09:13.610","Text":"The total, let\u0027s write this out as the sum of X_i,"},{"Start":"09:13.610 ","End":"09:21.280","Text":"where i equals 1 to 6 and we\u0027re looking for the expectation of the total."},{"Start":"09:21.280 ","End":"09:24.605","Text":"Now, from previous lessons,"},{"Start":"09:24.605 ","End":"09:28.505","Text":"we know that the expectation of the sum"},{"Start":"09:28.505 ","End":"09:33.200","Text":"equals the sum of the expectations and that\u0027s always."},{"Start":"09:33.200 ","End":"09:37.325","Text":"That equals to the expectation of X_1"},{"Start":"09:37.325 ","End":"09:43.205","Text":"plus the expectation of X_2 and so on and so forth until when?"},{"Start":"09:43.205 ","End":"09:46.460","Text":"Until the expectation of X_6,"},{"Start":"09:46.460 ","End":"09:48.570","Text":"we have 6 people."},{"Start":"09:49.000 ","End":"09:52.240","Text":"What is the expectation of X_1?"},{"Start":"09:52.240 ","End":"09:55.540","Text":"Well, that\u0027s 50.5."},{"Start":"09:55.540 ","End":"09:59.125","Text":"What about the expectation of X_2?"},{"Start":"09:59.125 ","End":"10:04.820","Text":"Again, that\u0027s 50.5 and so on and so forth, 6 times."},{"Start":"10:04.820 ","End":"10:11.555","Text":"We have 6 times 50.5 and that equals to 303."},{"Start":"10:11.555 ","End":"10:20.720","Text":"This is the expectation of the sum of the numbers selected by these 6 people."},{"Start":"10:21.580 ","End":"10:24.800","Text":"Let\u0027s take a look at the variance."},{"Start":"10:24.800 ","End":"10:32.130","Text":"What\u0027s the variance of the total or the sum of the number selected by the 6 people?"},{"Start":"10:32.130 ","End":"10:35.185","Text":"Well, that\u0027s the variance of the sum."},{"Start":"10:35.185 ","End":"10:40.460","Text":"Now, when does that equal to the sum of the variance?"},{"Start":"10:40.460 ","End":"10:44.750","Text":"That means that we\u0027re looking at the variance of X_1 plus"},{"Start":"10:44.750 ","End":"10:50.720","Text":"the variance of X_2 and so on and so forth until the variance of X_6."},{"Start":"10:50.720 ","End":"10:53.030","Text":"Now, when is that equal to this?"},{"Start":"10:53.030 ","End":"10:57.410","Text":"Under the condition that X_1 and X_2 and so forth,"},{"Start":"10:57.410 ","End":"11:01.610","Text":"all the 6 people are independent of each other and they are."},{"Start":"11:01.610 ","End":"11:08.665","Text":"They\u0027re selected randomly and they choose their numbers independently of each other."},{"Start":"11:08.665 ","End":"11:12.979","Text":"The variance of the sum here equals the sum of the variance."},{"Start":"11:12.979 ","End":"11:15.755","Text":"Again, just like in the expectation,"},{"Start":"11:15.755 ","End":"11:20.310","Text":"the variance of each X_i of each X,"},{"Start":"11:20.310 ","End":"11:21.525","Text":"X_1, X_2, X_3,"},{"Start":"11:21.525 ","End":"11:23.460","Text":"is the same and that\u0027s"},{"Start":"11:23.460 ","End":"11:32.160","Text":"833.25 plus 833.25 and so on and so forth."},{"Start":"11:32.160 ","End":"11:39.840","Text":"6 times 833.25 and"},{"Start":"11:39.840 ","End":"11:47.170","Text":"that turns out to be 4,999.5."},{"Start":"11:48.280 ","End":"11:52.190","Text":"But again, we\u0027re not asked for the variance,"},{"Start":"11:52.190 ","End":"11:54.980","Text":"we\u0027re asked for the standard deviation."},{"Start":"11:54.980 ","End":"11:56.600","Text":"Let\u0027s figure that out."},{"Start":"11:56.600 ","End":"12:00.095","Text":"Now, the standard deviation of the total, well,"},{"Start":"12:00.095 ","End":"12:03.290","Text":"that equals to the square root of the variance and that equals to"},{"Start":"12:03.290 ","End":"12:08.540","Text":"the square root of 4,999.5,"},{"Start":"12:08.540 ","End":"12:14.340","Text":"which equals to 70.71."}],"ID":14151},{"Watched":false,"Name":"Exercise 18","Duration":"7m 50s","ChapterTopicVideoID":13511,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.820","Text":"This question will be talking about throwing a die."},{"Start":"00:03.820 ","End":"00:06.690","Text":"Now X is a result of throwing 1 die,"},{"Start":"00:06.690 ","End":"00:10.075","Text":"and we\u0027re asked what is the probability distribution of X?"},{"Start":"00:10.075 ","End":"00:15.565","Text":"Well, we know that X is uniformly distributed,"},{"Start":"00:15.565 ","End":"00:19.945","Text":"where a equals 1 and b equals 6."},{"Start":"00:19.945 ","End":"00:21.535","Text":"Now, why is that?"},{"Start":"00:21.535 ","End":"00:25.060","Text":"Well, what is the values of a die?"},{"Start":"00:25.060 ","End":"00:27.310","Text":"Well, a die can have a value of 1,"},{"Start":"00:27.310 ","End":"00:28.894","Text":"or 2, or 3,"},{"Start":"00:28.894 ","End":"00:30.250","Text":"or 4, or 5 or 6."},{"Start":"00:30.250 ","End":"00:34.585","Text":"What is the probability of getting any one of the values?"},{"Start":"00:34.585 ","End":"00:37.150","Text":"Well, the probabilities are the same."},{"Start":"00:37.150 ","End":"00:38.770","Text":"It\u0027s a constant."},{"Start":"00:38.770 ","End":"00:42.870","Text":"Now, if X is uniformly distributed,"},{"Start":"00:42.870 ","End":"00:51.240","Text":"then the probability of X equaling k equals to 1/b minus a plus 1."},{"Start":"00:51.240 ","End":"00:56.235","Text":"In our case, that\u0027s 1/6 minus 1 plus 1,"},{"Start":"00:56.235 ","End":"00:59.320","Text":"that equals to 1/6."},{"Start":"01:00.040 ","End":"01:04.580","Text":"In section B, we\u0027re asked what\u0027s the expectation of X?"},{"Start":"01:04.580 ","End":"01:14.270","Text":"Well, if X is distributed uniformly then the expectation of X we have a formula for that,"},{"Start":"01:14.270 ","End":"01:17.660","Text":"That\u0027s b plus a/2."},{"Start":"01:17.660 ","End":"01:19.325","Text":"Now, let\u0027s plug that in."},{"Start":"01:19.325 ","End":"01:25.190","Text":"That\u0027s 6 plus 1/2 and that equals to 3.5."},{"Start":"01:25.190 ","End":"01:27.905","Text":"Now, having done that,"},{"Start":"01:27.905 ","End":"01:32.720","Text":"we calculated the expectation of X in a various number of ways."},{"Start":"01:32.720 ","End":"01:40.880","Text":"One of them would be the sum of X times the probability of X."},{"Start":"01:40.880 ","End":"01:43.925","Text":"Another technique would be to use a table."},{"Start":"01:43.925 ","End":"01:48.560","Text":"We have X and we have the probability of X,"},{"Start":"01:48.560 ","End":"01:51.980","Text":"where X would be 1, or 2, or 3,"},{"Start":"01:51.980 ","End":"01:53.570","Text":"or 4, or 5,"},{"Start":"01:53.570 ","End":"01:57.950","Text":"or 6 with a probability of a 6th,"},{"Start":"01:57.950 ","End":"02:00.560","Text":"for each one of the values."},{"Start":"02:00.560 ","End":"02:05.675","Text":"Now, I\u0027m not saying that these techniques are not valid,"},{"Start":"02:05.675 ","End":"02:09.890","Text":"but in this case they are redundant. Why is that?"},{"Start":"02:09.890 ","End":"02:15.095","Text":"Because we have a shortened version to calculate the expectation of X."},{"Start":"02:15.095 ","End":"02:18.875","Text":"In the case of a uniform distribution,"},{"Start":"02:18.875 ","End":"02:27.035","Text":"that short form that equation to calculate the expectation of X is this guy right here."},{"Start":"02:27.035 ","End":"02:29.467","Text":"These guys are good,"},{"Start":"02:29.467 ","End":"02:33.600","Text":"but in this case they\u0027re redundant."},{"Start":"02:34.860 ","End":"02:39.536","Text":"In this section, we\u0027re given that the die is thrown 4 times."},{"Start":"02:39.536 ","End":"02:42.370","Text":"We\u0027re asked what are the expectation and variance of"},{"Start":"02:42.370 ","End":"02:45.875","Text":"the sum of the results of the 4th rows?"},{"Start":"02:45.875 ","End":"02:49.800","Text":"Let\u0027s define X_i as"},{"Start":"02:49.800 ","End":"02:58.360","Text":"the result of the Ith throw."},{"Start":"02:59.810 ","End":"03:05.310","Text":"In essence, what we\u0027re looking at is throwing a dice 4"},{"Start":"03:05.310 ","End":"03:10.630","Text":"times and X_1 would be the result in the 1st row, X_2,"},{"Start":"03:10.630 ","End":"03:12.475","Text":"the result of the 2nd throw,"},{"Start":"03:12.475 ","End":"03:20.640","Text":"and so on and so forth for the 4th row so I equal 1-4."},{"Start":"03:20.640 ","End":"03:23.220","Text":"Now, what do we know about X_i?"},{"Start":"03:23.220 ","End":"03:25.725","Text":"Well, we know that X_i,"},{"Start":"03:25.725 ","End":"03:29.190","Text":"all the Xs are independent of each other."},{"Start":"03:29.190 ","End":"03:32.445","Text":"Each row is independent of the other,"},{"Start":"03:32.445 ","End":"03:37.950","Text":"and we also know that they have the same distribution."},{"Start":"03:38.780 ","End":"03:41.555","Text":"Now, what does that mean?"},{"Start":"03:41.555 ","End":"03:47.255","Text":"That means that each X_i is distributed uniformly,"},{"Start":"03:47.255 ","End":"03:51.905","Text":"where a equals 1 and b equals 6."},{"Start":"03:51.905 ","End":"03:58.550","Text":"Now, we know also that since X was distributed,"},{"Start":"03:58.550 ","End":"04:06.560","Text":"only 1 throw was distributed with a uniform distribution where a equals 1 and b equals 6."},{"Start":"04:06.560 ","End":"04:10.165","Text":"We know that the expectation of X,"},{"Start":"04:10.165 ","End":"04:12.950","Text":"that equal to 3.5."},{"Start":"04:12.950 ","End":"04:16.620","Text":"We calculated that in the last section."},{"Start":"04:16.750 ","End":"04:22.820","Text":"Now, what we\u0027re looking at is the expectation,"},{"Start":"04:22.820 ","End":"04:29.660","Text":"right here, and variance of the sum of the X_i\u0027s."},{"Start":"04:29.660 ","End":"04:32.885","Text":"That means that we\u0027re looking now at T,"},{"Start":"04:32.885 ","End":"04:36.850","Text":"total is equal to the sum, of X_i,"},{"Start":"04:36.850 ","End":"04:40.710","Text":"where X_i equals 1-4,"},{"Start":"04:40.710 ","End":"04:48.910","Text":"or X_1 plus X_2 plus X_3 plus X_4."},{"Start":"04:49.190 ","End":"04:51.320","Text":"Now what we want to know,"},{"Start":"04:51.320 ","End":"04:55.835","Text":"we want to know what\u0027s the expectation of the sum of the total?"},{"Start":"04:55.835 ","End":"04:59.735","Text":"Well, we know that the expectation of the sum"},{"Start":"04:59.735 ","End":"05:04.550","Text":"is always equals to the sum of the expectation."},{"Start":"05:04.550 ","End":"05:09.730","Text":"That means that we\u0027re looking at the expectation of X_1 plus the expectation of"},{"Start":"05:09.730 ","End":"05:17.860","Text":"X_2 plus the expectation of X_3 plus the expectation of X_4."},{"Start":"05:17.860 ","End":"05:22.295","Text":"Now, what did we say about the individual X_i\u0027s?"},{"Start":"05:22.295 ","End":"05:25.230","Text":"They have the same distribution."},{"Start":"05:25.230 ","End":"05:30.260","Text":"That means that the expectation is the same for each exercise."},{"Start":"05:30.260 ","End":"05:33.950","Text":"That means that\u0027s 3.5 plus"},{"Start":"05:33.950 ","End":"05:41.275","Text":"3.5 plus 3.5 plus 3.5 and 5."},{"Start":"05:41.275 ","End":"05:46.260","Text":"That equals to 14,"},{"Start":"05:46.260 ","End":"05:53.990","Text":"Which means that if we sum up all the results of the 4th rows when throwing a die,"},{"Start":"05:53.990 ","End":"05:58.620","Text":"we\u0027re expected to get a value of 14."},{"Start":"06:00.170 ","End":"06:05.180","Text":"Let\u0027s take a look now at the variance."},{"Start":"06:05.180 ","End":"06:07.265","Text":"The variance of X,"},{"Start":"06:07.265 ","End":"06:12.290","Text":"which is an individual throw that equals to b minus"},{"Start":"06:12.290 ","End":"06:18.568","Text":"a plus 1 squared minus 1/12."},{"Start":"06:18.568 ","End":"06:20.255","Text":"When we plug in, the numbers,"},{"Start":"06:20.255 ","End":"06:26.390","Text":"we\u0027ll get 6 minus 1 plus 1 squared minus 1/12,"},{"Start":"06:26.390 ","End":"06:31.845","Text":"and that equals to 2.917."},{"Start":"06:31.845 ","End":"06:33.995","Text":"Having said that, now,"},{"Start":"06:33.995 ","End":"06:36.785","Text":"what\u0027s the variance of the total?"},{"Start":"06:36.785 ","End":"06:42.485","Text":"That means that the variance of the sum of the X_i\u0027s, well,"},{"Start":"06:42.485 ","End":"06:47.630","Text":"that equals to the sum of the variances,"},{"Start":"06:47.630 ","End":"06:53.010","Text":"the variance of X_1 plus the variance of X_2 plus"},{"Start":"06:53.010 ","End":"06:58.315","Text":"the variance X_3 plus the variance of X_4."},{"Start":"06:58.315 ","End":"07:02.060","Text":"Now, that\u0027s true under which condition?"},{"Start":"07:02.060 ","End":"07:05.255","Text":"Well, the condition is that each X,"},{"Start":"07:05.255 ","End":"07:09.290","Text":"all the X_i\u0027s have to be independent."},{"Start":"07:09.290 ","End":"07:13.250","Text":"Now, because they\u0027re not only independent,"},{"Start":"07:13.250 ","End":"07:15.860","Text":"but they have the same distribution,"},{"Start":"07:15.860 ","End":"07:19.180","Text":"then they also have the same variance."},{"Start":"07:19.180 ","End":"07:25.820","Text":"The variance for X_1 equals 2.917 and that equals to"},{"Start":"07:25.820 ","End":"07:33.175","Text":"the variance of X_2 and X_3 and X_4 so we have to multiply this 4 times,"},{"Start":"07:33.175 ","End":"07:39.570","Text":"and that equals to 11.668."},{"Start":"07:39.570 ","End":"07:42.035","Text":"Here we go, we\u0027ve answered the question."},{"Start":"07:42.035 ","End":"07:50.110","Text":"We\u0027ve calculated the variance of the total and the expectation of the total."}],"ID":14152},{"Watched":false,"Name":"Exercise 19","Duration":"6m 4s","ChapterTopicVideoID":13512,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"In this question, we\u0027ll be talking about removing balls from a basket."},{"Start":"00:04.140 ","End":"00:06.585","Text":"Now, there are 10 balls in a basket,"},{"Start":"00:06.585 ","End":"00:08.400","Text":"only 1 of which is red."},{"Start":"00:08.400 ","End":"00:10.410","Text":"A person takes out 2 balls,"},{"Start":"00:10.410 ","End":"00:14.145","Text":"1 after the other without returns until he removes the red ball,"},{"Start":"00:14.145 ","End":"00:15.525","Text":"and then he stops."},{"Start":"00:15.525 ","End":"00:21.345","Text":"Now, what are the expectation and variance of the number of balls that were removed?"},{"Start":"00:21.345 ","End":"00:26.535","Text":"Now, this this isn\u0027t a trivial question."},{"Start":"00:26.535 ","End":"00:31.050","Text":"It requires a little bit of thought. Let\u0027s get to it."},{"Start":"00:31.050 ","End":"00:32.505","Text":"Well, first of all,"},{"Start":"00:32.505 ","End":"00:33.675","Text":"what are we looking at?"},{"Start":"00:33.675 ","End":"00:35.595","Text":"What\u0027s a random variable?"},{"Start":"00:35.595 ","End":"00:41.395","Text":"Well, our random variable is the number of balls that were removed."},{"Start":"00:41.395 ","End":"00:47.930","Text":"We\u0027re looking to calculate the expectation and variance of this random variable."},{"Start":"00:47.930 ","End":"00:50.255","Text":"Let\u0027s just write the sum,"},{"Start":"00:50.255 ","End":"00:57.370","Text":"x equals the number of balls that were removed."},{"Start":"00:57.770 ","End":"01:02.495","Text":"Now, I maintain that x has"},{"Start":"01:02.495 ","End":"01:09.350","Text":"a uniform distribution where a equals 1 and b equals 10."},{"Start":"01:09.350 ","End":"01:12.080","Text":"Now, to prove this,"},{"Start":"01:12.080 ","End":"01:15.005","Text":"I\u0027ll need the help of the probability tree."},{"Start":"01:15.005 ","End":"01:21.600","Text":"Let\u0027s construct this tree and see why this is so."},{"Start":"01:22.430 ","End":"01:25.410","Text":"This is basically the first step."},{"Start":"01:25.410 ","End":"01:27.945","Text":"When we get to the basket,"},{"Start":"01:27.945 ","End":"01:33.050","Text":"we initially have 10 balls and we only have 1 red ball."},{"Start":"01:33.050 ","End":"01:36.110","Text":"So the probability of getting a red ball is"},{"Start":"01:36.110 ","End":"01:42.125","Text":"1/10 and the probability of getting a ball that\u0027s not red is 9/10."},{"Start":"01:42.125 ","End":"01:44.630","Text":"Now, if we got the red ball right here,"},{"Start":"01:44.630 ","End":"01:51.780","Text":"we stop and if we didn\u0027t get a red ball, we continue."},{"Start":"01:51.910 ","End":"01:55.085","Text":"Now, this is the second step."},{"Start":"01:55.085 ","End":"01:58.055","Text":"In the second step we have 9 balls."},{"Start":"01:58.055 ","End":"02:02.345","Text":"Again, if we didn\u0027t get a red ball the first step,"},{"Start":"02:02.345 ","End":"02:08.055","Text":"then the probability of getting a red ball the second step is 1/9."},{"Start":"02:08.055 ","End":"02:11.175","Text":"If we got a red ball there, then we stop."},{"Start":"02:11.175 ","End":"02:14.270","Text":"If not, then we continue because"},{"Start":"02:14.270 ","End":"02:18.395","Text":"the probability of not getting a red ball right here is 8/9."},{"Start":"02:18.395 ","End":"02:23.985","Text":"Let\u0027s continue to the next step. Here\u0027s the next step."},{"Start":"02:23.985 ","End":"02:27.860","Text":"The next step, we have 8 balls and the probability of getting"},{"Start":"02:27.860 ","End":"02:33.230","Text":"a red ball would be 1/8 and if we get that red ball then we stop."},{"Start":"02:33.230 ","End":"02:40.635","Text":"If not, then the probability of getting a not red is 7/8 and so on and so forth."},{"Start":"02:40.635 ","End":"02:47.400","Text":"Now, let\u0027s calculate the probability of x being equal to 1."},{"Start":"02:47.400 ","End":"02:49.910","Text":"Well, when x equals to 1,"},{"Start":"02:49.910 ","End":"02:53.615","Text":"then we\u0027re looking at this branch right here."},{"Start":"02:53.615 ","End":"03:01.270","Text":"Now, that would mean that we got the red ball on the first try,"},{"Start":"03:01.270 ","End":"03:05.140","Text":"so the probability of that happening is 1/10."},{"Start":"03:05.140 ","End":"03:11.425","Text":"Now, let\u0027s take a look at the probability where x equals 2."},{"Start":"03:11.425 ","End":"03:16.240","Text":"X equals 2 means that we didn\u0027t get the ball on the first try,"},{"Start":"03:16.240 ","End":"03:19.540","Text":"but we did get the red ball on the second try."},{"Start":"03:19.540 ","End":"03:22.020","Text":"The probability of x equals 2,"},{"Start":"03:22.020 ","End":"03:28.590","Text":"that\u0027s 9/10 times 1/9."},{"Start":"03:28.590 ","End":"03:30.895","Text":"We didn\u0027t get it on the first try,"},{"Start":"03:30.895 ","End":"03:33.790","Text":"and we did get it under second try."},{"Start":"03:33.790 ","End":"03:36.190","Text":"That\u0027s the multiplication of these probabilities."},{"Start":"03:36.190 ","End":"03:41.500","Text":"Now, these 9s cancel out and we get a probability of 1/10."},{"Start":"03:41.500 ","End":"03:45.940","Text":"Well, let\u0027s take a look at what happens when x equals 3."},{"Start":"03:45.940 ","End":"03:49.120","Text":"Again, that means that we didn\u0027t get it in the first try,"},{"Start":"03:49.120 ","End":"03:51.220","Text":"we didn\u0027t get the ball in the second try,"},{"Start":"03:51.220 ","End":"03:54.035","Text":"but in the third try we did get the ball."},{"Start":"03:54.035 ","End":"03:56.530","Text":"Let\u0027s multiply the probabilities."},{"Start":"03:56.530 ","End":"04:04.905","Text":"That\u0027s 9/10 times 8/9 times 1/8."},{"Start":"04:04.905 ","End":"04:08.595","Text":"That\u0027s 9/10 times 8/9,"},{"Start":"04:08.595 ","End":"04:11.360","Text":"and we did get it on the third try, so that\u0027s 1/8."},{"Start":"04:11.360 ","End":"04:13.535","Text":"This is this branch right here."},{"Start":"04:13.535 ","End":"04:15.855","Text":"Again, the 9s cancel out,"},{"Start":"04:15.855 ","End":"04:16.940","Text":"the 8s cancel out,"},{"Start":"04:16.940 ","End":"04:18.830","Text":"and we get 1/10."},{"Start":"04:18.830 ","End":"04:25.220","Text":"The probability of x being equal to k, well,"},{"Start":"04:25.220 ","End":"04:27.785","Text":"that equals to 1/10,"},{"Start":"04:27.785 ","End":"04:30.110","Text":"where k equals 1, 2,"},{"Start":"04:30.110 ","End":"04:33.420","Text":"3, and so forth until 10."},{"Start":"04:34.360 ","End":"04:39.060","Text":"This is our probability tree,"},{"Start":"04:39.060 ","End":"04:43.100","Text":"and we saw that for every value of x,"},{"Start":"04:43.100 ","End":"04:46.265","Text":"the probability was 1/10."},{"Start":"04:46.265 ","End":"04:50.060","Text":"That makes this a uniform distribution."},{"Start":"04:50.060 ","End":"04:54.830","Text":"Now we know for sure we proved that x is distributed with"},{"Start":"04:54.830 ","End":"05:00.560","Text":"a uniform distribution where a equals 1 and b equals 10."},{"Start":"05:00.560 ","End":"05:02.420","Text":"Now, what are we asked?"},{"Start":"05:02.420 ","End":"05:06.580","Text":"We\u0027re asked for the expectation and variance of x."},{"Start":"05:06.580 ","End":"05:08.675","Text":"Let\u0027s figure that out."},{"Start":"05:08.675 ","End":"05:12.765","Text":"Well, the expectation of x,"},{"Start":"05:12.765 ","End":"05:19.280","Text":"where x is distributed with the uniform distribution is b plus a/2."},{"Start":"05:19.280 ","End":"05:21.830","Text":"Well, let\u0027s just plug in the numbers."},{"Start":"05:21.830 ","End":"05:27.390","Text":"That\u0027s 10 plus 1/2 and that\u0027s 5.5."},{"Start":"05:27.390 ","End":"05:30.065","Text":"What\u0027s the variance of x?"},{"Start":"05:30.065 ","End":"05:38.155","Text":"Well, that\u0027s b minus a plus 1 squared minus 1/12."},{"Start":"05:38.155 ","End":"05:40.535","Text":"Again, plugging in the numbers,"},{"Start":"05:40.535 ","End":"05:48.185","Text":"that\u0027s 10 minus 1 plus 1 squared minus 1/12,"},{"Start":"05:48.185 ","End":"05:53.305","Text":"and that equals to 8.25."},{"Start":"05:53.305 ","End":"05:57.285","Text":"This is the variance of x,"},{"Start":"05:57.285 ","End":"05:59.160","Text":"that\u0027s what we\u0027re asked for right here,"},{"Start":"05:59.160 ","End":"06:01.380","Text":"and this is the expectation,"},{"Start":"06:01.380 ","End":"06:03.730","Text":"that\u0027s were asked right here."}],"ID":14153},{"Watched":false,"Name":"Exercise 20","Duration":"5m 32s","ChapterTopicVideoID":13513,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.575","Text":"In this question, we\u0027ll be talking about randomly selecting numbers."},{"Start":"00:04.575 ","End":"00:09.715","Text":"Now, a number selected randomly between 1 and 50 inclusively."},{"Start":"00:09.715 ","End":"00:14.035","Text":"We\u0027re asked what are the chances that the number 4 will be selected?"},{"Start":"00:14.035 ","End":"00:22.018","Text":"Well, first of all, let\u0027s define x as being the number that was selected."},{"Start":"00:22.018 ","End":"00:30.980","Text":"Now, x has a uniform distribution where a equals 1 and b equals 50. Why is that?"},{"Start":"00:30.980 ","End":"00:36.004","Text":"Well, for every value that x has between 1 and 50,"},{"Start":"00:36.004 ","End":"00:41.255","Text":"each value has the same probability of being selected."},{"Start":"00:41.255 ","End":"00:46.579","Text":"We also know that because x is uniformly distributed,"},{"Start":"00:46.579 ","End":"00:50.570","Text":"that the probability of x being equal to k,"},{"Start":"00:50.570 ","End":"00:56.135","Text":"well, that equals to 1 over b minus a plus 1."},{"Start":"00:56.135 ","End":"01:00.835","Text":"Okay? Now, having said that, let\u0027s answer a."},{"Start":"01:00.835 ","End":"01:04.340","Text":"A asks us, what are the chances?"},{"Start":"01:04.340 ","End":"01:09.230","Text":"What\u0027s the probability that the number 4 will be selected x being equal to 4."},{"Start":"01:09.230 ","End":"01:13.490","Text":"Well, that equals to 1 over now,"},{"Start":"01:13.490 ","End":"01:16.350","Text":"b is 50, that\u0027s 50."},{"Start":"01:16.350 ","End":"01:17.700","Text":"Let\u0027s just plug in the numbers."},{"Start":"01:17.700 ","End":"01:21.260","Text":"Minus 1,A is 1 plus 1,"},{"Start":"01:21.260 ","End":"01:26.640","Text":"and that equals to 1/50."},{"Start":"01:27.020 ","End":"01:29.570","Text":"In section b, we\u0027re asked what are"},{"Start":"01:29.570 ","End":"01:33.280","Text":"the chances that the number selected is larger than 20?"},{"Start":"01:33.280 ","End":"01:36.495","Text":"Well, being saying,"},{"Start":"01:36.495 ","End":"01:40.385","Text":"what\u0027s the probability what x is greater than 20?"},{"Start":"01:40.385 ","End":"01:45.335","Text":"Well, if x is greater than 20, non-inclusive,"},{"Start":"01:45.335 ","End":"01:52.820","Text":"then we can write this as x is greater or equal to 21."},{"Start":"01:52.820 ","End":"01:57.805","Text":"What\u0027s the probability of x being greater or equal to 21?"},{"Start":"01:57.805 ","End":"02:02.570","Text":"That equals to the probability of x being greater than 20. Why is that?"},{"Start":"02:02.570 ","End":"02:09.420","Text":"Well, because x is a discrete random variable with increments of 1."},{"Start":"02:09.830 ","End":"02:13.155","Text":"We can write this out."},{"Start":"02:13.155 ","End":"02:18.000","Text":"What\u0027s the probability of x equaling any value?"},{"Start":"02:18.000 ","End":"02:19.545","Text":"That\u0027s 1/50."},{"Start":"02:19.545 ","End":"02:22.220","Text":"Let\u0027s put 50 in the denominator."},{"Start":"02:22.220 ","End":"02:24.860","Text":"Now, what do we have to do with the numerator?"},{"Start":"02:24.860 ","End":"02:27.935","Text":"Well, we have to know or calculate"},{"Start":"02:27.935 ","End":"02:33.650","Text":"how many values of x there are between 21 and 50 inclusive."},{"Start":"02:33.650 ","End":"02:39.030","Text":"Well, that\u0027s 50 minus 21 plus 1,"},{"Start":"02:39.030 ","End":"02:43.750","Text":"and that equals to 30/50."},{"Start":"02:44.300 ","End":"02:46.825","Text":"Section c asks us,"},{"Start":"02:46.825 ","End":"02:49.849","Text":"if a number larger than 20 is selected,"},{"Start":"02:49.849 ","End":"02:55.235","Text":"What\u0027s the probability that the number selected is smaller than 28?"},{"Start":"02:55.235 ","End":"02:58.640","Text":"Well, this is a conditional probability."},{"Start":"02:58.640 ","End":"03:02.060","Text":"Let\u0027s just set it up. What\u0027s the probability now?"},{"Start":"03:02.060 ","End":"03:03.920","Text":"What\u0027s the condition that come right here?"},{"Start":"03:03.920 ","End":"03:05.405","Text":"That\u0027s what\u0027s given to us."},{"Start":"03:05.405 ","End":"03:08.330","Text":"If a number larger than 20 selected,"},{"Start":"03:08.330 ","End":"03:11.030","Text":"that means x is greater than 20,"},{"Start":"03:11.030 ","End":"03:12.950","Text":"and what do we ask for?"},{"Start":"03:12.950 ","End":"03:16.280","Text":"The probability that the number selected is smarter than 28."},{"Start":"03:16.280 ","End":"03:19.925","Text":"That means that x is less than 28."},{"Start":"03:19.925 ","End":"03:22.650","Text":"Well, that equals,"},{"Start":"03:22.650 ","End":"03:26.780","Text":"now to the probability of what\u0027s given in the denominator,"},{"Start":"03:26.780 ","End":"03:30.960","Text":"that\u0027s x being greater than 20."},{"Start":"03:31.220 ","End":"03:36.545","Text":"In the numerator, the probability of the intersect."},{"Start":"03:36.545 ","End":"03:39.035","Text":"Now, what\u0027s the intersect here?"},{"Start":"03:39.035 ","End":"03:43.595","Text":"That x is greater than 20 and less than 28."},{"Start":"03:43.595 ","End":"03:48.365","Text":"We\u0027ll write this down x less than 28."},{"Start":"03:48.365 ","End":"03:52.940","Text":"Now, just like we did in section b,"},{"Start":"03:52.940 ","End":"03:56.120","Text":"we can rephrase these guys in order to"},{"Start":"03:56.120 ","End":"04:00.290","Text":"calculate the number of values that we have within a range,"},{"Start":"04:00.290 ","End":"04:03.090","Text":"so let\u0027s do that."},{"Start":"04:03.440 ","End":"04:09.560","Text":"That would equal the probability of x"},{"Start":"04:09.560 ","End":"04:18.760","Text":"being greater or equal to 21 and less than and equal to 27,"},{"Start":"04:18.760 ","End":"04:26.850","Text":"divided by the probability of x being greater or equal to 21."},{"Start":"04:27.650 ","End":"04:30.695","Text":"Let\u0027s take a look at the denominator."},{"Start":"04:30.695 ","End":"04:32.629","Text":"Well, in the denominator,"},{"Start":"04:32.629 ","End":"04:35.105","Text":"that\u0027s what we calculated in section b."},{"Start":"04:35.105 ","End":"04:38.735","Text":"That equals to 30/50, that\u0027s right here."},{"Start":"04:38.735 ","End":"04:40.685","Text":"We\u0027ll put that down here."},{"Start":"04:40.685 ","End":"04:46.085","Text":"Now, what is the probability of x,"},{"Start":"04:46.085 ","End":"04:49.865","Text":"x being between 21 and 27 inclusive?"},{"Start":"04:49.865 ","End":"04:55.960","Text":"Well, let\u0027s figure out how many values of x are there between 21 and 27 inclusive."},{"Start":"04:55.960 ","End":"05:02.890","Text":"Well, that\u0027s 27 minus 21 plus 1,"},{"Start":"05:02.890 ","End":"05:08.975","Text":"and again, because it\u0027s a probability that has to be over 50."},{"Start":"05:08.975 ","End":"05:17.500","Text":"Now that equals to 7/50 divided by 30/15."},{"Start":"05:17.500 ","End":"05:22.080","Text":"That comes out to 7/30."},{"Start":"05:22.080 ","End":"05:28.180","Text":"This is the answer for section c. This is the answer for section b."},{"Start":"05:28.180 ","End":"05:31.890","Text":"And this is the answer for Section a."}],"ID":14154},{"Watched":false,"Name":"Exercise 21","Duration":"4m 17s","ChapterTopicVideoID":13514,"CourseChapterTopicPlaylistID":64525,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this question we\u0027re asked to prove that if X is"},{"Start":"00:03.270 ","End":"00:07.365","Text":"distributed with a uniform distribution with parameters a and b,"},{"Start":"00:07.365 ","End":"00:12.130","Text":"then the expectation of X equals a plus b over 2."},{"Start":"00:12.500 ","End":"00:20.730","Text":"Let\u0027s get at it. We know that the general formula to calculate the expectation of X,"},{"Start":"00:20.730 ","End":"00:26.500","Text":"well, that equals to the sum of X times the probability of X."},{"Start":"00:26.500 ","End":"00:29.600","Text":"Now, because we know that X is"},{"Start":"00:29.600 ","End":"00:33.275","Text":"distributed with a uniform distribution with these parameters,"},{"Start":"00:33.275 ","End":"00:42.770","Text":"we know that the probability of X being equal to k equals 1 over b minus a plus 1,"},{"Start":"00:42.770 ","End":"00:46.520","Text":"where k equals a,"},{"Start":"00:46.520 ","End":"00:47.945","Text":"a plus 1,"},{"Start":"00:47.945 ","End":"00:50.910","Text":"and so on and so forth until b."},{"Start":"00:51.590 ","End":"00:56.270","Text":"Let\u0027s just plug this back into here."},{"Start":"00:56.270 ","End":"01:01.250","Text":"Now, that equals to the sum of X times the probability of X."},{"Start":"01:01.250 ","End":"01:02.480","Text":"Now the probability of X,"},{"Start":"01:02.480 ","End":"01:04.925","Text":"we said that\u0027s 1 over b minus a plus 1."},{"Start":"01:04.925 ","End":"01:09.365","Text":"That\u0027s 1 over b minus a plus 1."},{"Start":"01:09.365 ","End":"01:13.399","Text":"Now, this is a constant."},{"Start":"01:13.399 ","End":"01:18.020","Text":"It has nothing to do with the sum operator right here."},{"Start":"01:18.020 ","End":"01:20.630","Text":"We can take this out and write it like this,"},{"Start":"01:20.630 ","End":"01:28.080","Text":"1 over b minus a plus 1 times the sum of X,"},{"Start":"01:28.080 ","End":"01:34.270","Text":"where X starts with a and ends up with b."},{"Start":"01:35.810 ","End":"01:37.910","Text":"What do we have here?"},{"Start":"01:37.910 ","End":"01:43.865","Text":"We have a constant times the sum of a series."},{"Start":"01:43.865 ","End":"01:45.590","Text":"Now, what is that?"},{"Start":"01:45.590 ","End":"01:47.210","Text":"Why is this a series?"},{"Start":"01:47.210 ","End":"01:49.700","Text":"Well, let\u0027s take a look at this."},{"Start":"01:49.700 ","End":"01:55.310","Text":"That\u0027s 1 over b minus a plus 1 times,"},{"Start":"01:55.310 ","End":"01:57.320","Text":"well, let\u0027s take a look at this."},{"Start":"01:57.320 ","End":"02:02.030","Text":"That\u0027s a plus a plus 1,"},{"Start":"02:02.030 ","End":"02:04.850","Text":"plus a plus 2,"},{"Start":"02:04.850 ","End":"02:08.610","Text":"and so on and so forth, plus b."},{"Start":"02:09.170 ","End":"02:12.050","Text":"This is a series."},{"Start":"02:12.050 ","End":"02:19.880","Text":"Now, we have to remember what the formulas for the sum of a series."},{"Start":"02:19.880 ","End":"02:22.565","Text":"Let\u0027s just go over here to the side."},{"Start":"02:22.565 ","End":"02:24.725","Text":"The sum of a series."},{"Start":"02:24.725 ","End":"02:31.460","Text":"Well that equals to the first member in the series plus"},{"Start":"02:31.460 ","End":"02:40.610","Text":"the last member of the series times the number of members in the series divided by 2."},{"Start":"02:40.610 ","End":"02:47.990","Text":"In our case, a_1 will equal to a,"},{"Start":"02:47.990 ","End":"02:53.910","Text":"and a_n will equal to b. A_1,"},{"Start":"02:53.910 ","End":"02:56.970","Text":"that\u0027s the first value in the series."},{"Start":"02:56.970 ","End":"03:00.740","Text":"A_n, that\u0027s the last value in the series."},{"Start":"03:00.740 ","End":"03:02.570","Text":"Now, what\u0027s n?"},{"Start":"03:02.570 ","End":"03:09.080","Text":"N is the number of values or the number of members in the series."},{"Start":"03:09.080 ","End":"03:11.420","Text":"Well, how many members in the series do we have?"},{"Start":"03:11.420 ","End":"03:13.895","Text":"We have 1, 2, 3, and so on and so forth."},{"Start":"03:13.895 ","End":"03:18.940","Text":"But we have b minus a plus 1."},{"Start":"03:18.940 ","End":"03:22.760","Text":"Let\u0027s plug all this back into here."},{"Start":"03:22.760 ","End":"03:26.780","Text":"Well, that\u0027s 1 over b minus a plus 1."},{"Start":"03:26.780 ","End":"03:30.840","Text":"That\u0027s this guy right here times, now,"},{"Start":"03:30.840 ","End":"03:35.355","Text":"instead of this, let\u0027s plug-in these guys right here."},{"Start":"03:35.355 ","End":"03:40.695","Text":"A_1, that\u0027s a, plus a_n."},{"Start":"03:40.695 ","End":"03:42.855","Text":"Well, a_n is b."},{"Start":"03:42.855 ","End":"03:50.670","Text":"Now, divide by 2 times n. Now,"},{"Start":"03:50.670 ","End":"03:55.330","Text":"n is b minus a plus 1."},{"Start":"03:59.570 ","End":"04:06.315","Text":"This then cancels this out so we have a plus b over 2,"},{"Start":"04:06.315 ","End":"04:10.475","Text":"and that is the expectation of X,"},{"Start":"04:10.475 ","End":"04:17.790","Text":"where X is distributed with a uniform distribution with parameters a and b."}],"ID":14155}],"Thumbnail":null,"ID":64525},{"Name":"General Probabilities without Integrals","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"3m 22s","ChapterTopicVideoID":12924,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"In this chapter, we\u0027ll start talking about continuous random variables."},{"Start":"00:04.140 ","End":"00:08.625","Text":"For example, general probabilities without the use of integrals."},{"Start":"00:08.625 ","End":"00:10.680","Text":"As we said in this chapter,"},{"Start":"00:10.680 ","End":"00:16.470","Text":"we\u0027ll deal with a probability distribution of continuous random variables."},{"Start":"00:16.470 ","End":"00:18.480","Text":"For example, the height of a person,"},{"Start":"00:18.480 ","End":"00:21.030","Text":"reaction time, and so on and so forth."},{"Start":"00:21.030 ","End":"00:23.420","Text":"As opposed to discrete variables,"},{"Start":"00:23.420 ","End":"00:27.285","Text":"continuous variables have an infinite number of values in a given range."},{"Start":"00:27.285 ","End":"00:29.700","Text":"For example, if we roll the dice."},{"Start":"00:29.700 ","End":"00:31.530","Text":"The random variable X,"},{"Start":"00:31.530 ","End":"00:33.300","Text":"which will be the result,"},{"Start":"00:33.300 ","End":"00:38.435","Text":"could have basically 6 values from 1-6."},{"Start":"00:38.435 ","End":"00:42.260","Text":"But if we take a random variable,"},{"Start":"00:42.260 ","End":"00:44.090","Text":"for example, the height of a person,"},{"Start":"00:44.090 ","End":"00:46.595","Text":"well that\u0027s a continuous random variable."},{"Start":"00:46.595 ","End":"00:50.600","Text":"For example, if we take a range of values from,"},{"Start":"00:50.600 ","End":"00:55.190","Text":"let\u0027s say, 150 centimeters to 200 centimeters."},{"Start":"00:55.190 ","End":"00:57.980","Text":"Well, there\u0027s an infinite number of values between"},{"Start":"00:57.980 ","End":"01:01.075","Text":"that range that the height of a person can have."},{"Start":"01:01.075 ","End":"01:08.480","Text":"More so even if the range was between 170 and 171 centimeters in"},{"Start":"01:08.480 ","End":"01:11.720","Text":"that 1 centimeter range there still an infinite number of"},{"Start":"01:11.720 ","End":"01:16.505","Text":"values that the random variable X can have."},{"Start":"01:16.505 ","End":"01:20.300","Text":"Now, we\u0027ll describe the continuous random variable"},{"Start":"01:20.300 ","End":"01:24.085","Text":"by a function called the density function."},{"Start":"01:24.085 ","End":"01:30.770","Text":"In general, F of x denotes a density function of any continuous random variable,"},{"Start":"01:30.770 ","End":"01:37.115","Text":"where again, F of x is the density function and x is a continuous random variable."},{"Start":"01:37.115 ","End":"01:43.730","Text":"Now, the area under the density function gives the probability."},{"Start":"01:43.730 ","End":"01:48.745","Text":"That\u0027s between the density function and the x-axis."},{"Start":"01:48.745 ","End":"01:52.290","Text":"A density function must be non-negative"},{"Start":"01:52.290 ","End":"01:56.400","Text":"and the total area under the function is always 1."},{"Start":"01:56.510 ","End":"02:02.000","Text":"For example, X, the random variable can have negative numbers,"},{"Start":"02:02.000 ","End":"02:04.010","Text":"can be either negative or positive,"},{"Start":"02:04.010 ","End":"02:09.710","Text":"but the function F of x must be positive, must be non-negative."},{"Start":"02:09.710 ","End":"02:12.385","Text":"It has to be greater or equal to 0."},{"Start":"02:12.385 ","End":"02:19.370","Text":"The total area under this function is always equal to 1."},{"Start":"02:19.370 ","End":"02:23.945","Text":"Now, the current course doesn\u0027t use integration to calculate the areas."},{"Start":"02:23.945 ","End":"02:27.845","Text":"We\u0027ll be using common geometric forms,"},{"Start":"02:27.845 ","End":"02:34.580","Text":"to use as an example of density functions and will be given"},{"Start":"02:34.580 ","End":"02:42.270","Text":"also the equations to calculate the areas under these common geometric forms."},{"Start":"02:42.340 ","End":"02:51.050","Text":"Here, I\u0027ve given basically some equations of some general geometric forms."},{"Start":"02:51.050 ","End":"02:54.020","Text":"For example, the area of a triangle,"},{"Start":"02:54.020 ","End":"02:56.015","Text":"that\u0027s this guy right here."},{"Start":"02:56.015 ","End":"02:59.070","Text":"The area of a rectangle, that\u0027s this guy."},{"Start":"02:59.070 ","End":"03:03.575","Text":"I\u0027ve also given an equation for a straight line."},{"Start":"03:03.575 ","End":"03:06.440","Text":"That\u0027s all of these guys right here."},{"Start":"03:06.440 ","End":"03:10.470","Text":"What\u0027s the general expression for a straight line?"},{"Start":"03:10.470 ","End":"03:15.660","Text":"How to calculate the slope and the y-intercept."},{"Start":"03:15.880 ","End":"03:22.200","Text":"Let\u0027s get to an example and see how we can work with these density functions."}],"ID":32238},{"Watched":false,"Name":"Example - Parts a-b","Duration":"5m 18s","ChapterTopicVideoID":12928,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"In this question, X is a continuous variable having the density function,"},{"Start":"00:05.250 ","End":"00:07.050","Text":"right shown in this graph."},{"Start":"00:07.050 ","End":"00:11.880","Text":"We\u0027re asked to find the value of c. First of all,"},{"Start":"00:11.880 ","End":"00:14.504","Text":"if this is a density function,"},{"Start":"00:14.504 ","End":"00:19.435","Text":"then the area under the density function has to equal to 1."},{"Start":"00:19.435 ","End":"00:23.295","Text":"Now, the second thing here is that we see that"},{"Start":"00:23.295 ","End":"00:27.210","Text":"the density function is comprised of 2 rectangles."},{"Start":"00:27.210 ","End":"00:28.755","Text":"This rectangle right here,"},{"Start":"00:28.755 ","End":"00:31.440","Text":"and this rectangle right here."},{"Start":"00:31.440 ","End":"00:34.050","Text":"What\u0027s the area?"},{"Start":"00:34.050 ","End":"00:36.459","Text":"How do we calculate the area of a rectangle?"},{"Start":"00:36.459 ","End":"00:39.615","Text":"That\u0027s the base times the height."},{"Start":"00:39.615 ","End":"00:44.480","Text":"Let\u0027s just calculate the area underneath the density function."},{"Start":"00:44.480 ","End":"00:48.350","Text":"Well, that\u0027s the area of this rectangle."},{"Start":"00:48.350 ","End":"00:50.578","Text":"Well, that\u0027s 1, that\u0027s the base is 1,"},{"Start":"00:50.578 ","End":"00:53.920","Text":"and the height is 0.25."},{"Start":"00:53.920 ","End":"00:58.485","Text":"That\u0027s 1 times 0.25 plus,"},{"Start":"00:58.485 ","End":"01:00.090","Text":"what\u0027s the base here?"},{"Start":"01:00.090 ","End":"01:02.988","Text":"The base here is 4 times the height,"},{"Start":"01:02.988 ","End":"01:08.090","Text":"that\u0027s c. That has to equal to 1 because this is a density function."},{"Start":"01:08.090 ","End":"01:13.605","Text":"That means that 4_c equals 3 quarters,"},{"Start":"01:13.605 ","End":"01:19.150","Text":"and that means that c equals to 3 over 16."},{"Start":"01:22.500 ","End":"01:29.150","Text":"In this section, we\u0027re asked to construct a cumulative distribution function of X."},{"Start":"01:29.190 ","End":"01:31.240","Text":"Well, that\u0027s just like saying,"},{"Start":"01:31.240 ","End":"01:36.625","Text":"what\u0027s the probability of x being less than or equal to t,"},{"Start":"01:36.625 ","End":"01:40.465","Text":"and that equals to big F at t. Now,"},{"Start":"01:40.465 ","End":"01:43.405","Text":"in order to do that,"},{"Start":"01:43.405 ","End":"01:50.065","Text":"I\u0027m going to have to split the x-axis into various ranges."},{"Start":"01:50.065 ","End":"01:56.745","Text":"The first range is where t is less than 1,"},{"Start":"01:56.745 ","End":"01:59.920","Text":"that\u0027s this range right here."},{"Start":"02:00.170 ","End":"02:05.730","Text":"The next range is where t is between 1 and 2."},{"Start":"02:05.730 ","End":"02:07.545","Text":"Let\u0027s write that down,"},{"Start":"02:07.545 ","End":"02:11.355","Text":"t is between 1 and 2."},{"Start":"02:11.355 ","End":"02:18.165","Text":"The next range is where t is between 2 and 6."},{"Start":"02:18.165 ","End":"02:20.940","Text":"Let\u0027s write that down as well."},{"Start":"02:20.940 ","End":"02:24.540","Text":"T is between 2 and 6."},{"Start":"02:24.540 ","End":"02:29.050","Text":"The last range is where t is greater than 6,"},{"Start":"02:29.050 ","End":"02:32.135","Text":"that\u0027ll be this range right here."},{"Start":"02:32.135 ","End":"02:38.235","Text":"Let\u0027s begin. Where t is less than 1,"},{"Start":"02:38.235 ","End":"02:43.990","Text":"we have our density function equaling 0, it\u0027s right here."},{"Start":"02:43.990 ","End":"02:50.595","Text":"That means that the area between the density function and the x-axis is 0."},{"Start":"02:50.595 ","End":"02:57.520","Text":"The cumulative distribution, that equals to 0 as well."},{"Start":"02:57.520 ","End":"03:04.370","Text":"Now, let\u0027s look at the other end of the spectrum where t is greater than 6,"},{"Start":"03:04.370 ","End":"03:08.060","Text":"any value in the blue range right here."},{"Start":"03:08.060 ","End":"03:11.090","Text":"Well, when t is greater than 6,"},{"Start":"03:11.090 ","End":"03:14.480","Text":"we\u0027ve actually accumulated all of the area,"},{"Start":"03:14.480 ","End":"03:16.415","Text":"all of the density,"},{"Start":"03:16.415 ","End":"03:19.640","Text":"underneath the density function."},{"Start":"03:19.640 ","End":"03:21.650","Text":"That means that when we\u0027re here,"},{"Start":"03:21.650 ","End":"03:25.470","Text":"we\u0027ve gotten all of the area."},{"Start":"03:25.470 ","End":"03:29.550","Text":"That means that that equals to 1."},{"Start":"03:29.550 ","End":"03:32.120","Text":"Now, what happens in the middle here,"},{"Start":"03:32.120 ","End":"03:34.925","Text":"in this range between 1 and 2,"},{"Start":"03:34.925 ","End":"03:37.160","Text":"and then the other range between 2 and 6."},{"Start":"03:37.160 ","End":"03:40.050","Text":"Well, let\u0027s calculate them."},{"Start":"03:40.370 ","End":"03:43.335","Text":"In the range between 1 and 2,"},{"Start":"03:43.335 ","End":"03:51.050","Text":"let\u0027s just pick a random point we call that t. That\u0027s t right here."},{"Start":"03:51.050 ","End":"03:59.775","Text":"What we want to do is we want to calculate the area of this rectangle right here."},{"Start":"03:59.775 ","End":"04:05.479","Text":"Now, the area of a rectangle that equals to the base times the height."},{"Start":"04:05.479 ","End":"04:09.630","Text":"Now, what\u0027s the base here?"},{"Start":"04:09.630 ","End":"04:12.570","Text":"That\u0027s t minus 1."},{"Start":"04:12.570 ","End":"04:17.500","Text":"What\u0027s the height? Well, the height is 0.25."},{"Start":"04:18.200 ","End":"04:25.510","Text":"That\u0027s the expression for the CDF when we\u0027re in this range between 1 and 2."},{"Start":"04:25.510 ","End":"04:32.975","Text":"Now, what about the expression for the CDF when we\u0027re in the range between 2 and 6?"},{"Start":"04:32.975 ","End":"04:36.845","Text":"Well again, let\u0027s just pick a point,"},{"Start":"04:36.845 ","End":"04:43.430","Text":"we call that t. We\u0027re looking for the probability of x being less than or equal to t,"},{"Start":"04:43.430 ","End":"04:48.870","Text":"that means that we\u0027re looking for the probability,"},{"Start":"04:49.090 ","End":"04:53.990","Text":"this probability or this area right here."},{"Start":"05:00.640 ","End":"05:04.730","Text":"We know what the area of this rectangle is right here."},{"Start":"05:04.730 ","End":"05:11.600","Text":"That\u0027s 0.25, that\u0027s 1 times 0.25."},{"Start":"05:11.600 ","End":"05:16.415","Text":"We have to add that to the area of this rectangle right here."},{"Start":"05:16.415 ","End":"05:18.590","Text":"This area right here,"},{"Start":"05:18.590 ","End":"05:20.798","Text":"that\u0027s the base times the height."},{"Start":"05:20.798 ","End":"05:27.530","Text":"The base would be t minus 2 and the height would be c now."},{"Start":"05:27.530 ","End":"05:36.180","Text":"See we\u0027ve calculated that to be 3/16, so that\u0027s 3/16."},{"Start":"05:36.180 ","End":"05:42.450","Text":"This here is the cumulative distribution function of x."}],"ID":32239},{"Watched":false,"Name":"Example - Part c","Duration":"3m 3s","ChapterTopicVideoID":12925,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"Section c, we\u0027re asked to calculate"},{"Start":"00:02.040 ","End":"00:05.505","Text":"the chances of the waiting time being less than 2 minutes."},{"Start":"00:05.505 ","End":"00:12.525","Text":"That\u0027s like saying, what\u0027s the probability of x being less than 2?"},{"Start":"00:12.525 ","End":"00:15.840","Text":"Why is that? What\u0027s the probability of the chances?"},{"Start":"00:15.840 ","End":"00:17.310","Text":"X is the waiting time,"},{"Start":"00:17.310 ","End":"00:19.020","Text":"and less than 2 minutes."},{"Start":"00:19.020 ","End":"00:23.325","Text":"If we look at our density function,"},{"Start":"00:23.325 ","End":"00:27.450","Text":"we see that what we\u0027re looking for is the area"},{"Start":"00:27.450 ","End":"00:31.710","Text":"under the density function between 0 and 2."},{"Start":"00:31.710 ","End":"00:37.004","Text":"Why is that? Because the area under the density function is basically the probability."},{"Start":"00:37.004 ","End":"00:40.305","Text":"What we\u0027re looking for,"},{"Start":"00:40.305 ","End":"00:42.900","Text":"if this is the point 2,"},{"Start":"00:42.900 ","End":"00:44.470","Text":"x being equal to 2,"},{"Start":"00:44.470 ","End":"00:48.170","Text":"then we\u0027re looking for this area right here,"},{"Start":"00:48.170 ","End":"00:58.095","Text":"all the area underneath the density function between 0 and 2."},{"Start":"00:58.095 ","End":"01:04.234","Text":"Let\u0027s get started. Well, the area of this triangle,"},{"Start":"01:04.234 ","End":"01:09.630","Text":"but that equals the base times the height divided by 2."},{"Start":"01:09.630 ","End":"01:12.150","Text":"Now, the base is 2,"},{"Start":"01:12.150 ","End":"01:15.650","Text":"b equals 2, what\u0027s the height?"},{"Start":"01:15.650 ","End":"01:18.230","Text":"Well, in order to figure out the height,"},{"Start":"01:18.230 ","End":"01:22.460","Text":"but that\u0027s just the value of the density function where x equals 2."},{"Start":"01:22.460 ","End":"01:26.750","Text":"Now, the density function of x, if we recall,"},{"Start":"01:26.750 ","End":"01:28.590","Text":"was 1/8 times x,"},{"Start":"01:28.590 ","End":"01:35.050","Text":"where x was between 0 and 4, and 0 otherwise."},{"Start":"01:39.110 ","End":"01:41.280","Text":"Let\u0027s just plug that in."},{"Start":"01:41.280 ","End":"01:44.100","Text":"That\u0027s 1/8 times 2,"},{"Start":"01:44.100 ","End":"01:46.005","Text":"and that equals to 2/8,"},{"Start":"01:46.005 ","End":"01:49.980","Text":"and that equals to 1/4."},{"Start":"01:49.980 ","End":"01:54.150","Text":"Our height would be equal to 1/4."},{"Start":"01:54.150 ","End":"01:56.160","Text":"Let\u0027s plug that in."},{"Start":"01:56.160 ","End":"02:05.480","Text":"The area of this triangle would be 2 times 1/4 divided by 2,"},{"Start":"02:05.480 ","End":"02:08.045","Text":"and that equals to 1/4."},{"Start":"02:08.045 ","End":"02:14.755","Text":"This is the probability of the waiting time being less than 2 minutes."},{"Start":"02:14.755 ","End":"02:21.650","Text":"Now, another thing that we should know is that the probability of x being less than 2,"},{"Start":"02:21.650 ","End":"02:28.340","Text":"but that equals to the probability of x being less than or equal to 2."},{"Start":"02:28.340 ","End":"02:29.780","Text":"Why is that?"},{"Start":"02:29.780 ","End":"02:40.295","Text":"Because the area under the density function at a given point equals to 0."},{"Start":"02:40.295 ","End":"02:48.380","Text":"Whether we include the point 2 or not in the area,"},{"Start":"02:48.380 ","End":"02:52.700","Text":"it really doesn\u0027t matter for the area calculations here."},{"Start":"02:52.700 ","End":"02:55.550","Text":"Why is that? Again, because there\u0027s really"},{"Start":"02:55.550 ","End":"03:03.310","Text":"no area underneath the density function at a specific point."}],"ID":32240},{"Watched":false,"Name":"Example - Part d","Duration":"6m 43s","ChapterTopicVideoID":12926,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.980","Text":"An important subject when dealing with"},{"Start":"00:01.980 ","End":"00:06.660","Text":"continuous variables is the Cumulative Distribution Function,"},{"Start":"00:06.660 ","End":"00:09.615","Text":"or in short, CDF."},{"Start":"00:09.615 ","End":"00:14.730","Text":"Now, this is a function for a continuous variable that provides"},{"Start":"00:14.730 ","End":"00:19.845","Text":"the probability of a variable being less than or equal to a given value."},{"Start":"00:19.845 ","End":"00:21.810","Text":"Let\u0027s take, for example,"},{"Start":"00:21.810 ","End":"00:24.570","Text":"the value t. Now,"},{"Start":"00:24.570 ","End":"00:26.310","Text":"we write it out like this,"},{"Start":"00:26.310 ","End":"00:30.455","Text":"the CDF for t,"},{"Start":"00:30.455 ","End":"00:36.380","Text":"well that equals to the probability of x being less than or equal to t. Now similarly,"},{"Start":"00:36.380 ","End":"00:40.070","Text":"if we\u0027re looking for the probability of x being greater than t,"},{"Start":"00:40.070 ","End":"00:44.995","Text":"well that\u0027s 1 minus this guy or 1 minus F of t,"},{"Start":"00:44.995 ","End":"00:53.150","Text":"the CDF at t. What about the probability of x being between a and b?"},{"Start":"00:53.150 ","End":"00:57.620","Text":"Well, that\u0027s just the fair value of the CDF at point"},{"Start":"00:57.620 ","End":"01:02.815","Text":"b minus the value of the CDF at point a."},{"Start":"01:02.815 ","End":"01:10.175","Text":"Let\u0027s now go back to our example and see how we can work with this function."},{"Start":"01:10.175 ","End":"01:12.757","Text":"In section d, we\u0027re asked to build the Cumulative Distribution"},{"Start":"01:12.757 ","End":"01:17.495","Text":"Function of our continuous random variable x."},{"Start":"01:17.495 ","End":"01:21.590","Text":"Now, that means that we\u0027re looking for F of t,"},{"Start":"01:21.590 ","End":"01:28.370","Text":"which is defined as the probability of x being less than or equal to t. Now,"},{"Start":"01:28.370 ","End":"01:31.520","Text":"in a continuous random variable,"},{"Start":"01:31.520 ","End":"01:38.240","Text":"the probability is defined as the area underneath the density function."},{"Start":"01:38.240 ","End":"01:43.350","Text":"Let\u0027s take a look at our density function. Here it is."},{"Start":"01:43.350 ","End":"01:45.680","Text":"When we\u0027re looking at this guy,"},{"Start":"01:45.680 ","End":"01:50.810","Text":"we basically see 3 areas of x that we have to deal with;"},{"Start":"01:50.810 ","End":"01:52.955","Text":"where x is negative,"},{"Start":"01:52.955 ","End":"01:56.215","Text":"where x is above 4,"},{"Start":"01:56.215 ","End":"02:01.725","Text":"and where x is between 0 and 4."},{"Start":"02:01.725 ","End":"02:04.810","Text":"This is 0 right here."},{"Start":"02:04.940 ","End":"02:07.770","Text":"Let\u0027s just write this down."},{"Start":"02:07.770 ","End":"02:14.850","Text":"This equals to where x is negative,"},{"Start":"02:14.850 ","End":"02:19.560","Text":"where x is between 0 and 4,"},{"Start":"02:19.560 ","End":"02:22.575","Text":"and where x is greater than 4."},{"Start":"02:22.575 ","End":"02:29.435","Text":"Now, let\u0027s take a look at the range where x is negative now,"},{"Start":"02:29.435 ","End":"02:31.280","Text":"this guy right here."},{"Start":"02:31.280 ","End":"02:35.150","Text":"Now, assume that x is minus 1."},{"Start":"02:35.150 ","End":"02:41.000","Text":"Well, what would be the probability of x being less than minus 1?"},{"Start":"02:41.000 ","End":"02:48.070","Text":"Well, that\u0027s 0, because there\u0027s no area underneath the density function."},{"Start":"02:48.070 ","End":"02:54.875","Text":"Now, that would be for all xs that are negative until 0."},{"Start":"02:54.875 ","End":"03:02.975","Text":"There would be no area underneath the density function that we can accumulate."},{"Start":"03:02.975 ","End":"03:11.900","Text":"Therefore, the value of the CDF or the Cumulative Distribution Function is 0."},{"Start":"03:11.900 ","End":"03:16.940","Text":"Now, what about the other end of the spectrum where x is greater than 4,"},{"Start":"03:16.940 ","End":"03:18.890","Text":"that\u0027s this range right here?"},{"Start":"03:18.890 ","End":"03:25.100","Text":"Well, we could say that the probability of x being,"},{"Start":"03:25.100 ","End":"03:26.630","Text":"let\u0027s say 5,"},{"Start":"03:26.630 ","End":"03:28.565","Text":"assume t is 5, right?"},{"Start":"03:28.565 ","End":"03:33.110","Text":"It\u0027s right here. Then this function,"},{"Start":"03:33.110 ","End":"03:36.125","Text":"this Cumulative Distribution Function,"},{"Start":"03:36.125 ","End":"03:40.310","Text":"would be the probability of x being less than or equal to 5."},{"Start":"03:40.310 ","End":"03:46.815","Text":"Well, that would be all of this area until minus infinity."},{"Start":"03:46.815 ","End":"03:49.085","Text":"From minus infinity on,"},{"Start":"03:49.085 ","End":"03:52.625","Text":"we accumulated all the areas underneath"},{"Start":"03:52.625 ","End":"03:57.230","Text":"the density function until we arrived at the value of 5 here."},{"Start":"03:57.230 ","End":"04:03.980","Text":"Now, what\u0027s the total area underneath the density function that we\u0027ve accumulated?"},{"Start":"04:03.980 ","End":"04:10.370","Text":"Well, we\u0027ve accumulated all of the area underneath the function or all the probability,"},{"Start":"04:10.370 ","End":"04:13.755","Text":"so that means that would equal to 1."},{"Start":"04:13.755 ","End":"04:19.140","Text":"That means that for all values above 4,"},{"Start":"04:19.140 ","End":"04:23.520","Text":"the CDF would be equal to 1."},{"Start":"04:23.520 ","End":"04:25.245","Text":"Now, what about here?"},{"Start":"04:25.245 ","End":"04:28.745","Text":"In this area between the area of 0 and 4?"},{"Start":"04:28.745 ","End":"04:37.340","Text":"Well, let\u0027s take a point here of value of x and call it t. This is the same t here,"},{"Start":"04:37.340 ","End":"04:42.800","Text":"and we want to know what\u0027s the probability of x being less than or equal to t?"},{"Start":"04:42.800 ","End":"04:50.120","Text":"Well, that\u0027s basically asking what\u0027s the area of this triangle right here?"},{"Start":"04:50.120 ","End":"04:52.760","Text":"Well, let\u0027s just calculate that."},{"Start":"04:52.760 ","End":"04:56.540","Text":"Well, that\u0027s base times height divided by 2."},{"Start":"04:56.540 ","End":"05:00.090","Text":"Well, the base is t. Now, what\u0027s the height?"},{"Start":"05:00.090 ","End":"05:08.840","Text":"Well, that\u0027s the value of the density function at point t. Now, the density function,"},{"Start":"05:08.840 ","End":"05:12.920","Text":"if we recall, that was 1/8 times x,"},{"Start":"05:12.920 ","End":"05:18.410","Text":"where x was between 0 and 4, and 0 otherwise."},{"Start":"05:18.410 ","End":"05:22.500","Text":"So if we put t in as a specific point,"},{"Start":"05:22.500 ","End":"05:24.780","Text":"that will be 1/8th times t,"},{"Start":"05:24.780 ","End":"05:27.750","Text":"so that\u0027ll be 1/8th times t,"},{"Start":"05:27.750 ","End":"05:30.279","Text":"that\u0027s the base, that\u0027s the height,"},{"Start":"05:30.279 ","End":"05:32.850","Text":"and all this would be divided by 2."},{"Start":"05:32.850 ","End":"05:38.350","Text":"That equals to t squared over 16."},{"Start":"05:38.570 ","End":"05:43.640","Text":"This right here is the CDF or"},{"Start":"05:43.640 ","End":"05:49.310","Text":"the Cumulative Distribution Function of our random variable x."},{"Start":"05:49.310 ","End":"05:52.820","Text":"Now, just to point out, in the last section,"},{"Start":"05:52.820 ","End":"05:57.830","Text":"we asked what was the probability of x being less than 2?"},{"Start":"05:57.830 ","End":"06:02.630","Text":"We showed how to calculate this using the density function only."},{"Start":"06:02.630 ","End":"06:05.540","Text":"But now if we would have had the CDF,"},{"Start":"06:05.540 ","End":"06:07.880","Text":"the Cumulative Distribution Function,"},{"Start":"06:07.880 ","End":"06:11.120","Text":"all we would have had to do was to take this point,"},{"Start":"06:11.120 ","End":"06:13.020","Text":"this number 2,"},{"Start":"06:13.020 ","End":"06:15.755","Text":"look at what range it was in, and right now,"},{"Start":"06:15.755 ","End":"06:17.540","Text":"this is the range between 0 and 4,"},{"Start":"06:17.540 ","End":"06:19.123","Text":"2 is between 0 and 4,"},{"Start":"06:19.123 ","End":"06:23.000","Text":"and plug in 2 into this expression right here."},{"Start":"06:23.000 ","End":"06:27.245","Text":"That would have been 2 squared over 16,"},{"Start":"06:27.245 ","End":"06:29.300","Text":"which is 4/16,"},{"Start":"06:29.300 ","End":"06:31.370","Text":"and that equals to 1/4,"},{"Start":"06:31.370 ","End":"06:36.110","Text":"and that was the value or the probability that we"},{"Start":"06:36.110 ","End":"06:42.870","Text":"receive in section c and the probability that we receive right here, they\u0027re identical."}],"ID":32241},{"Watched":false,"Name":"Example - Part e","Duration":"3m 53s","ChapterTopicVideoID":12927,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.690","Text":"Another important concept is that of the percentile."},{"Start":"00:03.690 ","End":"00:09.930","Text":"Now, the P percentile is a value we denoted as x_p,"},{"Start":"00:09.930 ","End":"00:15.480","Text":"for which the chances of the variable being under it are P. In another words,"},{"Start":"00:15.480 ","End":"00:16.860","Text":"we write it like this."},{"Start":"00:16.860 ","End":"00:22.440","Text":"We write it as the probability of X being less than or equal to x_p,"},{"Start":"00:22.440 ","End":"00:28.410","Text":"but that would equal to p. We\u0027re looking for this specific number right"},{"Start":"00:28.410 ","End":"00:35.400","Text":"here under which the probability of X being under it is equal to p. Now,"},{"Start":"00:35.400 ","End":"00:37.945","Text":"if we look at this expression right here,"},{"Start":"00:37.945 ","End":"00:42.005","Text":"this is exactly the expression for the CDF,"},{"Start":"00:42.005 ","End":"00:44.965","Text":"the Cumulative Distribution Function."},{"Start":"00:44.965 ","End":"00:50.760","Text":"Let\u0027s go back to our example and see how we can work with this."},{"Start":"00:50.900 ","End":"00:54.040","Text":"In section e of our example, we\u0027re asked,"},{"Start":"00:54.040 ","End":"00:57.785","Text":"what\u0027s the 80th percentile of the distribution?"},{"Start":"00:57.785 ","End":"00:59.810","Text":"Well, as we said,"},{"Start":"00:59.810 ","End":"01:08.600","Text":"we\u0027re looking for the probability of x being less than or equal to x_p,"},{"Start":"01:08.600 ","End":"01:12.080","Text":"that being equal to p. Now,"},{"Start":"01:12.080 ","End":"01:17.210","Text":"we said that this is exactly like the CDF,"},{"Start":"01:17.210 ","End":"01:20.235","Text":"like the Cumulative Distribution Function of x."},{"Start":"01:20.235 ","End":"01:22.555","Text":"Let\u0027s just write this out,"},{"Start":"01:22.555 ","End":"01:25.550","Text":"the F of x,"},{"Start":"01:25.550 ","End":"01:31.505","Text":"which is equal to the probability of X being less than or equal to x."},{"Start":"01:31.505 ","End":"01:34.460","Text":"That equals to, now,"},{"Start":"01:34.460 ","End":"01:37.580","Text":"if we remember, we did it like this."},{"Start":"01:37.580 ","End":"01:39.950","Text":"X being less than 0."},{"Start":"01:39.950 ","End":"01:41.690","Text":"That was 0,"},{"Start":"01:41.690 ","End":"01:44.330","Text":"t squared over 16,"},{"Start":"01:44.330 ","End":"01:49.155","Text":"that was when x was between 0 and 4,"},{"Start":"01:49.155 ","End":"01:54.940","Text":"and 1 when x was greater than 4."},{"Start":"01:55.640 ","End":"01:58.140","Text":"What are we looking for?"},{"Start":"01:58.140 ","End":"02:01.865","Text":"Well, basically, we are looking at this expression right here,"},{"Start":"02:01.865 ","End":"02:06.250","Text":"and we\u0027re looking for P,"},{"Start":"02:06.250 ","End":"02:09.030","Text":"the percentile, P is the 80th percentile,"},{"Start":"02:09.030 ","End":"02:12.885","Text":"so P here has to equal to 0.8."},{"Start":"02:12.885 ","End":"02:17.420","Text":"We\u0027re looking at the probability of x being less than or"},{"Start":"02:17.420 ","End":"02:24.330","Text":"equal to sum x_0.8_ p,"},{"Start":"02:24.330 ","End":"02:26.580","Text":"that equals to 0.8."},{"Start":"02:26.580 ","End":"02:31.475","Text":"That means that all we have to do is to take this expression and equate it to"},{"Start":"02:31.475 ","End":"02:38.345","Text":"this percentile right here and this probability to get this percentile."},{"Start":"02:38.345 ","End":"02:40.730","Text":"Let\u0027s just do that."},{"Start":"02:40.730 ","End":"02:47.825","Text":"We\u0027re looking at t squared over 16 or divided by 16,"},{"Start":"02:47.825 ","End":"02:51.950","Text":"but that would equal to 0.8."},{"Start":"02:51.950 ","End":"03:00.570","Text":"Now, t squared then would equal to 12.8,"},{"Start":"03:00.570 ","End":"03:05.975","Text":"so t would be equal to the square root of 12.8."},{"Start":"03:05.975 ","End":"03:11.155","Text":"Now, we won\u0027t take the negative value."},{"Start":"03:11.155 ","End":"03:15.260","Text":"We\u0027ll take the positive value here because x is positive,"},{"Start":"03:15.260 ","End":"03:16.985","Text":"x is between 0 and 4,"},{"Start":"03:16.985 ","End":"03:21.870","Text":"so that would come out to 3.58."},{"Start":"03:21.880 ","End":"03:24.455","Text":"What does this mean?"},{"Start":"03:24.455 ","End":"03:30.015","Text":"It means that when x is equal to 3.58,"},{"Start":"03:30.015 ","End":"03:32.580","Text":"within this range, it\u0027s between 0 and 4,"},{"Start":"03:32.580 ","End":"03:35.765","Text":"so when x equals 3.58,"},{"Start":"03:35.765 ","End":"03:42.470","Text":"the probability of x being less than or equal to this number,"},{"Start":"03:42.470 ","End":"03:46.430","Text":"well, that would be 0.8 or 80 percent."},{"Start":"03:46.430 ","End":"03:53.430","Text":"This number right here is the 80th percentile of the distribution."}],"ID":32242},{"Watched":false,"Name":"Exercise 1 Parts a-b","Duration":"5m 42s","ChapterTopicVideoID":30188,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.250","Text":"In this question, X is a continuous variable having the density function,"},{"Start":"00:05.250 ","End":"00:07.050","Text":"right shown in this graph."},{"Start":"00:07.050 ","End":"00:11.880","Text":"We\u0027re asked to find the value of c. First of all,"},{"Start":"00:11.880 ","End":"00:14.504","Text":"if this is a density function,"},{"Start":"00:14.504 ","End":"00:19.435","Text":"then the area under the density function has to equal to 1."},{"Start":"00:19.435 ","End":"00:23.295","Text":"Now, the second thing here is that we see that"},{"Start":"00:23.295 ","End":"00:27.210","Text":"the density function is comprised of 2 rectangles."},{"Start":"00:27.210 ","End":"00:28.755","Text":"This rectangle right here,"},{"Start":"00:28.755 ","End":"00:31.440","Text":"and this rectangle right here."},{"Start":"00:31.440 ","End":"00:34.050","Text":"What\u0027s the area?"},{"Start":"00:34.050 ","End":"00:36.459","Text":"How do we calculate the area of a rectangle?"},{"Start":"00:36.459 ","End":"00:39.615","Text":"That\u0027s the base times the height."},{"Start":"00:39.615 ","End":"00:44.480","Text":"Let\u0027s just calculate the area underneath the density function."},{"Start":"00:44.480 ","End":"00:48.350","Text":"Well, that\u0027s the area of this rectangle."},{"Start":"00:48.350 ","End":"00:50.578","Text":"Well, that\u0027s 1, that\u0027s the base is 1,"},{"Start":"00:50.578 ","End":"00:53.920","Text":"and the height is 0.25."},{"Start":"00:53.920 ","End":"00:58.485","Text":"That\u0027s 1 times 0.25 plus,"},{"Start":"00:58.485 ","End":"01:00.090","Text":"what\u0027s the base here?"},{"Start":"01:00.090 ","End":"01:02.988","Text":"The base here is 4 times the height,"},{"Start":"01:02.988 ","End":"01:08.090","Text":"that\u0027s c. That has to equal to 1 because this is a density function."},{"Start":"01:08.090 ","End":"01:13.605","Text":"That means that 4_c equals 3 quarters,"},{"Start":"01:13.605 ","End":"01:19.150","Text":"and that means that c equals to 3 over 16."},{"Start":"01:22.500 ","End":"01:29.150","Text":"In this section, we\u0027re asked to construct a cumulative distribution function of X."},{"Start":"01:29.190 ","End":"01:31.240","Text":"Well, that\u0027s just like saying,"},{"Start":"01:31.240 ","End":"01:36.625","Text":"what\u0027s the probability of x being less than or equal to t,"},{"Start":"01:36.625 ","End":"01:40.465","Text":"and that equals to big F at t. Now,"},{"Start":"01:40.465 ","End":"01:43.405","Text":"in order to do that,"},{"Start":"01:43.405 ","End":"01:50.065","Text":"I\u0027m going to have to split the x-axis into various ranges."},{"Start":"01:50.065 ","End":"01:56.745","Text":"The first range is where t is less than 1,"},{"Start":"01:56.745 ","End":"01:59.920","Text":"that\u0027s this range right here."},{"Start":"02:00.170 ","End":"02:05.730","Text":"The next range is where t is between 1 and 2."},{"Start":"02:05.730 ","End":"02:07.545","Text":"Let\u0027s write that down,"},{"Start":"02:07.545 ","End":"02:11.355","Text":"t is between 1 and 2."},{"Start":"02:11.355 ","End":"02:18.165","Text":"The next range is where t is between 2 and 6."},{"Start":"02:18.165 ","End":"02:20.940","Text":"Let\u0027s write that down as well."},{"Start":"02:20.940 ","End":"02:24.540","Text":"T is between 2 and 6."},{"Start":"02:24.540 ","End":"02:29.050","Text":"The last range is where t is greater than 6,"},{"Start":"02:29.050 ","End":"02:32.135","Text":"that\u0027ll be this range right here."},{"Start":"02:32.135 ","End":"02:38.235","Text":"Let\u0027s begin. Where t is less than 1,"},{"Start":"02:38.235 ","End":"02:43.990","Text":"we have our density function equaling 0, it\u0027s right here."},{"Start":"02:43.990 ","End":"02:50.595","Text":"That means that the area between the density function and the x-axis is 0."},{"Start":"02:50.595 ","End":"02:57.520","Text":"The cumulative distribution, that equals to 0 as well."},{"Start":"02:57.520 ","End":"03:04.370","Text":"Now, let\u0027s look at the other end of the spectrum where t is greater than 6,"},{"Start":"03:04.370 ","End":"03:08.060","Text":"any value in the blue range right here."},{"Start":"03:08.060 ","End":"03:11.090","Text":"Well, when t is greater than 6,"},{"Start":"03:11.090 ","End":"03:14.480","Text":"we\u0027ve actually accumulated all of the area,"},{"Start":"03:14.480 ","End":"03:16.415","Text":"all of the density,"},{"Start":"03:16.415 ","End":"03:19.640","Text":"underneath the density function."},{"Start":"03:19.640 ","End":"03:21.650","Text":"That means that when we\u0027re here,"},{"Start":"03:21.650 ","End":"03:25.470","Text":"we\u0027ve gotten all of the area."},{"Start":"03:25.470 ","End":"03:29.550","Text":"That means that that equals to 1."},{"Start":"03:29.550 ","End":"03:32.120","Text":"Now, what happens in the middle here,"},{"Start":"03:32.120 ","End":"03:34.925","Text":"in this range between 1 and 2,"},{"Start":"03:34.925 ","End":"03:37.160","Text":"and then the other range between 2 and 6."},{"Start":"03:37.160 ","End":"03:40.050","Text":"Well, let\u0027s calculate them."},{"Start":"03:40.370 ","End":"03:43.335","Text":"In the range between 1 and 2,"},{"Start":"03:43.335 ","End":"03:51.050","Text":"let\u0027s just pick a random point we call that t. That\u0027s t right here."},{"Start":"03:51.050 ","End":"03:59.775","Text":"What we want to do is we want to calculate the area of this rectangle right here."},{"Start":"03:59.775 ","End":"04:05.479","Text":"Now, the area of a rectangle that equals to the base times the height."},{"Start":"04:05.479 ","End":"04:09.630","Text":"Now, what\u0027s the base here?"},{"Start":"04:09.630 ","End":"04:12.570","Text":"That\u0027s t minus 1."},{"Start":"04:12.570 ","End":"04:17.500","Text":"What\u0027s the height? Well, the height is 0.25."},{"Start":"04:18.200 ","End":"04:25.510","Text":"That\u0027s the expression for the CDF when we\u0027re in this range between 1 and 2."},{"Start":"04:25.510 ","End":"04:32.975","Text":"Now, what about the expression for the CDF when we\u0027re in the range between 2 and 6?"},{"Start":"04:32.975 ","End":"04:36.845","Text":"Well again, let\u0027s just pick a point,"},{"Start":"04:36.845 ","End":"04:43.430","Text":"we call that t. We\u0027re looking for the probability of x being less than or equal to t,"},{"Start":"04:43.430 ","End":"04:48.870","Text":"that means that we\u0027re looking for the probability,"},{"Start":"04:49.090 ","End":"04:53.990","Text":"this probability or this area right here."},{"Start":"05:00.640 ","End":"05:04.730","Text":"We know what the area of this rectangle is right here."},{"Start":"05:04.730 ","End":"05:11.600","Text":"That\u0027s 0.25, that\u0027s 1 times 0.25."},{"Start":"05:11.600 ","End":"05:16.415","Text":"We have to add that to the area of this rectangle right here."},{"Start":"05:16.415 ","End":"05:18.590","Text":"This area right here,"},{"Start":"05:18.590 ","End":"05:20.798","Text":"that\u0027s the base times the height."},{"Start":"05:20.798 ","End":"05:27.530","Text":"The base would be t minus 2 and the height would be c now."},{"Start":"05:27.530 ","End":"05:36.180","Text":"See we\u0027ve calculated that to be 3/16, so that\u0027s 3/16."},{"Start":"05:36.180 ","End":"05:42.450","Text":"This here is the cumulative distribution function of x."}],"ID":32243},{"Watched":false,"Name":"Exercise 1 - Part c","Duration":"3m 58s","ChapterTopicVideoID":12931,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.080","Text":"In Section C, we\u0027re asked to calculate the following probabilities."},{"Start":"00:04.080 ","End":"00:07.605","Text":"Well, let\u0027s use our CDF to do that."},{"Start":"00:07.605 ","End":"00:10.350","Text":"Our CDF is f of t,"},{"Start":"00:10.350 ","End":"00:18.240","Text":"that equals to the probability of x being less than or equal to t. What\u0027s our CDF?"},{"Start":"00:18.240 ","End":"00:21.450","Text":"Well, that equals to following."},{"Start":"00:21.450 ","End":"00:26.610","Text":"Now it\u0027s 0 when x was less than 1."},{"Start":"00:26.610 ","End":"00:30.285","Text":"When x was between 1 and 2,"},{"Start":"00:30.285 ","End":"00:37.365","Text":"that was t minus 1 times 0.25."},{"Start":"00:37.365 ","End":"00:42.555","Text":"When x was greater than 2 and less than 6,"},{"Start":"00:42.555 ","End":"00:52.335","Text":"that was 0.25 plus t minus 2 times 3/16."},{"Start":"00:52.335 ","End":"00:55.140","Text":"When x was greater than 6,"},{"Start":"00:55.140 ","End":"00:57.870","Text":"that was just equal to 1."},{"Start":"00:57.870 ","End":"01:03.090","Text":"Let\u0027s start calculating these probabilities."},{"Start":"01:03.160 ","End":"01:08.910","Text":"The probability of x being less than 4,"},{"Start":"01:08.910 ","End":"01:11.910","Text":"that\u0027s f at 4."},{"Start":"01:11.910 ","End":"01:14.565","Text":"That equals to what?"},{"Start":"01:14.565 ","End":"01:18.330","Text":"Well, 4 is in this range between 2 and 6."},{"Start":"01:18.330 ","End":"01:20.040","Text":"We\u0027ll be using this expression,"},{"Start":"01:20.040 ","End":"01:28.185","Text":"so that\u0027ll be 0.25 plus 4 minus 2 times 3/16,"},{"Start":"01:28.185 ","End":"01:32.080","Text":"and that equals to 5/8."},{"Start":"01:32.080 ","End":"01:37.250","Text":"What about the probability of x being greater than 1.5?"},{"Start":"01:37.250 ","End":"01:42.590","Text":"Well, that equals to 1 minus f at 1.5."},{"Start":"01:42.590 ","End":"01:44.645","Text":"That\u0027s 1 minus."},{"Start":"01:44.645 ","End":"01:49.040","Text":"Now, 1.5 is in this range between 1 and 2,"},{"Start":"01:49.040 ","End":"01:51.245","Text":"so we\u0027ll be using this expression."},{"Start":"01:51.245 ","End":"01:57.320","Text":"There\u0027ll be 1.5 minus 1 times 0.25,"},{"Start":"01:57.320 ","End":"02:02.160","Text":"and that will equal to 7/8."},{"Start":"02:02.160 ","End":"02:08.059","Text":"Let\u0027s take a look at this probability."},{"Start":"02:08.059 ","End":"02:15.470","Text":"Probability of x being between 1.5 and 5."},{"Start":"02:15.470 ","End":"02:23.660","Text":"Well, that equals to f at 5 minus f at 1.5."},{"Start":"02:23.660 ","End":"02:26.045","Text":"Now, f at 5,"},{"Start":"02:26.045 ","End":"02:28.370","Text":"we\u0027ll be using this expression."},{"Start":"02:28.370 ","End":"02:34.490","Text":"That\u0027ll be 0.25 plus 5 minus"},{"Start":"02:34.490 ","End":"02:41.330","Text":"2 times 3/16 minus f at 1.5."},{"Start":"02:41.330 ","End":"02:44.675","Text":"Well, then we\u0027ll be using this expression, minus,"},{"Start":"02:44.675 ","End":"02:51.910","Text":"that\u0027s 1.5 minus 1 times 0.25."},{"Start":"02:51.910 ","End":"03:02.190","Text":"Now, that will equal to 11/16."},{"Start":"03:02.190 ","End":"03:09.740","Text":"The last probability, that\u0027s the probability of x being between 5 and 10."},{"Start":"03:09.740 ","End":"03:12.800","Text":"Well, that\u0027s the probability,"},{"Start":"03:12.800 ","End":"03:17.610","Text":"or that\u0027s f of 10 minus f at 5."},{"Start":"03:17.610 ","End":"03:20.930","Text":"Now, f at 10,"},{"Start":"03:20.930 ","End":"03:24.515","Text":"but that\u0027s right here. That equals to 1."},{"Start":"03:24.515 ","End":"03:33.215","Text":"The probability of x being less than 10 is 1 minus f at 5."},{"Start":"03:33.215 ","End":"03:36.770","Text":"Now, f at 5 is right here."},{"Start":"03:36.770 ","End":"03:38.848","Text":"We\u0027ll be using this expression right here,"},{"Start":"03:38.848 ","End":"03:43.765","Text":"so that\u0027s 0.25 plus"},{"Start":"03:43.765 ","End":"03:50.390","Text":"5 minus 2 times 3/16,"},{"Start":"03:50.390 ","End":"03:59.560","Text":"and that equals to 3/16."}],"ID":32244},{"Watched":false,"Name":"Exercise 1 - Part d","Duration":"4m 22s","ChapterTopicVideoID":12929,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.005","Text":"In section d, we\u0027re asked to find the median of x."},{"Start":"00:04.005 ","End":"00:07.020","Text":"First of all, let\u0027s define the median."},{"Start":"00:07.020 ","End":"00:13.000","Text":"The median is a specific value of x where 1/2"},{"Start":"00:13.000 ","End":"00:18.795","Text":"of the probability lies below this value and half above it."},{"Start":"00:18.795 ","End":"00:23.340","Text":"We usually write the median as x_0.5,"},{"Start":"00:23.340 ","End":"00:26.670","Text":"or we just write md as well."},{"Start":"00:26.670 ","End":"00:32.590","Text":"This is a very special percentile, the median."},{"Start":"00:32.930 ","End":"00:34.980","Text":"We write it like this."},{"Start":"00:34.980 ","End":"00:37.500","Text":"That\u0027s the probability of x."},{"Start":"00:37.500 ","End":"00:41.770","Text":"Being less than or equal to a specific value,"},{"Start":"00:41.770 ","End":"00:45.345","Text":"that would be 0.5."},{"Start":"00:45.345 ","End":"00:48.608","Text":"This is a definition of a percentile,"},{"Start":"00:48.608 ","End":"00:56.095","Text":"where the probability that x would be less than that,"},{"Start":"00:56.095 ","End":"01:02.145","Text":"this specific value would be a 1/2."},{"Start":"01:02.145 ","End":"01:05.375","Text":"This also can be solved using the CDF."},{"Start":"01:05.375 ","End":"01:07.535","Text":"Let\u0027s write out the CDF."},{"Start":"01:07.535 ","End":"01:10.453","Text":"Now the CDF, F of t,"},{"Start":"01:10.453 ","End":"01:15.590","Text":"that\u0027s the probability of x being less than or equal to t. Well,"},{"Start":"01:15.590 ","End":"01:22.965","Text":"in our case, that was zero when x was less than 1."},{"Start":"01:22.965 ","End":"01:26.985","Text":"It was t minus 1 times a 1/4."},{"Start":"01:26.985 ","End":"01:31.125","Text":"When x was between 1 and 2,"},{"Start":"01:31.125 ","End":"01:38.565","Text":"it was a 1/4 plus t minus 2 times 3/16,"},{"Start":"01:38.565 ","End":"01:42.825","Text":"where x was between 2 and 6,"},{"Start":"01:42.825 ","End":"01:47.175","Text":"and 1 when x was greater than 6."},{"Start":"01:47.175 ","End":"01:53.993","Text":"How can we use this CDF to solve this probability,"},{"Start":"01:53.993 ","End":"01:57.645","Text":"or to find the median of x, or this distribution?"},{"Start":"01:57.645 ","End":"02:02.979","Text":"Well, let\u0027s use also the density function."},{"Start":"02:02.979 ","End":"02:09.400","Text":"Here\u0027s a density function and we can see the following."},{"Start":"02:09.400 ","End":"02:11.260","Text":"Well, first of all,"},{"Start":"02:11.260 ","End":"02:14.860","Text":"when we calculated the probabilities,"},{"Start":"02:14.860 ","End":"02:18.115","Text":"or the area underneath the density function,"},{"Start":"02:18.115 ","End":"02:26.575","Text":"we found that we calculated that the area under this first rectangle was 0.25."},{"Start":"02:26.575 ","End":"02:34.210","Text":"That means that in order to get 50 percent or 0.5 of the probabilities,"},{"Start":"02:34.210 ","End":"02:38.080","Text":"then the median has to be somewhere here."},{"Start":"02:38.080 ","End":"02:41.860","Text":"We\u0027ll call that t. That means that"},{"Start":"02:41.860 ","End":"02:48.720","Text":"all the probabilities below t have to equal to a 1/2."},{"Start":"02:48.720 ","End":"02:51.200","Text":"Now, how do we do that?"},{"Start":"02:51.200 ","End":"02:53.825","Text":"Since we know that t is in this range,"},{"Start":"02:53.825 ","End":"02:59.030","Text":"from 2-6, then we\u0027ll be using this expression right here."},{"Start":"02:59.030 ","End":"03:00.755","Text":"Let\u0027s just write this out."},{"Start":"03:00.755 ","End":"03:07.930","Text":"That will be a 1/4 plus t minus 2 times 3/16."},{"Start":"03:07.930 ","End":"03:15.700","Text":"Well, we know in order to find the median that has to be equal to a 1/2."},{"Start":"03:15.700 ","End":"03:24.755","Text":"That means now let\u0027s solve for t. That means that t minus 2 times 3/16,"},{"Start":"03:24.755 ","End":"03:28.529","Text":"that has to be equal to a 1/4,"},{"Start":"03:28.529 ","End":"03:33.155","Text":"and that means that t minus 2, well,"},{"Start":"03:33.155 ","End":"03:37.355","Text":"that equals to 16 over 12,"},{"Start":"03:37.355 ","End":"03:40.404","Text":"which equals to 4 over 3."},{"Start":"03:40.404 ","End":"03:47.000","Text":"That means that t now equals to 2 plus 4 over 3,"},{"Start":"03:47.000 ","End":"03:50.530","Text":"that equals to 3 and 1/3."},{"Start":"03:50.530 ","End":"03:52.715","Text":"What does that tell us?"},{"Start":"03:52.715 ","End":"03:57.785","Text":"That tells us that when t equals 3 and 1/3,"},{"Start":"03:57.785 ","End":"04:00.210","Text":"this point right here,"},{"Start":"04:00.250 ","End":"04:06.515","Text":"1/2 of the probability lies below this value,"},{"Start":"04:06.515 ","End":"04:08.930","Text":"and half lies above this value."},{"Start":"04:08.930 ","End":"04:13.135","Text":"That\u0027s the meaning, or that\u0027s the definition of a median."},{"Start":"04:13.135 ","End":"04:22.300","Text":"This is a solution of d. We found the median, that\u0027s right here."}],"ID":32245},{"Watched":false,"Name":"Exercise 2","Duration":"8m 51s","ChapterTopicVideoID":12932,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"In this question, we have a continuous random variable x"},{"Start":"00:03.240 ","End":"00:05.445","Text":"that has the following density function."},{"Start":"00:05.445 ","End":"00:07.905","Text":"Small of f of x, that\u0027s the density function,"},{"Start":"00:07.905 ","End":"00:09.390","Text":"equals c times x,"},{"Start":"00:09.390 ","End":"00:10.950","Text":"a constant times x,"},{"Start":"00:10.950 ","End":"00:15.285","Text":"where x is between 0 and b and 0 otherwise."},{"Start":"00:15.285 ","End":"00:20.880","Text":"Now it\u0027s also known that the probability of x being between 0 and 1, well, that\u0027s 1/4."},{"Start":"00:20.880 ","End":"00:24.840","Text":"We\u0027re asked to write the density function of x."},{"Start":"00:24.840 ","End":"00:28.510","Text":"Let\u0027s first draw this function."},{"Start":"00:28.790 ","End":"00:33.525","Text":"Here we have the density function right here."},{"Start":"00:33.525 ","End":"00:35.960","Text":"We\u0027re on the x-axis,"},{"Start":"00:35.960 ","End":"00:38.720","Text":"we see the points 0 and we see b,"},{"Start":"00:38.720 ","End":"00:41.780","Text":"so x is between 0 and b."},{"Start":"00:41.780 ","End":"00:44.035","Text":"That\u0027s what we have here."},{"Start":"00:44.035 ","End":"00:51.680","Text":"In that range, we define the density function fx equals c times x."},{"Start":"00:51.680 ","End":"00:53.540","Text":"Well, this is this guy right here."},{"Start":"00:53.540 ","End":"00:56.120","Text":"That\u0027s a straight line, c times x."},{"Start":"00:56.120 ","End":"01:00.620","Text":"Now, we can see that this straight line goes through"},{"Start":"01:00.620 ","End":"01:07.260","Text":"the origin because there\u0027s no y-intercept."},{"Start":"01:08.120 ","End":"01:10.490","Text":"What do we want to do?"},{"Start":"01:10.490 ","End":"01:13.700","Text":"We want to write the density function of x."},{"Start":"01:13.700 ","End":"01:19.085","Text":"That\u0027s fine. But what do we know here about density functions?"},{"Start":"01:19.085 ","End":"01:21.575","Text":"How can we calculate c?"},{"Start":"01:21.575 ","End":"01:24.680","Text":"Well, we know that the area under"},{"Start":"01:24.680 ","End":"01:30.870","Text":"the density function or the total area under the density function equals 1."},{"Start":"01:31.480 ","End":"01:33.830","Text":"What do we also know?"},{"Start":"01:33.830 ","End":"01:38.430","Text":"We also know that in the range 0 and 1,"},{"Start":"01:38.430 ","End":"01:41.760","Text":"the probability is 1/4."},{"Start":"01:41.760 ","End":"01:45.105","Text":"That means that in the range between 0 and 1,"},{"Start":"01:45.105 ","End":"01:47.215","Text":"which is right here,"},{"Start":"01:47.215 ","End":"01:51.800","Text":"the area under the density function"},{"Start":"01:51.800 ","End":"01:58.770","Text":"equals 1/4 because the area under the density function is the probability."},{"Start":"01:59.300 ","End":"02:05.630","Text":"First of all, let\u0027s calculate the areas of the 2 triangles right here."},{"Start":"02:05.630 ","End":"02:08.675","Text":"Well, let\u0027s take a look at the big triangle first."},{"Start":"02:08.675 ","End":"02:13.870","Text":"The base is b times the height where"},{"Start":"02:13.870 ","End":"02:19.085","Text":"the height is the value of the density function at point b."},{"Start":"02:19.085 ","End":"02:27.650","Text":"That would be c times b divided by 2 and that equals to 1."},{"Start":"02:27.650 ","End":"02:30.335","Text":"Now, what does that mean?"},{"Start":"02:30.335 ","End":"02:33.950","Text":"That means that b squared times c,"},{"Start":"02:33.950 ","End":"02:36.395","Text":"that equals to 2."},{"Start":"02:36.395 ","End":"02:40.490","Text":"Now, what else are we given?"},{"Start":"02:40.490 ","End":"02:43.835","Text":"Let\u0027s call this the first equation right here."},{"Start":"02:43.835 ","End":"02:50.150","Text":"Now, what about the second calculation of the area for the smaller triangle?"},{"Start":"02:50.150 ","End":"02:56.220","Text":"Well, the base is 1 times the height."},{"Start":"02:56.220 ","End":"02:59.630","Text":"Well, the height at 1 is basically"},{"Start":"02:59.630 ","End":"03:03.680","Text":"the value of the density function at 1 and that equals to c,"},{"Start":"03:03.680 ","End":"03:07.640","Text":"c times 1 equal c, divided by 2."},{"Start":"03:07.640 ","End":"03:10.740","Text":"That equals to 1/4."},{"Start":"03:10.740 ","End":"03:13.470","Text":"That\u0027s given to us right here."},{"Start":"03:13.470 ","End":"03:18.525","Text":"That means that c equals 1/2."},{"Start":"03:18.525 ","End":"03:25.780","Text":"We automatically got c here so let\u0027s just plug this value back into here."},{"Start":"03:26.270 ","End":"03:31.830","Text":"This means that b squared times c,"},{"Start":"03:31.830 ","End":"03:35.445","Text":"now, c is 1/2, equals 2."},{"Start":"03:35.445 ","End":"03:39.385","Text":"B squared equals 4."},{"Start":"03:39.385 ","End":"03:43.915","Text":"Now, when we take the square root of both sides,"},{"Start":"03:43.915 ","End":"03:47.590","Text":"we won\u0027t take the negative value of b,"},{"Start":"03:47.590 ","End":"03:49.600","Text":"we\u0027ll take only the positive value of b."},{"Start":"03:49.600 ","End":"03:51.970","Text":"That\u0027s the way it\u0027s being defined."},{"Start":"03:51.970 ","End":"03:58.790","Text":"That means that b equals 2."},{"Start":"03:58.970 ","End":"04:02.090","Text":"Now, let\u0027s take a look."},{"Start":"04:02.090 ","End":"04:06.500","Text":"Now let\u0027s write the density function of x."},{"Start":"04:06.500 ","End":"04:08.835","Text":"Well, that\u0027s f of x."},{"Start":"04:08.835 ","End":"04:14.790","Text":"Now, that would equal c. What\u0027s c? c is 1/2."},{"Start":"04:14.790 ","End":"04:17.925","Text":"So it\u0027s 1/2 times x,"},{"Start":"04:17.925 ","End":"04:26.850","Text":"where x is between 0 and 2 and 0 otherwise."},{"Start":"04:28.120 ","End":"04:31.955","Text":"In section b, we\u0027re asked to find the median of x."},{"Start":"04:31.955 ","End":"04:36.055","Text":"Well, the median of x is this special point,"},{"Start":"04:36.055 ","End":"04:39.015","Text":"we\u0027ll call this t for now,"},{"Start":"04:39.015 ","End":"04:47.780","Text":"where 1/2 of the probability lies below this point and 1/2 lies above this point."},{"Start":"04:47.780 ","End":"04:50.840","Text":"Now, when we talk about probabilities,"},{"Start":"04:50.840 ","End":"04:56.540","Text":"we\u0027re talking basically about the area underneath the density functions."},{"Start":"04:56.540 ","End":"05:02.750","Text":"We\u0027re looking for this area right here to be equal to 1/2."},{"Start":"05:02.750 ","End":"05:07.430","Text":"Now let\u0027s just recall what our density function was."},{"Start":"05:07.430 ","End":"05:10.400","Text":"Well, that was 1/2 times x,"},{"Start":"05:10.400 ","End":"05:18.140","Text":"where x was between 0 and 2 and 0 otherwise."},{"Start":"05:18.140 ","End":"05:23.600","Text":"Let\u0027s now calculate the area of this triangle right here."},{"Start":"05:23.600 ","End":"05:27.290","Text":"Well, that\u0027s the base times the height divided by 2."},{"Start":"05:27.290 ","End":"05:30.410","Text":"What\u0027s the base here of this smaller triangle right here?"},{"Start":"05:30.410 ","End":"05:34.085","Text":"Well, that\u0027s t. What\u0027s the height?"},{"Start":"05:34.085 ","End":"05:40.160","Text":"Well, the height is the value of the density function at point t. That equals"},{"Start":"05:40.160 ","End":"05:48.735","Text":"to 1/2 times t. We substitute x for t divided by 2."},{"Start":"05:48.735 ","End":"05:52.160","Text":"The base times the height divided by 2,"},{"Start":"05:52.160 ","End":"06:00.930","Text":"that\u0027s the area of this triangle and that has to equal to 1/2, that\u0027s 0.5."},{"Start":"06:00.930 ","End":"06:05.310","Text":"In essence, let\u0027s just solve for t here."},{"Start":"06:05.310 ","End":"06:08.040","Text":"That\u0027s t squared times 1/2,"},{"Start":"06:08.040 ","End":"06:11.595","Text":"that equals to 1 or t squared,"},{"Start":"06:11.595 ","End":"06:14.190","Text":"well, that\u0027s equals to 2."},{"Start":"06:14.190 ","End":"06:24.420","Text":"That means that t equals to the square root of 2 or 1.41 in approximation."},{"Start":"06:24.420 ","End":"06:29.535","Text":"This is the median of x."},{"Start":"06:29.535 ","End":"06:35.540","Text":"That means that we find 1/2 of"},{"Start":"06:35.540 ","End":"06:41.585","Text":"the probability below this point and 1/2 of the probability we find above this point."},{"Start":"06:41.585 ","End":"06:43.460","Text":"This is square root of 2."},{"Start":"06:43.460 ","End":"06:46.130","Text":"Here we found 0.5 of the probability,"},{"Start":"06:46.130 ","End":"06:49.270","Text":"that\u0027s this area right here."},{"Start":"06:49.270 ","End":"06:52.890","Text":"Here, that\u0027s the other 1/2, right here."},{"Start":"06:52.890 ","End":"06:55.510","Text":"That\u0027s your 0.5 right here."},{"Start":"06:56.110 ","End":"07:02.810","Text":"In section c, we\u0027re asked what are the chances of x being less than 0.5 or 1/2?"},{"Start":"07:02.810 ","End":"07:08.270","Text":"That\u0027s the probability of x being less than 1/2."},{"Start":"07:08.270 ","End":"07:10.115","Text":"This is what we have to find out."},{"Start":"07:10.115 ","End":"07:16.475","Text":"Now, if I would have had my CDF or my cumulative distribution function,"},{"Start":"07:16.475 ","End":"07:18.230","Text":"then I could just plug in the number,"},{"Start":"07:18.230 ","End":"07:19.655","Text":"but since I don\u0027t,"},{"Start":"07:19.655 ","End":"07:23.720","Text":"let\u0027s use the density function and find the area"},{"Start":"07:23.720 ","End":"07:29.150","Text":"under the density function from 0 to 1/2."},{"Start":"07:29.150 ","End":"07:32.775","Text":"Now, this is our point 1/2."},{"Start":"07:32.775 ","End":"07:34.890","Text":"This is where x equals 1/2."},{"Start":"07:34.890 ","End":"07:43.130","Text":"What we want to do is we want to find the area under the density function from 0-1/2."},{"Start":"07:43.130 ","End":"07:45.655","Text":"That\u0027s this guy right here."},{"Start":"07:45.655 ","End":"07:49.460","Text":"Let\u0027s just recall what our density function was."},{"Start":"07:49.460 ","End":"07:52.760","Text":"That was 1/2 times x,"},{"Start":"07:52.760 ","End":"08:00.480","Text":"where x was between 0 and b and 0 otherwise."},{"Start":"08:00.480 ","End":"08:03.110","Text":"What\u0027s this area right here?"},{"Start":"08:03.110 ","End":"08:05.915","Text":"Well, that\u0027s the area of a triangle,"},{"Start":"08:05.915 ","End":"08:09.140","Text":"which is the base times the height divided by 2,"},{"Start":"08:09.140 ","End":"08:12.230","Text":"where the base is 1/2."},{"Start":"08:12.230 ","End":"08:13.490","Text":"Now, what\u0027s the height?"},{"Start":"08:13.490 ","End":"08:19.430","Text":"The height is the value of the density function at this point right here,"},{"Start":"08:19.430 ","End":"08:22.490","Text":"at the point where x equals 1/2."},{"Start":"08:22.490 ","End":"08:30.525","Text":"That\u0027s 1/2 times 1/2 divided by 2."},{"Start":"08:30.525 ","End":"08:32.580","Text":"That\u0027s the base right here,"},{"Start":"08:32.580 ","End":"08:34.110","Text":"this is the height,"},{"Start":"08:34.110 ","End":"08:36.260","Text":"and we divide that by 2."},{"Start":"08:36.260 ","End":"08:38.615","Text":"Base times height divided by 2."},{"Start":"08:38.615 ","End":"08:45.230","Text":"That equals to 1/16 or 1 divided by 16th."},{"Start":"08:45.230 ","End":"08:51.330","Text":"This is the probability of x being less than 1/2."}],"ID":32246},{"Watched":false,"Name":"Exercise 3 - Parts a-b","Duration":"12m 12s","ChapterTopicVideoID":12933,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"The diagram below shows the density function of"},{"Start":"00:02.910 ","End":"00:06.284","Text":"the random variable Y, so this is a diagram."},{"Start":"00:06.284 ","End":"00:08.522","Text":"This axis right here,"},{"Start":"00:08.522 ","End":"00:11.100","Text":"well, that\u0027s the y-axis now."},{"Start":"00:11.100 ","End":"00:13.920","Text":"We\u0027re asked to calculate the value of c,"},{"Start":"00:13.920 ","End":"00:16.090","Text":"the c right here."},{"Start":"00:16.790 ","End":"00:20.685","Text":"We know that this is a density function."},{"Start":"00:20.685 ","End":"00:26.085","Text":"That means that the area underneath the density function has to equal to 1."},{"Start":"00:26.085 ","End":"00:28.980","Text":"We also know that the area of"},{"Start":"00:28.980 ","End":"00:32.730","Text":"a triangle and that equals to 1/2 times the base times the height,"},{"Start":"00:32.730 ","End":"00:42.210","Text":"so in our case the base would be 10 and the height would be c times a 1/2."},{"Start":"00:42.210 ","End":"00:45.510","Text":"That\u0027s the area of this triangle right here,"},{"Start":"00:45.510 ","End":"00:49.540","Text":"and because this is a density function that has to equal to 1."},{"Start":"00:49.540 ","End":"00:55.015","Text":"That means that 10C has to equal to 2,"},{"Start":"00:55.015 ","End":"01:00.620","Text":"or c has to equal to 2 over 10, that equals to 0.2."},{"Start":"01:00.620 ","End":"01:05.660","Text":"That\u0027s the value of c. In section b,"},{"Start":"01:05.660 ","End":"01:09.440","Text":"we\u0027re asked to calculate the cumulative distribution function of Y."},{"Start":"01:09.440 ","End":"01:17.450","Text":"Let\u0027s denote that as big F at t. That\u0027s the probability of y being"},{"Start":"01:17.450 ","End":"01:21.200","Text":"less than or equal to t. Now in order to calculate"},{"Start":"01:21.200 ","End":"01:25.745","Text":"big F of t or the cumulative distribution function at point t,"},{"Start":"01:25.745 ","End":"01:29.120","Text":"we\u0027re going to have to split y up into 4 ranges."},{"Start":"01:29.120 ","End":"01:33.985","Text":"The first range is where t is less than 0."},{"Start":"01:33.985 ","End":"01:36.315","Text":"That\u0027s this range right here."},{"Start":"01:36.315 ","End":"01:42.840","Text":"The second range is where t is between 0 and 5."},{"Start":"01:42.840 ","End":"01:49.035","Text":"The third range is where t is greater than 5 and less than 10,"},{"Start":"01:49.035 ","End":"01:50.805","Text":"between 5 and 10."},{"Start":"01:50.805 ","End":"01:55.755","Text":"The fourth range is where t is greater than 10."},{"Start":"01:55.755 ","End":"02:00.920","Text":"Now, for values of the CDF,"},{"Start":"02:00.920 ","End":"02:02.630","Text":"where t is less than 0,"},{"Start":"02:02.630 ","End":"02:06.320","Text":"while the value of the CDF equals 0 because we"},{"Start":"02:06.320 ","End":"02:10.480","Text":"haven\u0027t accumulated any area or any density,"},{"Start":"02:10.480 ","End":"02:12.560","Text":"in this range, and why is that?"},{"Start":"02:12.560 ","End":"02:16.745","Text":"Because the density function is defined between 0 and 10."},{"Start":"02:16.745 ","End":"02:23.294","Text":"There\u0027s no area to accumulate in this range or in this values of t,"},{"Start":"02:23.294 ","End":"02:25.220","Text":"so that equals 0."},{"Start":"02:25.220 ","End":"02:32.090","Text":"Now let\u0027s look at the other end of this density function where t is greater than 10."},{"Start":"02:32.090 ","End":"02:34.460","Text":"Well, in this range right here,"},{"Start":"02:34.460 ","End":"02:39.430","Text":"we\u0027ve actually accumulated all of the area under the density function."},{"Start":"02:39.430 ","End":"02:45.770","Text":"That means that the CDF or the cumulative distribution function equals 1."},{"Start":"02:45.770 ","End":"02:49.940","Text":"Now we have to figure out what\u0027s the value of"},{"Start":"02:49.940 ","End":"02:54.785","Text":"the CDF or what are the expressions of the CDF in these ranges?"},{"Start":"02:54.785 ","End":"02:59.160","Text":"Between 0 and 5 and between 5 and 10."},{"Start":"02:59.780 ","End":"03:05.060","Text":"Let\u0027s take a look now at this range right here between 0 and 5."},{"Start":"03:05.060 ","End":"03:10.550","Text":"In order to calculate the expression of F of t in this range, well,"},{"Start":"03:10.550 ","End":"03:12.380","Text":"we\u0027ll define a point,"},{"Start":"03:12.380 ","End":"03:14.170","Text":"we call that t,"},{"Start":"03:14.170 ","End":"03:17.460","Text":"and we\u0027ll figure it out."},{"Start":"03:17.460 ","End":"03:24.170","Text":"We\u0027ll calculate the area of the density function between 0 and t,"},{"Start":"03:24.170 ","End":"03:28.550","Text":"so in order to calculate the area of a triangle,"},{"Start":"03:28.550 ","End":"03:33.110","Text":"we know that that equals to 1/2 times the base times the height."},{"Start":"03:33.110 ","End":"03:38.000","Text":"Well, the base is t. But what about the height?"},{"Start":"03:38.000 ","End":"03:41.390","Text":"Well, the height equals the value of"},{"Start":"03:41.390 ","End":"03:45.650","Text":"the density function at point t. Now in order to calculate this value,"},{"Start":"03:45.650 ","End":"03:48.320","Text":"we need to find out what the expression is,"},{"Start":"03:48.320 ","End":"03:51.635","Text":"what the equation is of this straight line."},{"Start":"03:51.635 ","End":"03:54.135","Text":"Let\u0027s do that. Now,"},{"Start":"03:54.135 ","End":"03:58.850","Text":"we know that the equation of a straight line equals m,"},{"Start":"03:58.850 ","End":"04:03.530","Text":"the slope times x plus n, the y-intercept."},{"Start":"04:03.530 ","End":"04:12.744","Text":"In our case, n equals 0 because this line cuts the origin and goes through the origins."},{"Start":"04:12.744 ","End":"04:15.510","Text":"Now let\u0027s take a look at m. Well,"},{"Start":"04:15.510 ","End":"04:18.440","Text":"m equals 0.2 minus 0,"},{"Start":"04:18.440 ","End":"04:20.570","Text":"that\u0027s y_2 minus y_1."},{"Start":"04:20.570 ","End":"04:23.000","Text":"Remember that c equals 0.2."},{"Start":"04:23.000 ","End":"04:25.535","Text":"We\u0027ve calculated that in the last section,"},{"Start":"04:25.535 ","End":"04:29.525","Text":"divided by 5 minus 0,"},{"Start":"04:29.525 ","End":"04:33.065","Text":"that equals to 0.2 divided by 5,"},{"Start":"04:33.065 ","End":"04:36.080","Text":"that equals to 0.04."},{"Start":"04:36.080 ","End":"04:41.820","Text":"The expression for this line right here,"},{"Start":"04:41.850 ","End":"04:49.988","Text":"that equals to 0.04 times x plus 0,"},{"Start":"04:49.988 ","End":"04:58.655","Text":"so when we want to figure out right now what the area of this triangle is,"},{"Start":"04:58.655 ","End":"05:05.930","Text":"well that equals to the 1/2 times the base times the height."},{"Start":"05:05.930 ","End":"05:09.680","Text":"Well, the height is 0.04 times t,"},{"Start":"05:09.680 ","End":"05:16.355","Text":"so that\u0027s 0.04 times t. Now,"},{"Start":"05:16.355 ","End":"05:24.355","Text":"and that equals to 0.02 times t squared."},{"Start":"05:24.355 ","End":"05:27.750","Text":"We\u0027ll put that right here."},{"Start":"05:27.750 ","End":"05:32.600","Text":"That\u0027ll be 0.02t squared."},{"Start":"05:32.600 ","End":"05:35.750","Text":"Now, that\u0027s the expression of"},{"Start":"05:35.750 ","End":"05:42.930","Text":"the cumulative distribution function in the range between 0 and 5."},{"Start":"05:43.520 ","End":"05:50.135","Text":"Now let\u0027s calculate the cumulative distribution function in this range between 5 and 10."},{"Start":"05:50.135 ","End":"05:53.060","Text":"Now, just like we did in the previous range,"},{"Start":"05:53.060 ","End":"05:57.075","Text":"we\u0027ll pick a point t between 5 and 10,"},{"Start":"05:57.075 ","End":"06:00.470","Text":"and what we\u0027ll want to do is to calculate the area under"},{"Start":"06:00.470 ","End":"06:05.120","Text":"the density function between 0 and t. Now,"},{"Start":"06:05.120 ","End":"06:08.525","Text":"this looks like a pretty complicated area to calculate."},{"Start":"06:08.525 ","End":"06:11.960","Text":"But we can calculate this triangle right"},{"Start":"06:11.960 ","End":"06:15.785","Text":"here pretty easily between the range of t and 10."},{"Start":"06:15.785 ","End":"06:21.320","Text":"Now, the area of a triangle as we said that\u0027s a 1/2 times the base times the height."},{"Start":"06:21.320 ","End":"06:23.180","Text":"The base is pretty easy to calculate."},{"Start":"06:23.180 ","End":"06:25.265","Text":"That\u0027s 10 minus t, but what about the height?"},{"Start":"06:25.265 ","End":"06:27.530","Text":"Well, in order to calculate the height,"},{"Start":"06:27.530 ","End":"06:33.890","Text":"we need to calculate the equation for this line right here,"},{"Start":"06:33.890 ","End":"06:36.095","Text":"let\u0027s do that first."},{"Start":"06:36.095 ","End":"06:38.000","Text":"What\u0027s the slope? Well,"},{"Start":"06:38.000 ","End":"06:41.260","Text":"the slope is 0.2."},{"Start":"06:41.260 ","End":"06:43.900","Text":"That\u0027s y_2 minus y_1."},{"Start":"06:43.900 ","End":"06:48.665","Text":"Minus 0 divided by 5 minus 10,"},{"Start":"06:48.665 ","End":"06:52.910","Text":"that equals to 0.2 divided by minus 5,"},{"Start":"06:52.910 ","End":"06:55.925","Text":"and that equals to minus 0.04."},{"Start":"06:55.925 ","End":"06:59.360","Text":"Now, how do we calculate"},{"Start":"06:59.360 ","End":"07:03.755","Text":"the equation of a line where we know the slope and we know 1 point."},{"Start":"07:03.755 ","End":"07:08.103","Text":"Well, that equals to y minus y_1,"},{"Start":"07:08.103 ","End":"07:13.455","Text":"that would equal to the slope times x minus x_1."},{"Start":"07:13.455 ","End":"07:17.105","Text":"In our case, we\u0027ll take this point right here."},{"Start":"07:17.105 ","End":"07:19.145","Text":"That\u0027ll be y minus 0,"},{"Start":"07:19.145 ","End":"07:30.180","Text":"and that will be equal to minus 0.04 times x minus 10."},{"Start":"07:30.260 ","End":"07:36.080","Text":"This is the equation of this line right here."},{"Start":"07:36.080 ","End":"07:41.420","Text":"Now that we have that we can calculate the area of the triangle."},{"Start":"07:41.420 ","End":"07:45.910","Text":"Now the area of a triangle will be 1/2 times the base, well,"},{"Start":"07:45.910 ","End":"07:52.295","Text":"the base would be 10 minus t times the height."},{"Start":"07:52.295 ","End":"07:58.740","Text":"Well, the height is the value of the line,"},{"Start":"07:58.740 ","End":"08:04.700","Text":"this line right here at point t. That\u0027ll be times minus 0.04."},{"Start":"08:04.700 ","End":"08:09.965","Text":"We\u0027ll just input t into this guy times t minus 10."},{"Start":"08:09.965 ","End":"08:14.450","Text":"Now, we can simplify this,"},{"Start":"08:14.450 ","End":"08:21.710","Text":"and that will be 0.02"},{"Start":"08:21.710 ","End":"08:25.580","Text":"times 10 minus t squared."},{"Start":"08:25.580 ","End":"08:26.960","Text":"Now what did we do here?"},{"Start":"08:26.960 ","End":"08:29.930","Text":"10 minus t. We just switched this around,"},{"Start":"08:29.930 ","End":"08:31.490","Text":"but when we switch this around,"},{"Start":"08:31.490 ","End":"08:35.020","Text":"we had to make this a positive instead of a negative,"},{"Start":"08:35.020 ","End":"08:40.960","Text":"and 1/2 times 0.04 is 0.02,"},{"Start":"08:40.960 ","End":"08:44.150","Text":"and then we have 10 minus t squared."},{"Start":"08:44.150 ","End":"08:47.930","Text":"Now, this is the area of this triangle right here,"},{"Start":"08:47.930 ","End":"08:53.790","Text":"so all we need to do is subtract that from 1 to get this area right here."},{"Start":"08:53.790 ","End":"08:57.710","Text":"That will be the expression for the CDF in this range."},{"Start":"08:57.710 ","End":"08:59.390","Text":"Let\u0027s just write this out."},{"Start":"08:59.390 ","End":"09:07.700","Text":"That\u0027ll be 1 minus 0.02 times 10 minus t squared,"},{"Start":"09:07.700 ","End":"09:13.260","Text":"so this is the CDF of Y."}],"ID":32247},{"Watched":false,"Name":"Exercise 3 - Parts c-d","Duration":"8m 53s","ChapterTopicVideoID":12934,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"In Section C, we\u0027re asked to calculate the following probabilities,"},{"Start":"00:03.540 ","End":"00:07.515","Text":"the probability of Y being greater than 4 and so on and so forth."},{"Start":"00:07.515 ","End":"00:15.710","Text":"Well, let\u0027s first take a look at the CDF for Y. Here it is."},{"Start":"00:15.710 ","End":"00:18.320","Text":"Now, let\u0027s take the first one."},{"Start":"00:18.320 ","End":"00:21.020","Text":"Let\u0027s take this probability right here,"},{"Start":"00:21.020 ","End":"00:25.175","Text":"the probability of Y being equal to 7."},{"Start":"00:25.175 ","End":"00:27.680","Text":"Well, we don\u0027t have to do a lot of thinking here."},{"Start":"00:27.680 ","End":"00:29.600","Text":"That equals 0. Why is that?"},{"Start":"00:29.600 ","End":"00:34.565","Text":"Because the probability of equaling something is always"},{"Start":"00:34.565 ","End":"00:39.770","Text":"equal to 0 when we\u0027re dealing with a continuous random variable."},{"Start":"00:39.770 ","End":"00:41.960","Text":"Now, let\u0027s take a look at this."},{"Start":"00:41.960 ","End":"00:47.645","Text":"The probability of Y being less than or equal to 3."},{"Start":"00:47.645 ","End":"00:52.265","Text":"Well, less than or equal to 3 we\u0027re dealing here with this range right here."},{"Start":"00:52.265 ","End":"00:56.670","Text":"That\u0027s F at 3."},{"Start":"00:56.670 ","End":"01:01.400","Text":"Again, we\u0027re dealing with this range right here so let\u0027s just put 3"},{"Start":"01:01.400 ","End":"01:05.360","Text":"into t. Replace t with 3 here in this equation."},{"Start":"01:05.360 ","End":"01:09.530","Text":"That\u0027ll be 0.02 times 3 squared,"},{"Start":"01:09.530 ","End":"01:13.115","Text":"and that equals to 0.18."},{"Start":"01:13.115 ","End":"01:17.320","Text":"Now, what about this probability right here?"},{"Start":"01:17.320 ","End":"01:26.940","Text":"The probability of Y being between 7.5 and 15.5."},{"Start":"01:26.940 ","End":"01:37.110","Text":"Well, that equals to F at 15.5 minus F at 7.5."},{"Start":"01:37.110 ","End":"01:40.320","Text":"Well, F at 15.5,"},{"Start":"01:40.320 ","End":"01:43.980","Text":"well that\u0027s this range right here where Y is greater than 10,"},{"Start":"01:43.980 ","End":"01:46.950","Text":"so that\u0027ll equal to 1 minus."},{"Start":"01:46.950 ","End":"01:50.669","Text":"Now, what\u0027s F at 7.5? That\u0027s here."},{"Start":"01:50.669 ","End":"01:55.725","Text":"That\u0027s right here. All we need to do is plug in 7.5 here."},{"Start":"01:55.725 ","End":"02:07.180","Text":"That\u0027ll be 1 minus 10 minus 7.5 squared times 0.02."},{"Start":"02:08.390 ","End":"02:13.390","Text":"This comes out to 0.125."},{"Start":"02:15.010 ","End":"02:18.710","Text":"Now, what about this guy right here?"},{"Start":"02:18.710 ","End":"02:23.305","Text":"The probability of Y being greater than 4?"},{"Start":"02:23.305 ","End":"02:32.390","Text":"Well, that equals to 1 minus F at 4."},{"Start":"02:32.390 ","End":"02:36.665","Text":"Now, that equals to 1 minus, what\u0027s F at 4?"},{"Start":"02:36.665 ","End":"02:44.775","Text":"Again, that\u0027s here in this range so we\u0027ll be using this 1 minus 0.02 times 4 squared,"},{"Start":"02:44.775 ","End":"02:48.750","Text":"and that comes out to 0.32."},{"Start":"02:48.750 ","End":"02:57.810","Text":"These are the answer to the probabilities that were presented here."},{"Start":"02:57.940 ","End":"03:02.030","Text":"In section D, we\u0027re asked to calculate the bottom 10th percentile,"},{"Start":"03:02.030 ","End":"03:06.709","Text":"that\u0027s Y_0.1, the bottom 25th percentile,"},{"Start":"03:06.709 ","End":"03:10.190","Text":"the median, and the upper 10th percentile."},{"Start":"03:10.190 ","End":"03:14.405","Text":"Now, we can calculate this much easier once we see"},{"Start":"03:14.405 ","End":"03:20.100","Text":"the density function and the CDF. Here they are."},{"Start":"03:20.100 ","End":"03:23.065","Text":"That\u0027s the density function and this is a CDF."},{"Start":"03:23.065 ","End":"03:27.569","Text":"Now, first, let\u0027s take a look at the median."},{"Start":"03:29.020 ","End":"03:32.420","Text":"Now, what\u0027s the definition of the median?"},{"Start":"03:32.420 ","End":"03:38.660","Text":"The definition of the median is that we have a value of Y where"},{"Start":"03:38.660 ","End":"03:41.930","Text":"50 percent of the probability falls above"},{"Start":"03:41.930 ","End":"03:46.055","Text":"this value and 50 percent falls below this value."},{"Start":"03:46.055 ","End":"03:49.820","Text":"Now, here, when we look at the density function,"},{"Start":"03:49.820 ","End":"03:52.250","Text":"we see a very special characteristic here."},{"Start":"03:52.250 ","End":"03:56.275","Text":"We see that this function is symmetrical."},{"Start":"03:56.275 ","End":"03:59.620","Text":"We have the range from 0-10,"},{"Start":"03:59.620 ","End":"04:05.435","Text":"and it\u0027s symmetrical around the value 5. What does that mean?"},{"Start":"04:05.435 ","End":"04:09.410","Text":"That means that the area here in the range"},{"Start":"04:09.410 ","End":"04:16.105","Text":"between 0 and 5 equals to the area in the range between 5 and 10."},{"Start":"04:16.105 ","End":"04:20.870","Text":"Having said that, then this equals 0.5,"},{"Start":"04:20.870 ","End":"04:23.225","Text":"this area here equals 0.5,"},{"Start":"04:23.225 ","End":"04:26.615","Text":"and this area also equals 0.5."},{"Start":"04:26.615 ","End":"04:31.564","Text":"That\u0027s exactly the definition of a median,"},{"Start":"04:31.564 ","End":"04:33.725","Text":"5 is the median."},{"Start":"04:33.725 ","End":"04:36.470","Text":"We don\u0027t have to do a lot of calculations."},{"Start":"04:36.470 ","End":"04:41.330","Text":"We can say that the median of Y is 5 and we"},{"Start":"04:41.330 ","End":"04:47.340","Text":"can say that only because the density function is symmetrical."},{"Start":"04:47.510 ","End":"04:52.985","Text":"Let\u0027s talk now about the bottom 10th percentile."},{"Start":"04:52.985 ","End":"05:00.880","Text":"That\u0027s the probability of Y being less than or equal to Y_0.1,"},{"Start":"05:00.880 ","End":"05:04.935","Text":"and that has to equal to 0.1."},{"Start":"05:04.935 ","End":"05:11.869","Text":"We said that this portion of the area between 0 and 5,"},{"Start":"05:11.869 ","End":"05:15.170","Text":"then the area underneath the density function,"},{"Start":"05:15.170 ","End":"05:18.250","Text":"and this area has to be equal to 0.5."},{"Start":"05:18.250 ","End":"05:21.695","Text":"If we\u0027re looking for an area that equals to 0.1,"},{"Start":"05:21.695 ","End":"05:28.610","Text":"then the 10th percentile has to be in this range right here."},{"Start":"05:28.610 ","End":"05:31.595","Text":"We\u0027re looking at this guy right here."},{"Start":"05:31.595 ","End":"05:33.560","Text":"Now, if that\u0027s the case,"},{"Start":"05:33.560 ","End":"05:37.175","Text":"then we have to use this expression of the CDF,"},{"Start":"05:37.175 ","End":"05:38.795","Text":"so let\u0027s do that."},{"Start":"05:38.795 ","End":"05:45.230","Text":"We\u0027ll be using 0.02 times t squared and we want that to be equal to"},{"Start":"05:45.230 ","End":"05:52.920","Text":"0.1 in order to calculate the bottom 10th percentile."},{"Start":"05:52.920 ","End":"05:58.340","Text":"That equals to t squared being equal to 5."},{"Start":"05:58.340 ","End":"06:03.875","Text":"T then would be equal to the square root of 5,"},{"Start":"06:03.875 ","End":"06:07.559","Text":"and that equals to 2.24."},{"Start":"06:08.870 ","End":"06:13.340","Text":"Now, what about the bottom 25th percentile?"},{"Start":"06:13.340 ","End":"06:21.340","Text":"That\u0027s the probability of Y being less than or equal to Y_0.25,"},{"Start":"06:21.340 ","End":"06:24.200","Text":"and that has to equal to 0.25."},{"Start":"06:24.200 ","End":"06:29.390","Text":"Now again, the t that we\u0027re looking for is"},{"Start":"06:29.390 ","End":"06:34.910","Text":"the area equaling 25 percent and again,"},{"Start":"06:34.910 ","End":"06:37.865","Text":"that\u0027s in this range between 0 and 5."},{"Start":"06:37.865 ","End":"06:41.115","Text":"We\u0027re looking now for this area,"},{"Start":"06:41.115 ","End":"06:43.650","Text":"this t with this area here equals"},{"Start":"06:43.650 ","End":"06:47.810","Text":"25 percent so we\u0027ll be using the same expression for the CDF."},{"Start":"06:47.810 ","End":"06:51.950","Text":"That\u0027ll be 0.02 times t squared."},{"Start":"06:51.950 ","End":"06:56.960","Text":"That now will be equal to 0.25."},{"Start":"06:56.960 ","End":"07:04.910","Text":"That means that t squared equals 12.5 and t will be equal to"},{"Start":"07:04.910 ","End":"07:13.195","Text":"the square root of 12.5 or 3.54."},{"Start":"07:13.195 ","End":"07:17.135","Text":"Now, what about the upper 10th percentile right here?"},{"Start":"07:17.135 ","End":"07:19.700","Text":"Well, instead of doing a whole bunch of calculations,"},{"Start":"07:19.700 ","End":"07:27.890","Text":"we can infer what that percentile is and we can infer it from the bottom 10th percentile."},{"Start":"07:27.890 ","End":"07:29.969","Text":"Now, if you recall,"},{"Start":"07:30.650 ","End":"07:34.895","Text":"Y_0.1, that equal to 2.24."},{"Start":"07:34.895 ","End":"07:40.145","Text":"What that meant was that we found a point here where"},{"Start":"07:40.145 ","End":"07:47.435","Text":"the area under the density function from 0 to that point was equal to 0.1."},{"Start":"07:47.435 ","End":"07:49.550","Text":"That\u0027s the 10th percentile,"},{"Start":"07:49.550 ","End":"07:51.680","Text":"the bottom 10th percentile."},{"Start":"07:51.680 ","End":"07:56.340","Text":"Now, because the density function is symmetrical,"},{"Start":"07:56.340 ","End":"08:02.780","Text":"and only then can we infer about the upper 10th percentile."},{"Start":"08:02.780 ","End":"08:08.030","Text":"That means that we\u0027re looking for a point here where"},{"Start":"08:08.030 ","End":"08:16.160","Text":"the area under the density function from that point on to 10 would also be 0.1."},{"Start":"08:16.160 ","End":"08:19.300","Text":"Again, and I can\u0027t stress this enough,"},{"Start":"08:19.300 ","End":"08:22.700","Text":"because the density function is symmetrical,"},{"Start":"08:22.700 ","End":"08:28.325","Text":"then this distance right here should be equal to this distance right here."},{"Start":"08:28.325 ","End":"08:33.725","Text":"That means that we\u0027re looking at the 90th percentile,"},{"Start":"08:33.725 ","End":"08:40.155","Text":"the upper 10th percentile as being equal to 10 minus 2.24."},{"Start":"08:40.155 ","End":"08:43.220","Text":"That equals to 7.76."},{"Start":"08:43.220 ","End":"08:54.040","Text":"So this number here would be the upper 10th percentile or Y_0.9."}],"ID":32248},{"Watched":false,"Name":"Exercise 4","Duration":"8m 33s","ChapterTopicVideoID":30193,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.485","Text":"The diagram below shows the density function of the random variable X."},{"Start":"00:04.485 ","End":"00:07.200","Text":"We\u0027re asked to calculate the value of c,"},{"Start":"00:07.200 ","End":"00:10.830","Text":"that\u0027s this value right here that gives a density function."},{"Start":"00:10.830 ","End":"00:16.335","Text":"Now we know that the total area under the density function equals 1."},{"Start":"00:16.335 ","End":"00:21.900","Text":"All we have to do is calculate that the areas of each one of the sections"},{"Start":"00:21.900 ","End":"00:27.930","Text":"and equate that to 1 and we can solve for c, so let\u0027s get started."},{"Start":"00:27.930 ","End":"00:33.660","Text":"The first section or the base here is 3 and the height is 0.2,"},{"Start":"00:33.660 ","End":"00:37.020","Text":"so 3 times 0.2, that\u0027s 0.6."},{"Start":"00:37.020 ","End":"00:43.380","Text":"The next section, the base is 2 and the height is 0.05,"},{"Start":"00:43.380 ","End":"00:45.885","Text":"making the area 0.1."},{"Start":"00:45.885 ","End":"00:48.070","Text":"The next section, that\u0027s 1,"},{"Start":"00:48.070 ","End":"00:54.845","Text":"that\u0027s the base times the height that\u0027s 0.15 so the total area is 0.15."},{"Start":"00:54.845 ","End":"00:56.905","Text":"In the last section,"},{"Start":"00:56.905 ","End":"01:03.710","Text":"the base is c minus 7 and the height is 0.05."},{"Start":"01:03.710 ","End":"01:08.330","Text":"Now let\u0027s just sum up the areas equate to 1 and then"},{"Start":"01:08.330 ","End":"01:12.875","Text":"we can solve for c. That\u0027ll be 0.6 plus"},{"Start":"01:12.875 ","End":"01:23.235","Text":"0.1 plus 0.15 plus c minus 7 times 0.05 and that has to be equal to 1."},{"Start":"01:23.235 ","End":"01:26.040","Text":"That means that c minus 7,"},{"Start":"01:26.040 ","End":"01:32.841","Text":"times 0.05 has to equal to 0.15"},{"Start":"01:32.841 ","End":"01:41.530","Text":"and that means that c minus 7 equals to 3 and c equals 10."},{"Start":"01:41.980 ","End":"01:47.705","Text":"In this section, we\u0027re asked to calculate the cumulative distribution function of X."},{"Start":"01:47.705 ","End":"01:53.585","Text":"We\u0027re asked basically to find what\u0027s the probability of X being less than or equal to t?"},{"Start":"01:53.585 ","End":"01:58.850","Text":"Or F at t. In order to do that,"},{"Start":"01:58.850 ","End":"02:01.744","Text":"we\u0027re going to split X up into various sections."},{"Start":"02:01.744 ","End":"02:05.600","Text":"The first section is where t is less than 0."},{"Start":"02:05.600 ","End":"02:09.490","Text":"Well, in this range right here,"},{"Start":"02:09.490 ","End":"02:12.240","Text":"the density function isn\u0027t defined,"},{"Start":"02:12.240 ","End":"02:16.160","Text":"so the cumulative distribution function would be 0."},{"Start":"02:16.160 ","End":"02:21.920","Text":"Now, what about where t is between 0 and 3?"},{"Start":"02:21.920 ","End":"02:24.830","Text":"Let\u0027s define here point,"},{"Start":"02:24.830 ","End":"02:30.650","Text":"we\u0027ll call this t and the cumulative distribution function is basically"},{"Start":"02:30.650 ","End":"02:36.545","Text":"the area underneath the density function until point t. Now,"},{"Start":"02:36.545 ","End":"02:40.280","Text":"that means that the area here is t,"},{"Start":"02:40.280 ","End":"02:44.255","Text":"that\u0027s the base t times 0.2, that\u0027s the height."},{"Start":"02:44.255 ","End":"02:48.580","Text":"That\u0027ll be 0.2 times t. Now,"},{"Start":"02:48.580 ","End":"02:54.380","Text":"what about where t is between 3 and 5?"},{"Start":"02:54.570 ","End":"02:57.625","Text":"Let\u0027s just pick a point here."},{"Start":"02:57.625 ","End":"03:03.040","Text":"That\u0027ll be t and we want to calculate the area between 0 and t,"},{"Start":"03:03.040 ","End":"03:06.760","Text":"where the first area in the first section in yellow,"},{"Start":"03:06.760 ","End":"03:09.610","Text":"we know what that area is less 0.6."},{"Start":"03:09.610 ","End":"03:14.530","Text":"So the total area until the point t would be"},{"Start":"03:14.530 ","End":"03:21.700","Text":"0.6 plus now that\u0027ll be t minus 3 that\u0027s the base times the height."},{"Start":"03:21.700 ","End":"03:31.400","Text":"The height is 0.05. Now let\u0027s go to the next section where t is between 5 and 6."},{"Start":"03:31.680 ","End":"03:36.460","Text":"Again, let\u0027s just pick a point here, right in this range,"},{"Start":"03:36.460 ","End":"03:43.180","Text":"we\u0027ll call that t and we want to calculate the area between 0 and t. Now,"},{"Start":"03:43.180 ","End":"03:48.760","Text":"the total area in yellow is 0.7."},{"Start":"03:48.760 ","End":"03:54.145","Text":"That will be 0.6 for the first section and 0.1 for the second section."},{"Start":"03:54.145 ","End":"03:58.660","Text":"That\u0027ll be 0.7 plus now that\u0027ll be"},{"Start":"03:58.660 ","End":"04:03.940","Text":"t minus 5 that\u0027s the base of this rectangle right here, times the height."},{"Start":"04:03.940 ","End":"04:07.580","Text":"Well, the height is 0.15."},{"Start":"04:07.580 ","End":"04:09.750","Text":"Now let\u0027s go to the next section,"},{"Start":"04:09.750 ","End":"04:14.500","Text":"where t is between 6 and 7."},{"Start":"04:14.790 ","End":"04:18.175","Text":"Now, in this range right here,"},{"Start":"04:18.175 ","End":"04:19.945","Text":"right between 6 and 7,"},{"Start":"04:19.945 ","End":"04:23.080","Text":"the density function is 0."},{"Start":"04:23.080 ","End":"04:26.905","Text":"That means that the cumulative distribution function"},{"Start":"04:26.905 ","End":"04:30.640","Text":"is a constant and what\u0027s that constant?"},{"Start":"04:30.640 ","End":"04:38.155","Text":"That\u0027ll be 0.6 plus 0.1 plus 0.15."},{"Start":"04:38.155 ","End":"04:44.755","Text":"That\u0027s the area of the last section we\u0027ve calculated and that totals 0.85."},{"Start":"04:44.755 ","End":"04:54.570","Text":"Now, what about the last section right here where t is between 7 and 10?"},{"Start":"04:54.570 ","End":"04:56.625","Text":"C, don\u0027t forget was 10."},{"Start":"04:56.625 ","End":"04:59.400","Text":"Again, let\u0027s pick a point here,"},{"Start":"04:59.400 ","End":"05:10.370","Text":"t and we want to calculate the total area under the density function from 0 to t. Well,"},{"Start":"05:10.370 ","End":"05:14.775","Text":"the total area that\u0027s colored in yellow,"},{"Start":"05:14.775 ","End":"05:24.035","Text":"that\u0027s 0.85 plus now the area of this guy right here."},{"Start":"05:24.035 ","End":"05:29.480","Text":"Well, that\u0027s t minus 7 times the height,"},{"Start":"05:29.480 ","End":"05:37.180","Text":"that\u0027s the base times the height is 0.05."},{"Start":"05:37.180 ","End":"05:42.150","Text":"The last section is where t is greater than 10."},{"Start":"05:42.150 ","End":"05:44.820","Text":"Well, that equals to 1."},{"Start":"05:44.820 ","End":"05:52.730","Text":"This is the function or the cumulative distribution function of X."},{"Start":"05:52.730 ","End":"05:57.350","Text":"In this section, we\u0027re asked to calculate the following probabilities."},{"Start":"05:57.350 ","End":"06:01.010","Text":"Let\u0027s get started. Let\u0027s start with this probability right here."},{"Start":"06:01.010 ","End":"06:06.635","Text":"That\u0027s the probability of x being greater or equal to 4."},{"Start":"06:06.635 ","End":"06:14.315","Text":"Well, that equals to 1 minus the probability of x being less than or equal to 4."},{"Start":"06:14.315 ","End":"06:18.395","Text":"That equals to 1 minus F at 4."},{"Start":"06:18.395 ","End":"06:22.540","Text":"Now 4 is in this range right here so"},{"Start":"06:22.540 ","End":"06:27.940","Text":"we\u0027ll use this expression in order to calculate F at 4."},{"Start":"06:27.940 ","End":"06:31.340","Text":"Now that equals to 1 minus,"},{"Start":"06:31.340 ","End":"06:38.130","Text":"now that\u0027ll be 0.6 plus 4"},{"Start":"06:38.130 ","End":"06:47.945","Text":"minus 3 times 0.05 and that equals to 0.35."},{"Start":"06:47.945 ","End":"06:55.640","Text":"Now, what about the probability of x being greater or equal to minus 2?"},{"Start":"06:55.640 ","End":"07:02.290","Text":"Well, again, that\u0027s 1 minus the probability of x being less"},{"Start":"07:02.290 ","End":"07:09.085","Text":"than or equal to minus 2 or 1 minus F at minus 2."},{"Start":"07:09.085 ","End":"07:12.010","Text":"Well, that equals to 1 minus."},{"Start":"07:12.010 ","End":"07:17.740","Text":"Now, F at minus 2 is in this range right here,"},{"Start":"07:17.740 ","End":"07:20.140","Text":"and F at minus 2 is 0."},{"Start":"07:20.140 ","End":"07:25.315","Text":"That means that this probability equals 1."},{"Start":"07:25.315 ","End":"07:34.890","Text":"Now, what about the probability of x being between 1 and 5?"},{"Start":"07:34.890 ","End":"07:45.325","Text":"Well, that equals to F at 5 minus F at 1."},{"Start":"07:45.325 ","End":"07:48.670","Text":"Now, F at 5, well,"},{"Start":"07:48.670 ","End":"07:51.250","Text":"we can use this expression right here or"},{"Start":"07:51.250 ","End":"07:55.180","Text":"this expression right here because it\u0027s right on the border."},{"Start":"07:55.180 ","End":"07:57.925","Text":"Let\u0027s just use this guy right here."},{"Start":"07:57.925 ","End":"08:04.850","Text":"That\u0027ll be 0.6 plus 5 minus 3"},{"Start":"08:04.850 ","End":"08:13.570","Text":"times 0.05 minus F at 1."},{"Start":"08:13.570 ","End":"08:19.720","Text":"Now, F at 1 that\u0027s right here minus 0.2 times 1."},{"Start":"08:19.720 ","End":"08:26.565","Text":"That\u0027s covered right here and that comes out to 0.5."},{"Start":"08:26.565 ","End":"08:33.090","Text":"These are the probabilities that we were asked to calculate."}],"ID":32249},{"Watched":false,"Name":"Exercise 5","Duration":"5m 17s","ChapterTopicVideoID":12937,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.530","Text":"In this question, we\u0027re given"},{"Start":"00:01.530 ","End":"00:06.000","Text":"the following density function and we\u0027re asked what\u0027s the value of c?"},{"Start":"00:06.000 ","End":"00:09.090","Text":"Well, we can calculate the value of c by"},{"Start":"00:09.090 ","End":"00:14.100","Text":"calculating the area of this triangle right here,"},{"Start":"00:14.100 ","End":"00:15.840","Text":"which is the density function."},{"Start":"00:15.840 ","End":"00:22.350","Text":"Now, the area of this triangle is 1.5 times the base times the height."},{"Start":"00:22.350 ","End":"00:24.945","Text":"Now that equals to 1.5."},{"Start":"00:24.945 ","End":"00:27.615","Text":"Now the base is 10 times 10."},{"Start":"00:27.615 ","End":"00:31.185","Text":"This is the base and the height is c."},{"Start":"00:31.185 ","End":"00:36.285","Text":"Times c. We know that that equals 1 because it\u0027s a density function."},{"Start":"00:36.285 ","End":"00:40.275","Text":"That means that 10c equals 2."},{"Start":"00:40.275 ","End":"00:44.415","Text":"That means that c equals to 0.2."},{"Start":"00:44.415 ","End":"00:47.680","Text":"This here equals to 0.2."},{"Start":"00:48.740 ","End":"00:54.110","Text":"In section b, we\u0027re asked to find a symmetric interval around the value 5,"},{"Start":"00:54.110 ","End":"00:57.880","Text":"in which the probability equals 0.5."},{"Start":"00:57.880 ","End":"01:03.860","Text":"We\u0027re looking basically for 2 points around the value 5, which are symmetrical."},{"Start":"01:03.860 ","End":"01:07.490","Text":"That means that this line right here equals"},{"Start":"01:07.490 ","End":"01:12.770","Text":"this line right here and the length of these lines, let\u0027s call them a."},{"Start":"01:12.770 ","End":"01:15.065","Text":"We\u0027re looking for 2 points,"},{"Start":"01:15.065 ","End":"01:20.015","Text":"5 plus a and 5 minus a,"},{"Start":"01:20.015 ","End":"01:29.555","Text":"where the area under the density function between 5 minus a and 5 plus a,"},{"Start":"01:29.555 ","End":"01:32.605","Text":"but that has to equal to 0.5."},{"Start":"01:32.605 ","End":"01:40.495","Text":"That means that the area colored in yellow is 0.5."},{"Start":"01:40.495 ","End":"01:44.570","Text":"That means then that the area under"},{"Start":"01:44.570 ","End":"01:50.710","Text":"the density function outside of the yellow area also equals 0.5."},{"Start":"01:50.710 ","End":"01:54.860","Text":"That means that the area of this triangle plus"},{"Start":"01:54.860 ","End":"01:58.790","Text":"the area of this triangle has to equal to 0.5."},{"Start":"01:58.790 ","End":"02:05.300","Text":"Now because we have a symmetric interval around the value 5,"},{"Start":"02:05.300 ","End":"02:08.360","Text":"that means that these triangles are identical."},{"Start":"02:08.360 ","End":"02:17.040","Text":"If they\u0027re identical, then the value of the area of each triangle is 0.25."},{"Start":"02:19.160 ","End":"02:21.230","Text":"What are we asked to do?"},{"Start":"02:21.230 ","End":"02:23.390","Text":"We\u0027re asked to find a symmetric interval."},{"Start":"02:23.390 ","End":"02:27.430","Text":"Basically, we\u0027re asked to find the value of a here."},{"Start":"02:27.430 ","End":"02:29.240","Text":"In order to do that,"},{"Start":"02:29.240 ","End":"02:33.110","Text":"let\u0027s just take a look at this smaller triangle right here,"},{"Start":"02:33.110 ","End":"02:36.425","Text":"where it has an area of 0.25."},{"Start":"02:36.425 ","End":"02:46.400","Text":"But if we plug in the value of the base and the height and equate that to 0.25,"},{"Start":"02:46.400 ","End":"02:49.265","Text":"then we\u0027ll have the value of a."},{"Start":"02:49.265 ","End":"02:52.125","Text":"Let\u0027s do that. Now,"},{"Start":"02:52.125 ","End":"02:54.260","Text":"the base is 5 minus a,"},{"Start":"02:54.260 ","End":"02:55.340","Text":"but what about the height?"},{"Start":"02:55.340 ","End":"02:57.380","Text":"Well, in order to calculate the height,"},{"Start":"02:57.380 ","End":"03:02.020","Text":"I need to know the equation for this line."},{"Start":"03:02.020 ","End":"03:08.515","Text":"Now, the equation of a line is y equals mx plus n. Now,"},{"Start":"03:08.515 ","End":"03:11.540","Text":"n equals 0, because that\u0027s the y-intercept and"},{"Start":"03:11.540 ","End":"03:15.320","Text":"this line intercepts the y-axis at point 0."},{"Start":"03:15.320 ","End":"03:17.135","Text":"Now what about the slope?"},{"Start":"03:17.135 ","End":"03:23.280","Text":"Well, the slope is equal to 0.2 minus 0,"},{"Start":"03:23.280 ","End":"03:29.280","Text":"that\u0027s y2 minus y1 divided by x2 minus x1."},{"Start":"03:29.280 ","End":"03:31.800","Text":"That\u0027s 5 minus 0."},{"Start":"03:31.800 ","End":"03:35.085","Text":"That equals to 0.04."},{"Start":"03:35.085 ","End":"03:40.174","Text":"That means that the equation of this line equals y,"},{"Start":"03:40.174 ","End":"03:45.180","Text":"that would equal to 0.04 times x."},{"Start":"03:45.860 ","End":"03:51.440","Text":"Let\u0027s now calculate the area of this guy right here."},{"Start":"03:51.440 ","End":"03:54.425","Text":"Again, that\u0027ll be 1.5."},{"Start":"03:54.425 ","End":"04:00.950","Text":"Now the base is 5 minus a times the height."},{"Start":"04:00.950 ","End":"04:02.870","Text":"Now the height is right here,"},{"Start":"04:02.870 ","End":"04:05.870","Text":"that\u0027s 0.04 times x."},{"Start":"04:05.870 ","End":"04:08.460","Text":"Now x is 5 minus a."},{"Start":"04:08.570 ","End":"04:14.070","Text":"We know that that equals to 0.25."},{"Start":"04:14.070 ","End":"04:25.845","Text":"That means that we\u0027re looking at 5 minus a squared times 0.04. That equals to 0.5."},{"Start":"04:25.845 ","End":"04:30.380","Text":"That means that 5 minus a squared,"},{"Start":"04:30.380 ","End":"04:34.925","Text":"that equals to 12.5."},{"Start":"04:34.925 ","End":"04:38.080","Text":"That means that 5 minus a,"},{"Start":"04:38.080 ","End":"04:47.755","Text":"that equals to 3.54 and a then equals to 1.46."},{"Start":"04:47.755 ","End":"04:51.010","Text":"We\u0027ve calculated the value of a."},{"Start":"04:51.010 ","End":"04:57.000","Text":"The interval that we\u0027re looking at"},{"Start":"04:57.000 ","End":"05:06.465","Text":"is 5 minus 1.46 and 5 plus 1.46."},{"Start":"05:06.465 ","End":"05:12.440","Text":"That would be the interval under which the density function,"},{"Start":"05:12.440 ","End":"05:16.920","Text":"the area of the density function equals 0.5."}],"ID":32250},{"Watched":false,"Name":"Exercise 6","Duration":"6m 31s","ChapterTopicVideoID":12938,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.940","Text":"In this question, we\u0027ll be talking about the waiting time."},{"Start":"00:02.940 ","End":"00:07.590","Text":"The waiting time in minutes of a customer in line at the neighborhood supermarket has"},{"Start":"00:07.590 ","End":"00:11.385","Text":"the following cumulative distribution function F of t"},{"Start":"00:11.385 ","End":"00:16.275","Text":"equals 1 minus e to the power of minus 0.2 times t,"},{"Start":"00:16.275 ","End":"00:19.185","Text":"where e is the irrational number,"},{"Start":"00:19.185 ","End":"00:23.580","Text":"that approximately equals to 2.718."},{"Start":"00:23.580 ","End":"00:29.445","Text":"We\u0027re asked what are the chances that the waiting time will be at least 15 minutes?"},{"Start":"00:29.445 ","End":"00:37.530","Text":"That means that we\u0027re looking for the probability of x being greater or equal to 15."},{"Start":"00:37.530 ","End":"00:40.940","Text":"We already have the cumulative distribution function, that\u0027s this guy right here."},{"Start":"00:40.940 ","End":"00:50.995","Text":"That equals to 1 minus F at 15."},{"Start":"00:50.995 ","End":"00:57.400","Text":"Why 15? Because we\u0027re looking at the units of time in minutes."},{"Start":"00:57.400 ","End":"01:04.100","Text":"All we have to do now is plug in 15 into this equation right here because that\u0027s the CDF."},{"Start":"01:04.100 ","End":"01:14.010","Text":"1 minus 1 minus e to the power of minus 0.2 times 15."},{"Start":"01:14.240 ","End":"01:27.030","Text":"That equals to e to the power of minus 3 and that equals to 0.0498."},{"Start":"01:28.330 ","End":"01:33.680","Text":"In section B, we\u0027re asked what\u0027s the probability of a customer waiting in line a total of"},{"Start":"01:33.680 ","End":"01:38.545","Text":"less than 15 minutes if he\u0027s already waited in line for 10 minutes."},{"Start":"01:38.545 ","End":"01:41.990","Text":"This is a conditional probability, so let\u0027s set it up."},{"Start":"01:41.990 ","End":"01:44.495","Text":"We\u0027re looking for the probability now."},{"Start":"01:44.495 ","End":"01:45.965","Text":"What\u0027s given to us?"},{"Start":"01:45.965 ","End":"01:48.080","Text":"That he\u0027s waited in line for 10 minutes."},{"Start":"01:48.080 ","End":"01:51.710","Text":"That means that x is greater than 10 and we\u0027re looking for"},{"Start":"01:51.710 ","End":"01:57.490","Text":"the probability that he\u0027ll wait for less than 15."},{"Start":"01:57.490 ","End":"01:59.675","Text":"We know how to deal with this."},{"Start":"01:59.675 ","End":"02:01.640","Text":"What\u0027s in the denominator?"},{"Start":"02:01.640 ","End":"02:04.070","Text":"That\u0027s the probability of what\u0027s given,"},{"Start":"02:04.070 ","End":"02:05.885","Text":"x is greater than 10."},{"Start":"02:05.885 ","End":"02:07.175","Text":"In the numerator,"},{"Start":"02:07.175 ","End":"02:08.330","Text":"we have the intersects,"},{"Start":"02:08.330 ","End":"02:14.820","Text":"so that\u0027s the probability of x being between 10 and 15."},{"Start":"02:14.920 ","End":"02:17.210","Text":"We were given that"},{"Start":"02:17.210 ","End":"02:23.420","Text":"the cumulative distribution function equals 1 minus e to the power of minus 0.2"},{"Start":"02:23.420 ","End":"02:31.430","Text":"times t. Let\u0027s just plug in these numbers into this function."},{"Start":"02:31.430 ","End":"02:37.410","Text":"That equals to F at 15 minus F"},{"Start":"02:37.410 ","End":"02:44.290","Text":"at 10 divided by 1 minus F at 10."},{"Start":"02:46.250 ","End":"02:50.630","Text":"As I said, let\u0027s just plug in these numbers into this function."},{"Start":"02:50.630 ","End":"02:59.170","Text":"F at 15, that\u0027s 1 minus e to the power of minus 0.2 times 15,"},{"Start":"02:59.170 ","End":"03:02.280","Text":"minus F of 10."},{"Start":"03:02.280 ","End":"03:07.140","Text":"F of 10 is 1 minus e to the power of minus 0.2 times"},{"Start":"03:07.140 ","End":"03:13.720","Text":"10 divided by 1 minus F of 10."},{"Start":"03:14.120 ","End":"03:22.335","Text":"Now F of 10 is 1 minus e to the power of minus 0.2 times 10."},{"Start":"03:22.335 ","End":"03:25.035","Text":"Let\u0027s simplify that a bit."},{"Start":"03:25.035 ","End":"03:28.170","Text":"That 1 and 1 cancels each other out,"},{"Start":"03:28.170 ","End":"03:33.110","Text":"1 minus 1, and minus here turns this guy to a plus."},{"Start":"03:33.110 ","End":"03:40.280","Text":"That\u0027s e to the power of minus 2 minus e to the power of minus"},{"Start":"03:40.280 ","End":"03:48.220","Text":"3 divided by e to the power of minus 2."},{"Start":"03:48.220 ","End":"03:55.655","Text":"Because 1 cancels this out and then the minus here makes this a plus."},{"Start":"03:55.655 ","End":"04:07.110","Text":"That equals to 0.6321 and that\u0027s the answer to the conditional probability."},{"Start":"04:07.300 ","End":"04:10.580","Text":"In section c, we\u0027re asked what\u0027s the time under"},{"Start":"04:10.580 ","End":"04:13.640","Text":"which 90 percent of the customers have to wait?"},{"Start":"04:13.640 ","End":"04:18.035","Text":"Here we\u0027re looking for percentile, the 90th percentile."},{"Start":"04:18.035 ","End":"04:19.804","Text":"Now if we recall,"},{"Start":"04:19.804 ","End":"04:23.080","Text":"the definition of the percentile,"},{"Start":"04:23.080 ","End":"04:30.320","Text":"is the probability of x being less than or equal to x_0.9,"},{"Start":"04:30.320 ","End":"04:32.160","Text":"that\u0027s the 90th percentile."},{"Start":"04:32.160 ","End":"04:36.980","Text":"The probability of x being less than or equal to some value,"},{"Start":"04:36.980 ","End":"04:41.330","Text":"that would equal 0.9 when we\u0027re talking about the 90th percentile."},{"Start":"04:41.330 ","End":"04:47.135","Text":"Now let\u0027s recall, the cumulative distribution function,"},{"Start":"04:47.135 ","End":"04:56.060","Text":"that equal 1 minus e to the power of minus 0.2 t. What we want is for"},{"Start":"04:56.060 ","End":"04:58.670","Text":"this function to equal"},{"Start":"04:58.670 ","End":"05:05.829","Text":"this probability and then we\u0027ll be able to extract t. Let\u0027s do that."},{"Start":"05:05.829 ","End":"05:11.030","Text":"That would equal to 1 minus e to"},{"Start":"05:11.030 ","End":"05:17.930","Text":"the power of minus 0.2 t and that would equal to 0.9."},{"Start":"05:17.930 ","End":"05:23.650","Text":"Now, let\u0027s just switch things around and that equals to"},{"Start":"05:23.650 ","End":"05:32.040","Text":"e to the power of minus 0.2 t being equal to 0.1."},{"Start":"05:32.040 ","End":"05:34.530","Text":"In order to extract t,"},{"Start":"05:34.530 ","End":"05:41.909","Text":"we\u0027re going to have to take a natural logarithm of both sides."},{"Start":"05:41.980 ","End":"05:48.440","Text":"What we\u0027re doing is we\u0027re taking the natural logarithm of e to the power of minus"},{"Start":"05:48.440 ","End":"05:57.390","Text":"0.2 t and that will equal to the natural logarithm of 0.1."},{"Start":"05:58.130 ","End":"06:04.730","Text":"The natural logarithm of e to the power of something is that something itself."},{"Start":"06:04.730 ","End":"06:08.975","Text":"That would be minus 0.2 t,"},{"Start":"06:08.975 ","End":"06:14.070","Text":"that equals to the natural logarithm of 0.1."},{"Start":"06:14.090 ","End":"06:17.625","Text":"That means that t,"},{"Start":"06:17.625 ","End":"06:19.605","Text":"this t right here,"},{"Start":"06:19.605 ","End":"06:23.490","Text":"which equals to the 90th percentile,"},{"Start":"06:23.490 ","End":"06:28.000","Text":"that equals to 115.13."}],"ID":32251},{"Watched":false,"Name":"Exercise 7","Duration":"7m 32s","ChapterTopicVideoID":30192,"CourseChapterTopicPlaylistID":64526,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.170","Text":"In this question, we\u0027re given that the waiting time in minutes of a customer in line at"},{"Start":"00:04.170 ","End":"00:08.340","Text":"the neighborhood supermarket has the following cumulative distribution function."},{"Start":"00:08.340 ","End":"00:09.885","Text":"That\u0027s F at t,"},{"Start":"00:09.885 ","End":"00:14.460","Text":"and that equals to 1 minus e to the power of minus 0.2t."},{"Start":"00:14.460 ","End":"00:18.495","Text":"We\u0027re asked to draw the cumulative distribution function."},{"Start":"00:18.495 ","End":"00:22.710","Text":"Well, the first thing that we need to know is that when we\u0027re dealing with time,"},{"Start":"00:22.710 ","End":"00:25.920","Text":"then t is always greater or equal to 0."},{"Start":"00:25.920 ","End":"00:31.080","Text":"What happens to the function when t equals 0?"},{"Start":"00:31.080 ","End":"00:35.370","Text":"Well, that\u0027ll be 1 minus e^0."},{"Start":"00:35.370 ","End":"00:37.460","Text":"E to the power of 0 is 1,"},{"Start":"00:37.460 ","End":"00:40.460","Text":"so 1 minus 1, that equals to 0."},{"Start":"00:40.460 ","End":"00:45.590","Text":"Let\u0027s take the first derivative of the function."},{"Start":"00:45.590 ","End":"00:49.280","Text":"Well, that equals to minus e to the power of minus"},{"Start":"00:49.280 ","End":"00:54.620","Text":"0.2t times the internal derivative, that\u0027s minus 0.2."},{"Start":"00:54.620 ","End":"00:58.140","Text":"That is greater than 0."},{"Start":"00:58.140 ","End":"01:02.660","Text":"That means that this function is monotonically increasing."},{"Start":"01:02.660 ","End":"01:05.750","Text":"The other thing and the last thing that we want to"},{"Start":"01:05.750 ","End":"01:09.710","Text":"do is we want to take the limit of this function, well,"},{"Start":"01:09.710 ","End":"01:13.780","Text":"that\u0027ll be limit of F at t,"},{"Start":"01:13.780 ","End":"01:17.065","Text":"where t goes to plus infinity."},{"Start":"01:17.065 ","End":"01:21.010","Text":"Well, that equals to the limit of t going to"},{"Start":"01:21.010 ","End":"01:26.590","Text":"plus infinity of 1 minus e to the power of minus 0.2t."},{"Start":"01:26.590 ","End":"01:29.140","Text":"Well, if t goes to plus infinity,"},{"Start":"01:29.140 ","End":"01:31.870","Text":"then this expression right here goes to 0,"},{"Start":"01:31.870 ","End":"01:34.025","Text":"so the limit equals 1."},{"Start":"01:34.025 ","End":"01:36.660","Text":"I think we have enough now."},{"Start":"01:36.660 ","End":"01:44.700","Text":"If we want to draw the CDF let\u0027s assume here that this is 1."},{"Start":"01:44.700 ","End":"01:48.180","Text":"The function starts at 0,"},{"Start":"01:48.180 ","End":"01:50.536","Text":"and it\u0027s monotonically increasing,"},{"Start":"01:50.536 ","End":"01:53.130","Text":"so it looks something like this."},{"Start":"01:53.130 ","End":"01:58.720","Text":"This is what the CDF looks like."},{"Start":"01:58.870 ","End":"02:02.240","Text":"In this section, we\u0027re asked what are the chances"},{"Start":"02:02.240 ","End":"02:04.820","Text":"that the waiting time will be at least 15 minutes?"},{"Start":"02:04.820 ","End":"02:10.895","Text":"We\u0027re looking at the probability of t being greater or equal to 15."},{"Start":"02:10.895 ","End":"02:14.910","Text":"We\u0027re given F at t,"},{"Start":"02:14.910 ","End":"02:21.410","Text":"and that equal to 1 minus e to the power of minus 0.2t but that equals to"},{"Start":"02:21.410 ","End":"02:29.705","Text":"the probability of x being less than or equal to t. If we take 1 minus this,"},{"Start":"02:29.705 ","End":"02:31.550","Text":"then it will be fine."},{"Start":"02:31.550 ","End":"02:41.100","Text":"That\u0027ll be 1 minus the probability of t being less than 15."},{"Start":"02:41.100 ","End":"02:47.130","Text":"That equals to 1 minus F at 15."},{"Start":"02:47.130 ","End":"02:56.615","Text":"That equals 1 minus e to the power of minus 0.2 times 15."},{"Start":"02:56.615 ","End":"03:02.300","Text":"That equals to e to the power of minus 3,"},{"Start":"03:02.300 ","End":"03:08.245","Text":"which equals to 0.0498."},{"Start":"03:08.245 ","End":"03:14.795","Text":"That\u0027s the probability that the waiting time will be at least 15 minutes."},{"Start":"03:14.795 ","End":"03:17.030","Text":"In this section, we\u0027re asked what\u0027s"},{"Start":"03:17.030 ","End":"03:19.760","Text":"the probability of a customer waiting in line a total of"},{"Start":"03:19.760 ","End":"03:24.950","Text":"less than 15 minutes if he has already waited in line for 10 minutes."},{"Start":"03:24.950 ","End":"03:28.235","Text":"We can see that this is a conditional probability."},{"Start":"03:28.235 ","End":"03:32.060","Text":"Let\u0027s set it up. We\u0027re looking for the probability."},{"Start":"03:32.060 ","End":"03:33.245","Text":"What\u0027s given to us?"},{"Start":"03:33.245 ","End":"03:36.590","Text":"We\u0027re given that the customers waiting in line for 10 minutes."},{"Start":"03:36.590 ","End":"03:38.845","Text":"That means X is greater than 10,"},{"Start":"03:38.845 ","End":"03:43.220","Text":"and we\u0027re looking for the probability that he\u0027ll wait less than 15 minutes,"},{"Start":"03:43.220 ","End":"03:47.000","Text":"so X is less than 15."},{"Start":"03:47.000 ","End":"03:50.000","Text":"We know how to do this."},{"Start":"03:50.000 ","End":"03:51.695","Text":"Well, in the denominator,"},{"Start":"03:51.695 ","End":"03:53.900","Text":"that\u0027s the probability of what\u0027s given,"},{"Start":"03:53.900 ","End":"03:58.285","Text":"that\u0027s the probability of X being greater than 10."},{"Start":"03:58.285 ","End":"04:00.875","Text":"In the numerator, well, that\u0027s the intersect."},{"Start":"04:00.875 ","End":"04:05.855","Text":"That means that we\u0027re looking for the probability of X"},{"Start":"04:05.855 ","End":"04:11.940","Text":"being greater than 10 and less than 15."},{"Start":"04:12.580 ","End":"04:16.695","Text":"That equals to what?"},{"Start":"04:16.695 ","End":"04:21.210","Text":"Well, here, we\u0027re looking at F at"},{"Start":"04:21.210 ","End":"04:30.840","Text":"15 minus F at 10 divided by 1 minus F at 10."},{"Start":"04:30.840 ","End":"04:35.265","Text":"Let\u0027s just plug in the numbers."},{"Start":"04:35.265 ","End":"04:39.865","Text":"Our cumulative distribution function is this."},{"Start":"04:39.865 ","End":"04:48.350","Text":"For 15, that\u0027ll be 1 minus e to the power of minus 0.2 times 15"},{"Start":"04:48.350 ","End":"04:56.360","Text":"minus,1 minus e to the power of"},{"Start":"04:56.360 ","End":"05:05.295","Text":"minus 0.2 times 10 divided by 1 minus F at 10."},{"Start":"05:05.295 ","End":"05:12.260","Text":"Well, that\u0027s 1 minus e to the power of minus 0.2 times 10."},{"Start":"05:12.260 ","End":"05:21.460","Text":"That equals to e to the power of minus 2 minus e to the power of minus 3,"},{"Start":"05:21.460 ","End":"05:25.405","Text":"divided by e to the power of minus 2,"},{"Start":"05:25.405 ","End":"05:30.350","Text":"and that equals to 0.6321."},{"Start":"05:30.500 ","End":"05:36.280","Text":"This is the probability of a person or a customer waiting in line"},{"Start":"05:36.280 ","End":"05:42.680","Text":"a total of less than 15 minutes if he\u0027s already waited in line for 10 minutes."},{"Start":"05:42.870 ","End":"05:46.045","Text":"In this section, we\u0027re asked what\u0027s the time"},{"Start":"05:46.045 ","End":"05:49.075","Text":"under which 90 percent of the customers have to wait?"},{"Start":"05:49.075 ","End":"05:51.460","Text":"Well, here we\u0027re dealing with a percentile,"},{"Start":"05:51.460 ","End":"05:54.125","Text":"and we\u0027re looking for the 90th percentile."},{"Start":"05:54.125 ","End":"06:00.820","Text":"We\u0027re looking for the probability of X being less than or equal to x_ 0.9."},{"Start":"06:00.820 ","End":"06:05.395","Text":"We want this probability to be equal to 0.9."},{"Start":"06:05.395 ","End":"06:10.180","Text":"Our cumulative distribution function was"},{"Start":"06:10.180 ","End":"06:16.300","Text":"1 minus e to the power of minus 0.2 times t. If we take this function,"},{"Start":"06:16.300 ","End":"06:18.610","Text":"we equate it to 0.9,"},{"Start":"06:18.610 ","End":"06:20.260","Text":"we extract t,"},{"Start":"06:20.260 ","End":"06:23.375","Text":"they\u0027ll be a 90th percentile."},{"Start":"06:23.375 ","End":"06:27.510","Text":"Let\u0027s do that. That\u0027ll be 0.9,"},{"Start":"06:27.510 ","End":"06:33.635","Text":"that will be equal to 1 minus e to the power of minus 0.2t."},{"Start":"06:33.635 ","End":"06:36.860","Text":"That means if this is 0.1,"},{"Start":"06:36.860 ","End":"06:42.485","Text":"then that\u0027ll equal to e to the power of minus 0.2t."},{"Start":"06:42.485 ","End":"06:44.900","Text":"Let\u0027s take ln of both sides."},{"Start":"06:44.900 ","End":"06:46.850","Text":"It\u0027ll be ln 0.1."},{"Start":"06:46.850 ","End":"06:52.890","Text":"That will be equal to ln of e to the power of minus 0.2t."},{"Start":"06:53.000 ","End":"07:03.730","Text":"Ln 0.1, and that equals to minus 0.2t times ln of e. Ln of e is 1."},{"Start":"07:04.100 ","End":"07:09.140","Text":"Let\u0027s just divide both sides by minus 0.2."},{"Start":"07:09.140 ","End":"07:18.525","Text":"We\u0027ll get that t will be equal to ln of 0.1 divided by minus 0.2,"},{"Start":"07:18.525 ","End":"07:22.475","Text":"and that equals to 11.51."},{"Start":"07:22.475 ","End":"07:31.470","Text":"11.51, that\u0027s the time in minutes under which 90 percent of the customers have to wait."}],"ID":32252}],"Thumbnail":null,"ID":64526},{"Name":"General Probabilities with Integrals","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"9m 11s","ChapterTopicVideoID":30208,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.645","Text":"In this chapter, we\u0027ll be talking about continuous random variables,"},{"Start":"00:03.645 ","End":"00:06.104","Text":"general probabilities with integrals,"},{"Start":"00:06.104 ","End":"00:09.090","Text":"as opposed to the previous chapter where we talked"},{"Start":"00:09.090 ","End":"00:12.495","Text":"about general probabilities without integrals."},{"Start":"00:12.495 ","End":"00:15.375","Text":"This chapter will deal with"},{"Start":"00:15.375 ","End":"00:20.580","Text":"the probability distribution of continuous random variables such as height,"},{"Start":"00:20.580 ","End":"00:23.010","Text":"weight, time, and so on and so forth."},{"Start":"00:23.010 ","End":"00:25.785","Text":"As opposed to discrete variables,"},{"Start":"00:25.785 ","End":"00:30.695","Text":"continuous variables have an infinite number of values in a given range."},{"Start":"00:30.695 ","End":"00:33.230","Text":"For example, in the range between 1 and 10,"},{"Start":"00:33.230 ","End":"00:35.885","Text":"we have an infinite number of values,"},{"Start":"00:35.885 ","End":"00:39.740","Text":"and the same for the range between 0 and 1."},{"Start":"00:39.740 ","End":"00:43.400","Text":"Even in this range, we have an infinite number of values."},{"Start":"00:43.400 ","End":"00:49.714","Text":"We\u0027ll describe the continuous random variable by a function called the density function."},{"Start":"00:49.714 ","End":"00:57.350","Text":"In general, f of x denotes the density function of any continuous variable."},{"Start":"00:57.350 ","End":"01:03.035","Text":"The area under the density function represents the probability."},{"Start":"01:03.035 ","End":"01:07.190","Text":"A density function must be non-negative and"},{"Start":"01:07.190 ","End":"01:11.540","Text":"the total area under the function is always 1."},{"Start":"01:11.540 ","End":"01:15.995","Text":"Let\u0027s take a look at an example right now."},{"Start":"01:15.995 ","End":"01:22.610","Text":"Let\u0027s define a density function f x as being equal to x squared,"},{"Start":"01:22.610 ","End":"01:31.360","Text":"where x is between 0 and the cube root of 3, and 0 otherwise."},{"Start":"01:31.360 ","End":"01:36.165","Text":"Let\u0027s just take a look at how this looks on a graph,"},{"Start":"01:36.165 ","End":"01:38.690","Text":"and here it is. What do we see here?"},{"Start":"01:38.690 ","End":"01:41.000","Text":"We will see the x-axis right here,"},{"Start":"01:41.000 ","End":"01:44.840","Text":"and the y-axis, that\u0027s f of x."},{"Start":"01:44.840 ","End":"01:47.923","Text":"These are the origins,"},{"Start":"01:47.923 ","End":"01:49.730","Text":"and this point right here,"},{"Start":"01:49.730 ","End":"01:52.535","Text":"that\u0027s the cube root of 3."},{"Start":"01:52.535 ","End":"01:54.080","Text":"How do I know that?"},{"Start":"01:54.080 ","End":"01:58.820","Text":"Well, this is the range of the density function between 0 and the cube root of 3,"},{"Start":"01:58.820 ","End":"02:00.590","Text":"so that\u0027s there."},{"Start":"02:00.590 ","End":"02:02.255","Text":"What else?"},{"Start":"02:02.255 ","End":"02:06.020","Text":"We know that the density function is non-negative."},{"Start":"02:06.020 ","End":"02:07.690","Text":"It could equal to 0,"},{"Start":"02:07.690 ","End":"02:11.520","Text":"but otherwise, it\u0027s always positive,"},{"Start":"02:11.520 ","End":"02:14.405","Text":"above the x-axis. What else?"},{"Start":"02:14.405 ","End":"02:19.430","Text":"We know that the total area underneath the function always equals to 1."},{"Start":"02:19.430 ","End":"02:24.050","Text":"That means that the area between the x-axis and the density function,"},{"Start":"02:24.050 ","End":"02:27.095","Text":"the total area always equals to 1."},{"Start":"02:27.095 ","End":"02:32.450","Text":"Let\u0027s just for argument\u0027s sake say that we want to"},{"Start":"02:32.450 ","End":"02:38.860","Text":"calculate the probability of x being less than or equal to 1."},{"Start":"02:38.860 ","End":"02:42.005","Text":"Well, where is 1 on the x-axis?"},{"Start":"02:42.005 ","End":"02:44.837","Text":"Let\u0027s say it\u0027s right here, that\u0027s 1."},{"Start":"02:44.837 ","End":"02:46.210","Text":"What we want to do,"},{"Start":"02:46.210 ","End":"02:47.960","Text":"we want to calculate the probability,"},{"Start":"02:47.960 ","End":"02:51.335","Text":"that means that we want to calculate the area"},{"Start":"02:51.335 ","End":"02:57.795","Text":"underneath the density function in the range between 0 and 1."},{"Start":"02:57.795 ","End":"03:01.005","Text":"Well, in order to do that, we\u0027ll use the integral,"},{"Start":"03:01.005 ","End":"03:04.860","Text":"that\u0027s the integral between 0 and 1 of the function."},{"Start":"03:04.860 ","End":"03:08.250","Text":"Well, that\u0027s x squared dx."},{"Start":"03:08.250 ","End":"03:16.160","Text":"The integral of x squared is x cubed divided by 3 in the range 0 and 1."},{"Start":"03:16.160 ","End":"03:25.415","Text":"That means that\u0027ll equal to 1 cubed divided by 3 minus 0 cubed divided by 3,"},{"Start":"03:25.415 ","End":"03:29.259","Text":"and that equals to 1/3."},{"Start":"03:29.259 ","End":"03:35.630","Text":"So 1/3 is the probability that x will be less than or equal to 1."},{"Start":"03:35.630 ","End":"03:40.760","Text":"Let\u0027s talk now about the cumulative distribution function."},{"Start":"03:40.760 ","End":"03:44.630","Text":"This is a function for a continuous variable that provides"},{"Start":"03:44.630 ","End":"03:49.505","Text":"the probability of a variable being less than or equal to a given value."},{"Start":"03:49.505 ","End":"03:54.515","Text":"That means that we\u0027re looking at the probability of x being less than or equal to"},{"Start":"03:54.515 ","End":"03:59.610","Text":"t. We can also write it as F of"},{"Start":"03:59.610 ","End":"04:04.580","Text":"t. That\u0027s the cumulative distribution function at point t. This is"},{"Start":"04:04.580 ","End":"04:14.010","Text":"defined as the integral between minus infinity and t of the density function."},{"Start":"04:14.030 ","End":"04:18.080","Text":"Here we have the density function of x,"},{"Start":"04:18.080 ","End":"04:22.054","Text":"and here we have the graph for the density function."},{"Start":"04:22.054 ","End":"04:23.900","Text":"What do we want to do right now,"},{"Start":"04:23.900 ","End":"04:30.127","Text":"let\u0027s just calculate the CDF or the cumulative distribution function for X."},{"Start":"04:30.127 ","End":"04:36.905","Text":"We\u0027re looking at F at point t. Now that will equal to what?"},{"Start":"04:36.905 ","End":"04:38.300","Text":"In order to do that,"},{"Start":"04:38.300 ","End":"04:44.300","Text":"we\u0027re going to have to divide the variable x into 3 ranges."},{"Start":"04:44.300 ","End":"04:49.370","Text":"The first range is where t would be less than 0,"},{"Start":"04:49.370 ","End":"04:55.670","Text":"next range is where t is between 0 and the cube root of 3,"},{"Start":"04:55.670 ","End":"05:01.340","Text":"and then the last range where t is greater than the cube root of 3."},{"Start":"05:01.340 ","End":"05:04.535","Text":"Let\u0027s take a look at this range right here,"},{"Start":"05:04.535 ","End":"05:08.645","Text":"where t is less than 0, t is negative."},{"Start":"05:08.645 ","End":"05:14.090","Text":"What\u0027s the value of the CDF here?"},{"Start":"05:14.090 ","End":"05:17.810","Text":"Well, the CDF is defined as the integral of"},{"Start":"05:17.810 ","End":"05:22.535","Text":"the density function between now minus infinity and 0."},{"Start":"05:22.535 ","End":"05:26.600","Text":"What we\u0027re asked to do here with the integral is calculate"},{"Start":"05:26.600 ","End":"05:30.658","Text":"the area underneath the density function."},{"Start":"05:30.658 ","End":"05:36.905","Text":"But since here there\u0027s no density function or the density function is 0,"},{"Start":"05:36.905 ","End":"05:44.500","Text":"that means that the area underneath this function is also equal to 0."},{"Start":"05:44.500 ","End":"05:52.070","Text":"What about this range right here where X is greater than the cube root of 3?"},{"Start":"05:52.070 ","End":"05:57.845","Text":"For any value above the cube root of 3,"},{"Start":"05:57.845 ","End":"06:05.240","Text":"we basically accumulated all of the area underneath the density function."},{"Start":"06:05.240 ","End":"06:10.280","Text":"That means that for every t above the cube root of 3,"},{"Start":"06:10.280 ","End":"06:17.130","Text":"the CDF equals 1 because we\u0027ve already accumulated all this area,"},{"Start":"06:17.130 ","End":"06:21.695","Text":"all the area underneath the density function."},{"Start":"06:21.695 ","End":"06:27.440","Text":"What about the range in the middle between 0 and the cube root of 3?"},{"Start":"06:27.440 ","End":"06:29.390","Text":"Well, let\u0027s pick a point,"},{"Start":"06:29.390 ","End":"06:31.348","Text":"we\u0027ll call it t,"},{"Start":"06:31.348 ","End":"06:35.030","Text":"and we want to know what the integral of"},{"Start":"06:35.030 ","End":"06:41.670","Text":"the density function between 0 and t. Let\u0027s just write this out."},{"Start":"06:41.670 ","End":"06:46.380","Text":"That\u0027s the integral between 0 and t of our density function,"},{"Start":"06:46.380 ","End":"06:49.685","Text":"where the density function of x squared dx,"},{"Start":"06:49.685 ","End":"06:56.490","Text":"and that equals to x cubed divided by 3 in the range between 0 and"},{"Start":"06:56.490 ","End":"07:04.805","Text":"t. That equals to t cubed divided by 3 minus 0 cubed divided by 3,"},{"Start":"07:04.805 ","End":"07:09.130","Text":"and that equals to t cubed divided by 3."},{"Start":"07:09.130 ","End":"07:15.860","Text":"This is the expression for the cumulative distribution function in this range."},{"Start":"07:15.860 ","End":"07:17.585","Text":"So let\u0027s write this out."},{"Start":"07:17.585 ","End":"07:22.305","Text":"That will be t cubed divided by 3."},{"Start":"07:22.305 ","End":"07:27.935","Text":"In order to show that this is correct, well,"},{"Start":"07:27.935 ","End":"07:31.400","Text":"remember what we did previously when we tried to calculate"},{"Start":"07:31.400 ","End":"07:37.970","Text":"the probability of X being less than or equal to 1 and we received the value of 1/3."},{"Start":"07:37.970 ","End":"07:42.435","Text":"Now 1 is in this range right here."},{"Start":"07:42.435 ","End":"07:46.188","Text":"Let\u0027s just plug in 1 into this expression."},{"Start":"07:46.188 ","End":"07:53.475","Text":"It\u0027ll be 1^3 divided by 3, and that\u0027s 1/3."},{"Start":"07:53.475 ","End":"07:56.363","Text":"We see that we\u0027re okay here."},{"Start":"07:56.363 ","End":"08:04.220","Text":"This then is the cumulative distribution function of the random variable x."},{"Start":"08:05.000 ","End":"08:09.685","Text":"Once we have the cumulative distribution function,"},{"Start":"08:09.685 ","End":"08:17.710","Text":"we can say that the probability of X being greater than t,"},{"Start":"08:17.710 ","End":"08:23.335","Text":"well, that would equal to 1 minus F at t,"},{"Start":"08:23.335 ","End":"08:24.790","Text":"and here it is."},{"Start":"08:24.790 ","End":"08:27.410","Text":"That\u0027s this guy right here."},{"Start":"08:28.160 ","End":"08:32.950","Text":"You can also say that if we want to know the probability where"},{"Start":"08:32.950 ","End":"08:36.970","Text":"X is between a and b, well,"},{"Start":"08:36.970 ","End":"08:43.175","Text":"that equals to F at b minus F at a,"},{"Start":"08:43.175 ","End":"08:47.110","Text":"and again, we see that right here."},{"Start":"08:50.030 ","End":"08:55.055","Text":"Now we have this full spectrum of how to calculate"},{"Start":"08:55.055 ","End":"09:04.745","Text":"the probability of a random variable being less than or equal to a specific value,"},{"Start":"09:04.745 ","End":"09:07.145","Text":"greater than a specific value,"},{"Start":"09:07.145 ","End":"09:11.070","Text":"or between 2 values."}],"ID":32253},{"Watched":false,"Name":"Tutorial 2","Duration":"8m 44s","ChapterTopicVideoID":30209,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"Let\u0027s talk about the expectation of a continuous variable."},{"Start":"00:03.570 ","End":"00:06.930","Text":"Now the expectation is defined as the integral between"},{"Start":"00:06.930 ","End":"00:12.795","Text":"minus infinity and plus infinity of X times the density function of x."},{"Start":"00:12.795 ","End":"00:17.106","Text":"That\u0027s also represented by the Greek letter Mu."},{"Start":"00:17.106 ","End":"00:19.215","Text":"In our case,"},{"Start":"00:19.215 ","End":"00:24.935","Text":"the density function is equal to x squared,"},{"Start":"00:24.935 ","End":"00:33.505","Text":"where x was between 0 and the cube root of 3, and 0 otherwise."},{"Start":"00:33.505 ","End":"00:38.390","Text":"Let\u0027s try to calculate the expectation of X."},{"Start":"00:38.390 ","End":"00:41.405","Text":"That\u0027s the integral."},{"Start":"00:41.405 ","End":"00:44.930","Text":"Instead of from minus infinity to plus infinity,"},{"Start":"00:44.930 ","End":"00:50.420","Text":"that\u0027ll be from 0 to the cube root of 3, well,"},{"Start":"00:50.420 ","End":"01:00.140","Text":"that\u0027s 3^1/3, and that\u0027s the integral of x times the density function."},{"Start":"01:00.140 ","End":"01:02.975","Text":"That\u0027s x squared dx."},{"Start":"01:02.975 ","End":"01:08.450","Text":"That equals to the integral of x cubed dx,"},{"Start":"01:08.450 ","End":"01:13.745","Text":"so between 0 and 3^1/3."},{"Start":"01:13.745 ","End":"01:17.375","Text":"What\u0027s the integral of x cubed?"},{"Start":"01:17.375 ","End":"01:23.060","Text":"That equals to x^4 divided by"},{"Start":"01:23.060 ","End":"01:30.485","Text":"4 in the range 0 and 3^1/3."},{"Start":"01:30.485 ","End":"01:32.539","Text":"Let\u0027s just plug in the numbers."},{"Start":"01:32.539 ","End":"01:39.664","Text":"That\u0027s 3^1/3^4 divided by 4."},{"Start":"01:39.664 ","End":"01:48.685","Text":"That equals to 3^4/3 divided by 4."},{"Start":"01:48.685 ","End":"01:53.545","Text":"That will be the expectation of X."},{"Start":"01:53.545 ","End":"01:58.880","Text":"What about the variance of a continuous random variable?"},{"Start":"01:58.880 ","End":"02:05.540","Text":"The variance is defined as the integral between minus infinity and plus infinity of x"},{"Start":"02:05.540 ","End":"02:12.120","Text":"squared times the density function of x minus the expectation squared or Mu squared,"},{"Start":"02:12.120 ","End":"02:15.830","Text":"and that\u0027s represented by Sigma squared."},{"Start":"02:15.830 ","End":"02:19.460","Text":"Let\u0027s calculate that."},{"Start":"02:19.460 ","End":"02:20.720","Text":"Now in order to do that,"},{"Start":"02:20.720 ","End":"02:27.235","Text":"we need to calculate the expectation of X squared."},{"Start":"02:27.235 ","End":"02:35.965","Text":"That equals to the integral between 0 and 3^1/3 of what?"},{"Start":"02:35.965 ","End":"02:40.060","Text":"Of x squared times the density function of x,"},{"Start":"02:40.060 ","End":"02:43.210","Text":"which is x squared dx."},{"Start":"02:43.210 ","End":"02:53.245","Text":"That equals to the integral between 0 and 3^1/3 of x^4 dx."},{"Start":"02:53.245 ","End":"02:58.570","Text":"That integral equals to x^5 divided by"},{"Start":"02:58.570 ","End":"03:04.131","Text":"5 in the range between 0 and 3^1/3."},{"Start":"03:04.131 ","End":"03:07.625","Text":"Let\u0027s plug in this number,"},{"Start":"03:07.625 ","End":"03:15.335","Text":"and that would be equal to 3^1/3^5 divided by 5."},{"Start":"03:15.335 ","End":"03:22.775","Text":"That equals to 3^5/3 divided by 5."},{"Start":"03:22.775 ","End":"03:29.150","Text":"That equals to the expectation of X squared."},{"Start":"03:29.150 ","End":"03:31.610","Text":"Once we have this,"},{"Start":"03:31.610 ","End":"03:34.760","Text":"we can calculate the variance."},{"Start":"03:34.760 ","End":"03:38.240","Text":"The variance of X, as we said,"},{"Start":"03:38.240 ","End":"03:44.825","Text":"that was the expectation of x squared minus the expectation squared of x."},{"Start":"03:44.825 ","End":"03:50.985","Text":"That equals to the expectation of x squared is right here."},{"Start":"03:50.985 ","End":"04:01.020","Text":"That\u0027s 3^5/3 divided by 5 minus the expectation squared of x,"},{"Start":"04:01.020 ","End":"04:04.365","Text":"where the expectation of x is this expression right here,"},{"Start":"04:04.365 ","End":"04:12.560","Text":"that\u0027s 3^4/3 divided by 4 squared."},{"Start":"04:12.560 ","End":"04:21.140","Text":"That equals, after calculation that comes out to 0.078."},{"Start":"04:21.140 ","End":"04:26.790","Text":"That is the variance of X."},{"Start":"04:26.830 ","End":"04:32.000","Text":"Let\u0027s talk now about the expectation of a function of x."},{"Start":"04:32.000 ","End":"04:36.365","Text":"What we want to do is we want to calculate the expectation of g of x,"},{"Start":"04:36.365 ","End":"04:38.020","Text":"a function of x,"},{"Start":"04:38.020 ","End":"04:43.010","Text":"and that\u0027s defined as the integral between minus infinity and plus"},{"Start":"04:43.010 ","End":"04:48.845","Text":"infinity of g of x times the density function f of x dx."},{"Start":"04:48.845 ","End":"04:53.525","Text":"We\u0027ll use our density function and"},{"Start":"04:53.525 ","End":"04:59.180","Text":"we\u0027ll assume just as an example that g of x equals 1 divided by x."},{"Start":"04:59.180 ","End":"05:05.840","Text":"What we want to do is we want to calculate the expectation of 1 divided by x."},{"Start":"05:05.840 ","End":"05:10.310","Text":"Well, according to our definition, that\u0027s the integral."},{"Start":"05:10.310 ","End":"05:13.400","Text":"Let\u0027s use the density functions range,"},{"Start":"05:13.400 ","End":"05:19.010","Text":"that\u0027s from 0 to 3^1/3 times g of x."},{"Start":"05:19.010 ","End":"05:23.480","Text":"Now g of x is 1 divided by x times f of x."},{"Start":"05:23.480 ","End":"05:26.660","Text":"Well, that\u0027s x squared dx."},{"Start":"05:26.660 ","End":"05:29.975","Text":"That equals to the integral."},{"Start":"05:29.975 ","End":"05:36.110","Text":"Again, the range here is from 0 to 3^1/3."},{"Start":"05:36.110 ","End":"05:40.610","Text":"Now 1 over x divided by x squared, that\u0027s x dx."},{"Start":"05:40.610 ","End":"05:42.770","Text":"That equals to,"},{"Start":"05:42.770 ","End":"05:46.165","Text":"the integral of x is x squared over 2,"},{"Start":"05:46.165 ","End":"05:51.409","Text":"in the range between 0 and 3^1/3."},{"Start":"05:51.409 ","End":"05:54.650","Text":"Let\u0027s plug in the numbers."},{"Start":"05:54.650 ","End":"06:04.490","Text":"That\u0027s 3^1/3 squared divided by 2 minus 0 squared divided by 2."},{"Start":"06:04.490 ","End":"06:11.200","Text":"Well, that equals to 3^2/3 divided by 2."},{"Start":"06:11.200 ","End":"06:13.880","Text":"Or we can write it out like this."},{"Start":"06:13.880 ","End":"06:19.680","Text":"That\u0027s the cube root of 9 divided by 2."},{"Start":"06:19.680 ","End":"06:25.820","Text":"That would be the expectation of the new function g of x,"},{"Start":"06:25.820 ","End":"06:28.980","Text":"which is 1 divided by x."},{"Start":"06:29.380 ","End":"06:32.645","Text":"Let\u0027s talk now about percentiles."},{"Start":"06:32.645 ","End":"06:37.760","Text":"The p percentile is a value and it\u0027s denoted by X_p,"},{"Start":"06:37.760 ","End":"06:42.605","Text":"for which the chances of the variable being under it are p. In other words,"},{"Start":"06:42.605 ","End":"06:46.370","Text":"the probability of x being less than or equal to X_p."},{"Start":"06:46.370 ","End":"06:51.905","Text":"Well, that equals to P. Let\u0027s try to make things a little bit clearer."},{"Start":"06:51.905 ","End":"06:59.450","Text":"Now assume that P here would be equal to 0.25."},{"Start":"06:59.450 ","End":"07:01.160","Text":"What are we doing?"},{"Start":"07:01.160 ","End":"07:06.500","Text":"We\u0027re actually saying, what is the 25th percentile?"},{"Start":"07:06.500 ","End":"07:08.945","Text":"We\u0027re looking for a value,"},{"Start":"07:08.945 ","End":"07:11.075","Text":"let\u0027s right here,"},{"Start":"07:11.075 ","End":"07:15.890","Text":"a value of x, x_0.25,"},{"Start":"07:15.890 ","End":"07:24.515","Text":"where the area under the density function between 0 and this value x_0.25,"},{"Start":"07:24.515 ","End":"07:28.950","Text":"that area here equals to 0.25."},{"Start":"07:29.870 ","End":"07:39.940","Text":"In essence, we\u0027re looking for the probability of x being less than or equal to x_0.25,"},{"Start":"07:39.940 ","End":"07:44.350","Text":"that equals to a quarter, 0.25."},{"Start":"07:44.350 ","End":"07:49.100","Text":"Now this is the cumulative distribution function."},{"Start":"07:49.100 ","End":"07:53.135","Text":"Let\u0027s look at this function right here."},{"Start":"07:53.135 ","End":"08:01.760","Text":"The 25th percentile falls in this range between 0 and the cube root of 3."},{"Start":"08:01.760 ","End":"08:04.805","Text":"We\u0027ll use this expression right here."},{"Start":"08:04.805 ","End":"08:09.600","Text":"We want t cubed divided by 3,"},{"Start":"08:09.600 ","End":"08:13.035","Text":"and we want that to equal 1/4."},{"Start":"08:13.035 ","End":"08:19.845","Text":"That means that t cubed equals 3/4,"},{"Start":"08:19.845 ","End":"08:26.135","Text":"and that means that t would be equal to the cube root of 3/4."},{"Start":"08:26.135 ","End":"08:28.790","Text":"This number right here,"},{"Start":"08:28.790 ","End":"08:32.702","Text":"that\u0027s the 25th percentile."},{"Start":"08:32.702 ","End":"08:39.800","Text":"The area under the density function between 0 and the cube root of 3/4,"},{"Start":"08:39.800 ","End":"08:44.190","Text":"that area would equal 25 percent."}],"ID":32254},{"Watched":false,"Name":"Exercise 1 - Parts a-b","Duration":"6m 23s","ChapterTopicVideoID":30194,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.015","Text":"A density function of a random variable X"},{"Start":"00:03.015 ","End":"00:06.930","Text":"equals 2 over x and is defined from 1 to k and we\u0027re"},{"Start":"00:06.930 ","End":"00:09.285","Text":"asked to calculate the value of K."},{"Start":"00:09.285 ","End":"00:14.280","Text":"The first thing I want to do is to see how this function looks."},{"Start":"00:14.280 ","End":"00:16.230","Text":"Let\u0027s draw this."},{"Start":"00:16.230 ","End":"00:18.780","Text":"This is how it looks."},{"Start":"00:18.780 ","End":"00:21.890","Text":"We see that we have a hyperbola."},{"Start":"00:21.890 ","End":"00:27.470","Text":"Great. Now, the next thing that we want to know is where is this function defined?"},{"Start":"00:27.470 ","End":"00:31.525","Text":"Was defined from 1 to K. That\u0027s the range."},{"Start":"00:31.525 ","End":"00:37.695","Text":"Let\u0027s assume that this is 1 and this is K right here."},{"Start":"00:37.695 ","End":"00:41.060","Text":"We can disregard any other portion of"},{"Start":"00:41.060 ","End":"00:46.000","Text":"this function right outside of this range. Let\u0027s do that."},{"Start":"00:46.000 ","End":"00:52.640","Text":"Great. We\u0027ve basically will be erased all other sections of this function,"},{"Start":"00:52.640 ","End":"00:56.000","Text":"except in the range between 1 and K. We\u0027re"},{"Start":"00:56.000 ","End":"00:59.960","Text":"asked to calculate the value of K. Now, what else do we know?"},{"Start":"00:59.960 ","End":"01:02.195","Text":"We know that in this range,"},{"Start":"01:02.195 ","End":"01:05.960","Text":"this function is defined as a density function."},{"Start":"01:05.960 ","End":"01:11.300","Text":"That means that the area here underneath this graph from 1 to K,"},{"Start":"01:11.300 ","End":"01:12.774","Text":"well that has to equal 1."},{"Start":"01:12.774 ","End":"01:17.720","Text":"Let\u0027s just calculate this area and solve for K."},{"Start":"01:17.720 ","End":"01:26.605","Text":"That means that we\u0027re looking at the integral between 1 and K of this function,"},{"Start":"01:26.605 ","End":"01:29.120","Text":"2 divided by x dx."},{"Start":"01:29.120 ","End":"01:32.015","Text":"Now, we know that has to equal to 1."},{"Start":"01:32.015 ","End":"01:34.700","Text":"Now, what\u0027s this integral?"},{"Start":"01:34.700 ","End":"01:38.130","Text":"The integral at 2 over x?"},{"Start":"01:38.130 ","End":"01:43.565","Text":"Well, that equals to 2 ln x,"},{"Start":"01:43.565 ","End":"01:48.440","Text":"in this range right here from 1 to K. Now,"},{"Start":"01:48.440 ","End":"01:50.360","Text":"that has to equal to 1."},{"Start":"01:50.360 ","End":"01:52.880","Text":"Let\u0027s just plug in the numbers here."},{"Start":"01:52.880 ","End":"02:02.610","Text":"That\u0027ll be 2 ln k minus 2 ln 1 and that has to equal to 1."},{"Start":"02:02.610 ","End":"02:04.620","Text":"Now, ln 1 equals 0,"},{"Start":"02:04.620 ","End":"02:08.190","Text":"so we\u0027re looking at 2 ln k,"},{"Start":"02:08.190 ","End":"02:15.840","Text":"that equals to 1 ln k equals 1/2."},{"Start":"02:15.840 ","End":"02:23.345","Text":"That means that k now equals to e^1/2."},{"Start":"02:23.345 ","End":"02:28.580","Text":"That\u0027s the value of k. In this section,"},{"Start":"02:28.580 ","End":"02:32.300","Text":"we\u0027re asked to calculate the cumulative distribution function of X."},{"Start":"02:32.300 ","End":"02:36.410","Text":"Here is our density function."},{"Start":"02:36.410 ","End":"02:43.580","Text":"We said that the density function equals 2 divided by x,"},{"Start":"02:43.580 ","End":"02:47.210","Text":"where x is between 1 and k,"},{"Start":"02:47.210 ","End":"02:52.535","Text":"and k equal e^0.5."},{"Start":"02:52.535 ","End":"02:56.449","Text":"Now, about the cumulative distribution function,"},{"Start":"02:56.449 ","End":"02:59.315","Text":"the definition of the CDF,"},{"Start":"02:59.315 ","End":"03:05.960","Text":"that\u0027s the probability of x being less than or equal to some value t,"},{"Start":"03:05.960 ","End":"03:09.635","Text":"and that would equal to the integral between"},{"Start":"03:09.635 ","End":"03:18.095","Text":"minus infinity to this point t of the density function d of x."},{"Start":"03:18.095 ","End":"03:21.920","Text":"Now, we can also denote that as big F of"},{"Start":"03:21.920 ","End":"03:29.285","Text":"t. Let\u0027s get started and see if we can calculate the CDF."},{"Start":"03:29.285 ","End":"03:32.975","Text":"Well, in order to do that,"},{"Start":"03:32.975 ","End":"03:37.130","Text":"we\u0027re going to have to split the x-axis into its various ranges,"},{"Start":"03:37.130 ","End":"03:38.419","Text":"where t is less than 1."},{"Start":"03:38.419 ","End":"03:40.890","Text":"T is"},{"Start":"03:44.480 ","End":"03:50.190","Text":"between 1 and e^0.5."},{"Start":"03:50.190 ","End":"03:56.310","Text":"The last range is where t is less than e^0.5."},{"Start":"03:56.310 ","End":"03:57.820","Text":"That\u0027s these ranges right here."},{"Start":"03:57.820 ","End":"03:59.020","Text":"That\u0027s the first range,"},{"Start":"03:59.020 ","End":"04:00.775","Text":"that\u0027s the second range,"},{"Start":"04:00.775 ","End":"04:02.455","Text":"and that\u0027s the third range."},{"Start":"04:02.455 ","End":"04:08.050","Text":"Now, for t less than 1, well,"},{"Start":"04:08.050 ","End":"04:14.215","Text":"the density function isn\u0027t defined in this range right here from minus infinity into 1."},{"Start":"04:14.215 ","End":"04:19.855","Text":"There\u0027s no area that we can calculate under the density function."},{"Start":"04:19.855 ","End":"04:26.279","Text":"That means that the value of the CDF here is 0."},{"Start":"04:26.279 ","End":"04:34.040","Text":"Now let\u0027s take a look at the other side where t is greater than e^0.5."},{"Start":"04:34.040 ","End":"04:39.290","Text":"That\u0027s all the values from this point on wards to plus infinity."},{"Start":"04:39.290 ","End":"04:41.930","Text":"Well, in all that range,"},{"Start":"04:41.930 ","End":"04:48.575","Text":"we\u0027ve already accumulated all the area under the density function."},{"Start":"04:48.575 ","End":"04:53.420","Text":"That means that the CDF is 1."},{"Start":"04:53.420 ","End":"05:00.380","Text":"Now, what happens in this range right here between 1 and e^0.5?"},{"Start":"05:00.380 ","End":"05:03.920","Text":"Well, let\u0027s take a point here,"},{"Start":"05:03.920 ","End":"05:08.195","Text":"call that t. What we want to do is want to"},{"Start":"05:08.195 ","End":"05:13.595","Text":"calculate this area right here between minus infinity to t,"},{"Start":"05:13.595 ","End":"05:17.540","Text":"or in order to simplify things between 1 and"},{"Start":"05:17.540 ","End":"05:25.160","Text":"t. So we\u0027re looking at calculating the area colored in yellow."},{"Start":"05:25.160 ","End":"05:31.650","Text":"Well, that would be the integral from 1 till"},{"Start":"05:31.650 ","End":"05:39.965","Text":"the point t of the density function is 2 over x 2 divided by x dx."},{"Start":"05:39.965 ","End":"05:47.130","Text":"Now that equals to 2 ln x."},{"Start":"05:47.130 ","End":"05:52.590","Text":"In the range between 1 and t. Now that"},{"Start":"05:52.590 ","End":"06:00.675","Text":"equals to 2 ln t minus 2 ln 1."},{"Start":"06:00.675 ","End":"06:04.920","Text":"2 ln 1 equals 0 because ln 1 equals 0."},{"Start":"06:04.920 ","End":"06:09.590","Text":"That means that the cumulative distribution function"},{"Start":"06:09.590 ","End":"06:13.100","Text":"is only 2 ln t. Let\u0027s write that down here."},{"Start":"06:13.100 ","End":"06:17.520","Text":"Will be 2 times ln t. This"},{"Start":"06:17.520 ","End":"06:23.430","Text":"then right here would be the cumulative distribution function of x."}],"ID":32255},{"Watched":false,"Name":"Exercise 1 - Parts c-e","Duration":"3m 54s","ChapterTopicVideoID":30195,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"This section we\u0027re asked to calculate the chances that X is at least 1."},{"Start":"00:04.990 ","End":"00:12.300","Text":"That means that we\u0027re looking for the probability of X being greater than 1.5."},{"Start":"00:12.300 ","End":"00:20.485","Text":"Now, that also equals to 1 minus the probability of X being less than or equal to 1.5,"},{"Start":"00:20.485 ","End":"00:27.405","Text":"and that is 1 minus F, at 1.5."},{"Start":"00:27.405 ","End":"00:34.832","Text":"Now, 1.5 is in this range right here between 1 and e to the power of f,"},{"Start":"00:34.832 ","End":"00:39.710","Text":"so that means that this is equal to 1"},{"Start":"00:39.710 ","End":"00:46.425","Text":"minus 2 times log of 1.5,"},{"Start":"00:46.425 ","End":"00:53.230","Text":"and calculating that, that turns out to be 0.189."},{"Start":"00:53.830 ","End":"00:56.660","Text":"In this section, we\u0027re asked to calculate"},{"Start":"00:56.660 ","End":"01:00.395","Text":"the bottom 10th percentile of the probability distribution."},{"Start":"01:00.395 ","End":"01:07.880","Text":"That means that we\u0027re looking for the probability that X is less than x_0.1,"},{"Start":"01:07.880 ","End":"01:11.890","Text":"we want that probability to be equal to 0.1,"},{"Start":"01:11.890 ","End":"01:13.310","Text":"that\u0027s the 10th percentile,"},{"Start":"01:13.310 ","End":"01:15.005","Text":"the bottom 10th percentile."},{"Start":"01:15.005 ","End":"01:21.270","Text":"Well, this equals to f at x_0.1,"},{"Start":"01:21.270 ","End":"01:23.695","Text":"we want that to be equal to 0.1."},{"Start":"01:23.695 ","End":"01:29.930","Text":"That means that we\u0027ll take this expression of the cumulative distribution function,"},{"Start":"01:29.930 ","End":"01:35.230","Text":"and it will equate that to 0.1 to extract"},{"Start":"01:35.230 ","End":"01:41.190","Text":"t which in our case is x_0.1, so let\u0027s do that."},{"Start":"01:41.190 ","End":"01:48.075","Text":"That\u0027ll be 2 times ln x_0.1,"},{"Start":"01:48.075 ","End":"01:50.505","Text":"and that has to equal to 0.1,"},{"Start":"01:50.505 ","End":"01:54.710","Text":"that means that ln x_0.1,"},{"Start":"01:54.710 ","End":"01:57.290","Text":"that has to equal to 0.05."},{"Start":"01:57.290 ","End":"02:01.970","Text":"Now, let\u0027s take the inverse function of ln on both sides,"},{"Start":"02:01.970 ","End":"02:05.585","Text":"and they\u0027ll be x_0.1,"},{"Start":"02:05.585 ","End":"02:15.900","Text":"that will equal to e^0.05 and that equals to 1.051."},{"Start":"02:16.390 ","End":"02:20.645","Text":"In this section, we\u0027re asked what\u0027s the expectation of X?"},{"Start":"02:20.645 ","End":"02:26.540","Text":"Well, the expectation of X is defined as the integral from"},{"Start":"02:26.540 ","End":"02:33.245","Text":"minus infinity to plus infinity of the density function times X."},{"Start":"02:33.245 ","End":"02:36.409","Text":"Now, in our case,"},{"Start":"02:36.409 ","End":"02:41.855","Text":"that\u0027s the integral not for minus infinity to plus infinity,"},{"Start":"02:41.855 ","End":"02:48.440","Text":"but the range was from 1 to e^0.5,"},{"Start":"02:48.440 ","End":"02:54.113","Text":"of 2 divided by x times x dx,"},{"Start":"02:54.113 ","End":"02:57.600","Text":"so these things cancel out."},{"Start":"02:57.600 ","End":"03:05.445","Text":"We\u0027ll have the integral between 1 and e^0.5 of 2dx."},{"Start":"03:05.445 ","End":"03:10.650","Text":"Now that equals to 2x,"},{"Start":"03:10.650 ","End":"03:16.250","Text":"in the range between 1 and e^0.5."},{"Start":"03:16.250 ","End":"03:25.430","Text":"Now that equals to 2 times e^0.5 minus 2,"},{"Start":"03:25.430 ","End":"03:33.935","Text":"and that equals to 2 times e^0.5 minus 1."},{"Start":"03:33.935 ","End":"03:37.070","Text":"If we want a numerical value for this,"},{"Start":"03:37.070 ","End":"03:41.195","Text":"that equals to 1.297."},{"Start":"03:41.195 ","End":"03:49.295","Text":"Now, all we have to check is whether this falls into the range between 1 and e^0.5,"},{"Start":"03:49.295 ","End":"03:54.030","Text":"and it does, so this is the expectation of X."}],"ID":32256},{"Watched":false,"Name":"Exercise 2 - Parts a-b","Duration":"9m 21s","ChapterTopicVideoID":30197,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.105","Text":"In this question, we\u0027re given the following density function."},{"Start":"00:03.105 ","End":"00:05.880","Text":"AX squared times 10 minus X,"},{"Start":"00:05.880 ","End":"00:08.280","Text":"where X is defined between 0 and 10,"},{"Start":"00:08.280 ","End":"00:10.695","Text":"and where A is a positive constant."},{"Start":"00:10.695 ","End":"00:13.980","Text":"We\u0027re asked to calculate A."},{"Start":"00:13.980 ","End":"00:19.020","Text":"First of all, let\u0027s see how f of x or the density function behaves."},{"Start":"00:19.020 ","End":"00:21.660","Text":"When x equals 0,"},{"Start":"00:21.660 ","End":"00:24.945","Text":"the density function equals 0."},{"Start":"00:24.945 ","End":"00:27.750","Text":"It\u0027ll be here, that\u0027ll be 0."},{"Start":"00:27.750 ","End":"00:30.600","Text":"What happens when x equals 10?"},{"Start":"00:30.600 ","End":"00:33.940","Text":"Again, it equals 0."},{"Start":"00:33.940 ","End":"00:36.710","Text":"What else do we know?"},{"Start":"00:36.710 ","End":"00:38.540","Text":"Since this is a density function,"},{"Start":"00:38.540 ","End":"00:41.665","Text":"then the function has to be positive,"},{"Start":"00:41.665 ","End":"00:42.949","Text":"so it looks something."},{"Start":"00:42.949 ","End":"00:45.860","Text":"It really doesn\u0027t matter what the shape of the function is,"},{"Start":"00:45.860 ","End":"00:48.065","Text":"as long as we know that it is positive,"},{"Start":"00:48.065 ","End":"00:49.280","Text":"and what else do we know?"},{"Start":"00:49.280 ","End":"00:51.110","Text":"We know that the density function,"},{"Start":"00:51.110 ","End":"00:53.584","Text":"the area under the density function,"},{"Start":"00:53.584 ","End":"00:56.345","Text":"that has to equal to 1."},{"Start":"00:56.345 ","End":"01:00.645","Text":"Having said all that, let\u0027s calculate A."},{"Start":"01:00.645 ","End":"01:05.165","Text":"The way to do that is to take the integral between 0 and 10,"},{"Start":"01:05.165 ","End":"01:09.800","Text":"that\u0027s where the density function is defined of the density function,"},{"Start":"01:09.800 ","End":"01:16.000","Text":"that\u0027ll be AX squared times 10 minus x dx,"},{"Start":"01:16.000 ","End":"01:17.980","Text":"and that has to equal to 1."},{"Start":"01:17.980 ","End":"01:20.904","Text":"Let\u0027s just simplify this out."},{"Start":"01:20.904 ","End":"01:22.030","Text":"That\u0027ll be the integral"},{"Start":"01:22.030 ","End":"01:27.189","Text":"between 0 and 10 of A times"},{"Start":"01:27.189 ","End":"01:34.330","Text":"10x squared minus Ax cubed,"},{"Start":"01:34.330 ","End":"01:38.335","Text":"dx, and that has to equal to 1."},{"Start":"01:38.335 ","End":"01:41.220","Text":"Let\u0027s do the integral."},{"Start":"01:41.220 ","End":"01:43.850","Text":"That\u0027ll be A times,"},{"Start":"01:43.850 ","End":"01:46.360","Text":"the integral of 10x squared,"},{"Start":"01:46.360 ","End":"01:49.675","Text":"that\u0027ll be 10x cubed divided by 3,"},{"Start":"01:49.675 ","End":"01:55.325","Text":"minus x^4 divided by 4,"},{"Start":"01:55.325 ","End":"01:59.220","Text":"between 0 and 10."},{"Start":"01:59.220 ","End":"02:01.740","Text":"That has to equal to 1."},{"Start":"02:01.740 ","End":"02:06.480","Text":"Let\u0027s just plug in the numbers here."},{"Start":"02:06.480 ","End":"02:16.740","Text":"We\u0027ll have A times 10^4 divided"},{"Start":"02:16.740 ","End":"02:27.940","Text":"by 3 minus 10^4 divided by 4,"},{"Start":"02:28.360 ","End":"02:31.580","Text":"minus when x is 0,"},{"Start":"02:31.580 ","End":"02:34.265","Text":"well then all this section here is 0."},{"Start":"02:34.265 ","End":"02:37.325","Text":"That has to equal to 1."},{"Start":"02:37.325 ","End":"02:39.890","Text":"Let\u0027s take a common denominator,"},{"Start":"02:39.890 ","End":"02:44.055","Text":"A times 4 times"},{"Start":"02:44.055 ","End":"02:51.690","Text":"10^4 minus 3 times 10^4 divided by 12,"},{"Start":"02:51.690 ","End":"02:54.165","Text":"and that has to equal to 1."},{"Start":"02:54.165 ","End":"03:04.815","Text":"Therefore, A equals 12 divided by 10^4,"},{"Start":"03:04.815 ","End":"03:11.480","Text":"and that equals to 0.0012."},{"Start":"03:11.480 ","End":"03:15.455","Text":"In this section, we\u0027re asked to calculate the probability of x greater than 5,"},{"Start":"03:15.455 ","End":"03:18.095","Text":"given that x is greater than 2."},{"Start":"03:18.095 ","End":"03:21.770","Text":"This is a conditional probability and we know how to solve that."},{"Start":"03:21.770 ","End":"03:28.520","Text":"That\u0027s the probability of the intersect x being greater than 5,"},{"Start":"03:28.520 ","End":"03:31.625","Text":"intersect x being greater than 2,"},{"Start":"03:31.625 ","End":"03:36.575","Text":"divided by the probability of what\u0027s given,"},{"Start":"03:36.575 ","End":"03:38.635","Text":"x being greater than 2."},{"Start":"03:38.635 ","End":"03:42.010","Text":"That equals 2."},{"Start":"03:42.010 ","End":"03:44.495","Text":"What\u0027s the intersect here?"},{"Start":"03:44.495 ","End":"03:49.985","Text":"Let\u0027s take a look at the intersect of x being greater than 5 and x being greater than 2."},{"Start":"03:49.985 ","End":"03:55.350","Text":"Well, if this is x and here\u0027s 2,"},{"Start":"03:55.350 ","End":"03:59.705","Text":"so x greater than 2 is all this area right here."},{"Start":"03:59.705 ","End":"04:06.610","Text":"Here\u0027s 5, x greater than 5,"},{"Start":"04:06.610 ","End":"04:09.860","Text":"that means that we have all the values of the x above 5."},{"Start":"04:09.860 ","End":"04:11.329","Text":"What\u0027s the intersect?"},{"Start":"04:11.329 ","End":"04:16.115","Text":"The intersect is this guy right here. What does that mean?"},{"Start":"04:16.115 ","End":"04:22.235","Text":"That this intersect, is just x greater than 5."},{"Start":"04:22.235 ","End":"04:25.230","Text":"Let\u0027s write that out."},{"Start":"04:25.250 ","End":"04:30.350","Text":"That equals to the probability of x being greater than"},{"Start":"04:30.350 ","End":"04:36.125","Text":"5 divided by probability of x being greater than 2."},{"Start":"04:36.125 ","End":"04:38.990","Text":"That\u0027s this conditional probability."},{"Start":"04:38.990 ","End":"04:45.325","Text":"That equals to 1 minus the probability of x less than 5,"},{"Start":"04:45.325 ","End":"04:50.778","Text":"divided by 1 minus the probability of x being less than 2,"},{"Start":"04:50.778 ","End":"04:55.125","Text":"and that equals to 1 minus F at 5,"},{"Start":"04:55.125 ","End":"04:59.138","Text":"divided by 1 minus F at 2."},{"Start":"04:59.138 ","End":"05:00.920","Text":"What\u0027s F right here?"},{"Start":"05:00.920 ","End":"05:06.255","Text":"That\u0027s the cumulative distribution function of x."},{"Start":"05:06.255 ","End":"05:10.950","Text":"We have to build that in order to calculate these guys."},{"Start":"05:10.950 ","End":"05:12.831","Text":"Let\u0027s do that."},{"Start":"05:12.831 ","End":"05:15.075","Text":"In order to do that,"},{"Start":"05:15.075 ","End":"05:19.250","Text":"I\u0027m going to have to split x up into ranges."},{"Start":"05:19.250 ","End":"05:21.380","Text":"What are the ranges for?"},{"Start":"05:21.380 ","End":"05:29.825","Text":"First of all, the first range is T being less than 0."},{"Start":"05:29.825 ","End":"05:33.320","Text":"That\u0027s the range where x is negative."},{"Start":"05:33.320 ","End":"05:39.830","Text":"The next range is where t is between 0 and 10."},{"Start":"05:39.830 ","End":"05:43.595","Text":"That\u0027s this range where the density function is defined,"},{"Start":"05:43.595 ","End":"05:49.420","Text":"and the last range is where t is greater than 10, right here."},{"Start":"05:49.420 ","End":"05:52.220","Text":"When t is negative,"},{"Start":"05:52.220 ","End":"05:54.770","Text":"the density function hasn\u0027t been defined there,"},{"Start":"05:54.770 ","End":"05:58.180","Text":"so that means that we have a 0."},{"Start":"05:58.180 ","End":"06:05.420","Text":"We haven\u0027t accumulated any area underneath the density function from minus infinity to 0."},{"Start":"06:05.420 ","End":"06:09.440","Text":"Let\u0027s take a look at this range right here,"},{"Start":"06:09.440 ","End":"06:11.285","Text":"where t is greater than 10."},{"Start":"06:11.285 ","End":"06:13.160","Text":"Well, when t is greater than 10,"},{"Start":"06:13.160 ","End":"06:18.740","Text":"we\u0027ve actually accumulated all of the area underneath the density function,"},{"Start":"06:18.740 ","End":"06:23.840","Text":"so that means that the cumulative distribution function is 1."},{"Start":"06:23.840 ","End":"06:27.545","Text":"What happens here in this range right here."},{"Start":"06:27.545 ","End":"06:31.490","Text":"Now, in order to calculate this expression,"},{"Start":"06:31.490 ","End":"06:34.205","Text":"we\u0027re going to have to pick a point right here,"},{"Start":"06:34.205 ","End":"06:38.765","Text":"call it t. What we want to do,"},{"Start":"06:38.765 ","End":"06:46.985","Text":"is we want to calculate the area under this function from 0 to t, so let\u0027s do that."},{"Start":"06:46.985 ","End":"06:54.410","Text":"That\u0027ll be the integral from 0 to t of the density function."},{"Start":"06:54.410 ","End":"06:59.300","Text":"Now, the density function is 0.0012."},{"Start":"06:59.300 ","End":"07:03.350","Text":"That\u0027s the A that we calculated in the last section,"},{"Start":"07:03.350 ","End":"07:05.230","Text":"times x squared,"},{"Start":"07:05.230 ","End":"07:09.960","Text":"times 10 minus x dx."},{"Start":"07:09.960 ","End":"07:12.920","Text":"Now, that equals,"},{"Start":"07:12.920 ","End":"07:15.380","Text":"let\u0027s just multiply things in here."},{"Start":"07:15.380 ","End":"07:23.525","Text":"That\u0027s the integral from 0 to t of 0.012x squared,"},{"Start":"07:23.525 ","End":"07:30.665","Text":"minus 0.0012x cubed dx."},{"Start":"07:30.665 ","End":"07:35.450","Text":"Now, let\u0027s just do the integrals."},{"Start":"07:35.450 ","End":"07:41.854","Text":"That will be 0.012x cubed"},{"Start":"07:41.854 ","End":"07:49.905","Text":"divided by 3 minus 0.0012x^4 divided by 4,"},{"Start":"07:49.905 ","End":"07:54.265","Text":"between 0 and t. Now,"},{"Start":"07:54.265 ","End":"07:56.960","Text":"let\u0027s just plug in the numbers here."},{"Start":"07:56.960 ","End":"08:03.695","Text":"That\u0027ll be 0.012t cubed divided by 3"},{"Start":"08:03.695 ","End":"08:11.880","Text":"minus 0.0012t^4 divided by 4,"},{"Start":"08:11.880 ","End":"08:14.940","Text":"all this minus, where x is 0."},{"Start":"08:14.940 ","End":"08:18.984","Text":"Where x is 0, both these expressions are 0s,"},{"Start":"08:18.984 ","End":"08:20.735","Text":"so this is what we have left,"},{"Start":"08:20.735 ","End":"08:27.240","Text":"and this is the expression that has to go in here."},{"Start":"08:28.400 ","End":"08:32.135","Text":"Getting back to our conditional probability,"},{"Start":"08:32.135 ","End":"08:37.910","Text":"now that we\u0027ve calculated the expression for the cumulative distribution function,"},{"Start":"08:37.910 ","End":"08:42.520","Text":"we can plug in 5 and 2 into this expression,"},{"Start":"08:42.520 ","End":"08:45.875","Text":"and then we can calculate this conditional probability."},{"Start":"08:45.875 ","End":"08:51.785","Text":"Let\u0027s do that. That will be 1 minus F at 5,"},{"Start":"08:51.785 ","End":"08:55.820","Text":"divided by 1 minus F at 2,"},{"Start":"08:55.820 ","End":"09:07.200","Text":"and that would equal to 1 minus 0.3125 divided by 1 minus 0.0272,"},{"Start":"09:08.590 ","End":"09:11.820","Text":"and that equals to"},{"Start":"09:11.950 ","End":"09:21.870","Text":"0.7067. so this is the conditional probability that we were looking for right here."}],"ID":32257},{"Watched":false,"Name":"Exercise 2 - Part c","Duration":"4m 24s","ChapterTopicVideoID":30196,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.515","Text":"In this section we\u0027re asked what are the expectation variance of X."},{"Start":"00:04.515 ","End":"00:09.360","Text":"The expectation of x, that equals,"},{"Start":"00:09.360 ","End":"00:14.460","Text":"that\u0027s defined as the integral from minus infinity to plus infinity"},{"Start":"00:14.460 ","End":"00:20.535","Text":"of x times its density function f x dx."},{"Start":"00:20.535 ","End":"00:22.110","Text":"Now in our case,"},{"Start":"00:22.110 ","End":"00:29.310","Text":"that\u0027s the integral from 0-10 of x times the density function."},{"Start":"00:29.310 ","End":"00:39.060","Text":"Our density function is 0.0012 x squared times 10 minus x dx."},{"Start":"00:39.280 ","End":"00:44.680","Text":"That equals to the integral from 0-10,"},{"Start":"00:44.680 ","End":"00:51.245","Text":"of 0.012 x cubed"},{"Start":"00:51.245 ","End":"00:58.930","Text":"minus 0.0012 x to the fourth dx."},{"Start":"00:58.930 ","End":"01:01.220","Text":"That\u0027s this guy right here."},{"Start":"01:01.220 ","End":"01:05.030","Text":"Now, that equals, let\u0027s do the integral."},{"Start":"01:05.030 ","End":"01:12.590","Text":"That equals to 0.012 times x to the 4 divided by"},{"Start":"01:12.590 ","End":"01:22.225","Text":"4 minus 0.0012 x to the 5 divided by 5,"},{"Start":"01:22.225 ","End":"01:25.700","Text":"in the range between 0 and 10."},{"Start":"01:25.720 ","End":"01:32.660","Text":"Now, that comes out to 120 divided by"},{"Start":"01:32.660 ","End":"01:38.960","Text":"4 minus 120 divided by 5."},{"Start":"01:38.960 ","End":"01:42.420","Text":"That\u0027s when we plug in 10 in here,"},{"Start":"01:42.420 ","End":"01:43.890","Text":"and when we plug in 0,"},{"Start":"01:43.890 ","End":"01:45.510","Text":"well this expression equals 0."},{"Start":"01:45.510 ","End":"01:57.270","Text":"This is what we have, and this turns out to be 6 and thus the expectation of X."},{"Start":"01:57.270 ","End":"02:00.035","Text":"Now let\u0027s take a look at the variance of X."},{"Start":"02:00.035 ","End":"02:04.970","Text":"Now the variance of X is defined as the integral from minus infinity to plus"},{"Start":"02:04.970 ","End":"02:11.310","Text":"infinity of x squared times the density function of x,"},{"Start":"02:11.310 ","End":"02:14.535","Text":"minus the expectation squared."},{"Start":"02:14.535 ","End":"02:17.870","Text":"That\u0027s Mu squared."},{"Start":"02:17.870 ","End":"02:19.970","Text":"All that equals Sigma squared,"},{"Start":"02:19.970 ","End":"02:21.650","Text":"and that\u0027s the variance."},{"Start":"02:21.650 ","End":"02:25.220","Text":"We\u0027ve already calculated the expectation."},{"Start":"02:25.220 ","End":"02:30.920","Text":"That\u0027s 6, so let\u0027s just concentrate on this integral right now."},{"Start":"02:32.480 ","End":"02:39.515","Text":"The integral now, instead of minus infinity to plus infinity, that\u0027s from 0-10."},{"Start":"02:39.515 ","End":"02:45.050","Text":"But this integral right here would be x squared times the density function,"},{"Start":"02:45.050 ","End":"02:54.425","Text":"of density function is 0.0012 x squared times 10 minus x dx."},{"Start":"02:54.425 ","End":"03:02.990","Text":"Now that equals to the integral from 0-10 of"},{"Start":"03:02.990 ","End":"03:08.090","Text":"0.012 x to the fourth"},{"Start":"03:08.090 ","End":"03:16.330","Text":"minus 0.0012 x to the fifth dx."},{"Start":"03:16.330 ","End":"03:24.610","Text":"So that equals to 0.012 x to the fifth divided by"},{"Start":"03:24.610 ","End":"03:31.495","Text":"5 minus 0.0012 x"},{"Start":"03:31.495 ","End":"03:35.705","Text":"to the sixth divided by 6,"},{"Start":"03:35.705 ","End":"03:39.040","Text":"in the range between 0 and 10."},{"Start":"03:39.040 ","End":"03:43.450","Text":"Now, that equals to 40."},{"Start":"03:43.450 ","End":"03:52.270","Text":"Now, that is the expectation of x squared. We\u0027re not done yet."},{"Start":"03:52.270 ","End":"03:54.970","Text":"We still have to calculate the variance."},{"Start":"03:54.970 ","End":"03:57.060","Text":"Now what is that?"},{"Start":"03:57.060 ","End":"04:05.510","Text":"Now, the variance is the expectation of x squared minus the expectation squared of x."},{"Start":"04:05.510 ","End":"04:07.775","Text":"That means that we have 40."},{"Start":"04:07.775 ","End":"04:13.270","Text":"We have 40 minus 6 squared."},{"Start":"04:13.270 ","End":"04:17.955","Text":"That equals to 40 minus 36,"},{"Start":"04:17.955 ","End":"04:24.370","Text":"and therefore the variance of X equals 4."}],"ID":32258},{"Watched":false,"Name":"Exercise 3 - Parts a-b","Duration":"6m 34s","ChapterTopicVideoID":30198,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"In this question, we\u0027re given that the density function of"},{"Start":"00:02.730 ","End":"00:08.130","Text":"a continuous random variable X is 0.5 times e^2x,"},{"Start":"00:08.130 ","End":"00:10.817","Text":"where X is between minus infinity and"},{"Start":"00:10.817 ","End":"00:15.090","Text":"ln c. We\u0027re asked to calculate the value of c. Now,"},{"Start":"00:15.090 ","End":"00:19.260","Text":"before we do that, let\u0027s just see what this function looks like."},{"Start":"00:19.260 ","End":"00:20.610","Text":"In order to do that,"},{"Start":"00:20.610 ","End":"00:23.295","Text":"it\u0027s enough just to take the first derivative."},{"Start":"00:23.295 ","End":"00:30.950","Text":"The first derivative of this function would be 0.5e^2x times the internal derivative,"},{"Start":"00:30.950 ","End":"00:34.235","Text":"that\u0027s 2, that equals to e^2x."},{"Start":"00:34.235 ","End":"00:37.945","Text":"We can see that this is greater than 0."},{"Start":"00:37.945 ","End":"00:43.159","Text":"That means that the density function is monotonically increasing."},{"Start":"00:43.159 ","End":"00:47.755","Text":"Now, the function will look something like this then."},{"Start":"00:47.755 ","End":"00:52.895","Text":"Where on 1 hand this goes to minus infinity,"},{"Start":"00:52.895 ","End":"00:55.010","Text":"but at this side,"},{"Start":"00:55.010 ","End":"01:00.710","Text":"it stops at the value on c. What else do we know?"},{"Start":"01:00.710 ","End":"01:04.850","Text":"Well, we know that the area under the density function,"},{"Start":"01:04.850 ","End":"01:07.730","Text":"that equals to 1."},{"Start":"01:07.730 ","End":"01:10.940","Text":"That will help us to extract c. Now,"},{"Start":"01:10.940 ","End":"01:13.100","Text":"how will we do that?"},{"Start":"01:13.100 ","End":"01:18.655","Text":"Well, let\u0027s first take the integral from minus infinity till"},{"Start":"01:18.655 ","End":"01:26.475","Text":"ln c of the density function 0.5e^2x dx,"},{"Start":"01:26.475 ","End":"01:29.410","Text":"and that has to equal to 1."},{"Start":"01:30.170 ","End":"01:33.330","Text":"Let\u0027s just take the integral here."},{"Start":"01:33.330 ","End":"01:38.415","Text":"That\u0027ll be 0.5e^2x,"},{"Start":"01:38.415 ","End":"01:41.550","Text":"but now we divide by 2."},{"Start":"01:41.550 ","End":"01:50.230","Text":"Where x is between minus infinity and ln c. That has to equal to 1."},{"Start":"01:50.230 ","End":"01:52.870","Text":"Now, let\u0027s just plug in the numbers."},{"Start":"01:52.870 ","End":"02:00.090","Text":"We\u0027ll have 0.5e^2 times ln c divided by"},{"Start":"02:00.090 ","End":"02:08.020","Text":"2 minus 0.5e^2 times minus infinity divided by 2,"},{"Start":"02:08.020 ","End":"02:10.535","Text":"and that has to equal to 1."},{"Start":"02:10.535 ","End":"02:14.110","Text":"Now, let\u0027s take a look at this expression right here."},{"Start":"02:14.110 ","End":"02:18.385","Text":"Specifically, e^2 times minus infinity."},{"Start":"02:18.385 ","End":"02:24.640","Text":"Well, that equals to 1 over e^2 times infinity."},{"Start":"02:24.640 ","End":"02:27.365","Text":"Now, that equals 0."},{"Start":"02:27.365 ","End":"02:30.435","Text":"We can disregard this,"},{"Start":"02:30.435 ","End":"02:33.920","Text":"and let\u0027s rewrite this expression."},{"Start":"02:33.920 ","End":"02:39.020","Text":"Now, that\u0027ll be 0.5e to the power of,"},{"Start":"02:39.020 ","End":"02:41.260","Text":"instead of 2 times ln c,"},{"Start":"02:41.260 ","End":"02:46.115","Text":"we can write ln of c squared divided by 2,"},{"Start":"02:46.115 ","End":"02:48.470","Text":"and that has to equal to 1."},{"Start":"02:48.470 ","End":"02:51.920","Text":"Now, let\u0027s just simplify this."},{"Start":"02:51.920 ","End":"02:54.920","Text":"We\u0027ll multiply both sides by 2 and divide by a half."},{"Start":"02:54.920 ","End":"03:01.508","Text":"That means that we have e^ln c squared."},{"Start":"03:01.508 ","End":"03:04.085","Text":"That equals to 4."},{"Start":"03:04.085 ","End":"03:11.540","Text":"Let\u0027s just recall our basic algebra and look at this side of the equation."},{"Start":"03:11.540 ","End":"03:18.660","Text":"Now, if we remember a^log_a of b,"},{"Start":"03:18.660 ","End":"03:20.735","Text":"that equals to b."},{"Start":"03:20.735 ","End":"03:22.205","Text":"Now, in our case,"},{"Start":"03:22.205 ","End":"03:28.584","Text":"we have e^log_e of c squared."},{"Start":"03:28.584 ","End":"03:30.837","Text":"That equals to c squared,"},{"Start":"03:30.837 ","End":"03:36.610","Text":"so that means that this side of the equation equals c squared. Let\u0027s write that."},{"Start":"03:36.610 ","End":"03:41.974","Text":"We have here c squared that equals to 4."},{"Start":"03:41.974 ","End":"03:46.550","Text":"Now, that means that c can be either plus or minus 2."},{"Start":"03:46.550 ","End":"03:50.405","Text":"But if we look at the ln operator,"},{"Start":"03:50.405 ","End":"03:52.970","Text":"it only receives positive arguments."},{"Start":"03:52.970 ","End":"03:56.885","Text":"That means that c can only be plus 2."},{"Start":"03:56.885 ","End":"04:00.950","Text":"That\u0027s the value of c. In this section,"},{"Start":"04:00.950 ","End":"04:04.685","Text":"we\u0027re asked to calculate the variable\u0027s cumulative distribution function,"},{"Start":"04:04.685 ","End":"04:09.745","Text":"so we\u0027re looking at the probability of x being less than or equal to some value t,"},{"Start":"04:09.745 ","End":"04:12.750","Text":"and we can also write it as f of t,"},{"Start":"04:12.750 ","End":"04:21.980","Text":"and it\u0027s defined as the integral from minus infinity till the point t of f of x dx."},{"Start":"04:21.980 ","End":"04:23.840","Text":"Now, in our case,"},{"Start":"04:23.840 ","End":"04:27.150","Text":"let\u0027s define a point t here."},{"Start":"04:27.860 ","End":"04:34.604","Text":"We\u0027ll calculate the area under the density function until the point t,"},{"Start":"04:34.604 ","End":"04:38.405","Text":"and that will be our cumulative distribution function."},{"Start":"04:38.405 ","End":"04:40.865","Text":"First, let\u0027s do that."},{"Start":"04:40.865 ","End":"04:46.700","Text":"That\u0027s the integral from minus infinity to t of the density function."},{"Start":"04:46.700 ","End":"04:52.459","Text":"The density function is 0.5e^2x dx."},{"Start":"04:52.459 ","End":"04:58.055","Text":"Now, that equals to 0.5."},{"Start":"04:58.055 ","End":"05:02.930","Text":"We\u0027re doing the integral, e^2x now divided by 2"},{"Start":"05:02.930 ","End":"05:08.615","Text":"in the range between minus infinity and t. Now,"},{"Start":"05:08.615 ","End":"05:10.310","Text":"let\u0027s just plug in the numbers."},{"Start":"05:10.310 ","End":"05:16.085","Text":"That\u0027ll be 0.5e^2 times t divided by 2"},{"Start":"05:16.085 ","End":"05:25.400","Text":"minus 0.5e^2 times minus infinity divided by 2."},{"Start":"05:25.400 ","End":"05:29.030","Text":"Now, this actually is equal to 0."},{"Start":"05:29.030 ","End":"05:32.680","Text":"We\u0027ve discussed that in the last section,"},{"Start":"05:32.680 ","End":"05:39.765","Text":"so we can now write the cumulative distribution function F t,"},{"Start":"05:39.765 ","End":"05:43.200","Text":"that\u0027ll equal to this."},{"Start":"05:43.200 ","End":"05:48.050","Text":"Now, we\u0027ll have to divide the x-axis into 2 ranges"},{"Start":"05:48.050 ","End":"05:53.143","Text":"where t is less than or equal to ln 2,"},{"Start":"05:53.143 ","End":"05:57.140","Text":"and when t is greater than ln 2."},{"Start":"05:57.140 ","End":"05:59.210","Text":"Now, when t is less than ln 2,"},{"Start":"05:59.210 ","End":"06:01.535","Text":"this is this range right here."},{"Start":"06:01.535 ","End":"06:06.830","Text":"We\u0027ve calculated the expression for the cumulative distribution function."},{"Start":"06:06.830 ","End":"06:08.405","Text":"That\u0027s this guy right here."},{"Start":"06:08.405 ","End":"06:10.970","Text":"Now, 0.5 divided by 2,"},{"Start":"06:10.970 ","End":"06:13.340","Text":"that\u0027s a half divided by 2, that\u0027s a quarter."},{"Start":"06:13.340 ","End":"06:18.360","Text":"That\u0027ll be 1/4e^2t."},{"Start":"06:19.460 ","End":"06:23.670","Text":"For any value above ln t,"},{"Start":"06:23.670 ","End":"06:27.605","Text":"the cumulative distribution function would be 1."},{"Start":"06:27.605 ","End":"06:34.510","Text":"This is the cumulative distribution of x."}],"ID":32259},{"Watched":false,"Name":"Exercise 3 - Parts c-d","Duration":"3m 38s","ChapterTopicVideoID":30199,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.470","Text":"In this section we\u0027re asked to calculate the probability that X is greater than 0."},{"Start":"00:05.470 ","End":"00:08.685","Text":"Probability of X being greater than 0,"},{"Start":"00:08.685 ","End":"00:13.695","Text":"that equals to 1 minus probability of X being less than or equal to 0,"},{"Start":"00:13.695 ","End":"00:17.775","Text":"or 1 minus F at 0."},{"Start":"00:17.775 ","End":"00:19.750","Text":"Now, what\u0027s F at 0?"},{"Start":"00:19.750 ","End":"00:22.130","Text":"0 is in this range,"},{"Start":"00:22.130 ","End":"00:25.960","Text":"0 is smaller then the number line 2."},{"Start":"00:25.960 ","End":"00:30.770","Text":"We\u0027ll use this expression to calculate F at 0."},{"Start":"00:30.770 ","End":"00:35.410","Text":"That\u0027ll be 1 minus, now,"},{"Start":"00:35.410 ","End":"00:41.705","Text":"a 1/4 e times e to the power of 2 times 0."},{"Start":"00:41.705 ","End":"00:47.390","Text":"That equals to 1 minus now 8 to the power of 2 times 0, that\u0027s 1,"},{"Start":"00:47.390 ","End":"00:49.010","Text":"so 1 times 1/4,"},{"Start":"00:49.010 ","End":"00:51.370","Text":"that\u0027ll be a 1/4,"},{"Start":"00:51.370 ","End":"00:55.880","Text":"so the probability equals to 3/4."},{"Start":"00:55.880 ","End":"00:59.075","Text":"That means that the probability of x being greater than 0,"},{"Start":"00:59.075 ","End":"01:01.800","Text":"well, that\u0027s 3/4 quarters."},{"Start":"01:01.890 ","End":"01:07.395","Text":"In this section, we\u0027re asked what\u0027s the upper 25th percentile of X?"},{"Start":"01:07.395 ","End":"01:11.365","Text":"We\u0027re basically looking for a point here."},{"Start":"01:11.365 ","End":"01:18.345","Text":"We\u0027ll call it t, where the area under the density function from t onwards,"},{"Start":"01:18.345 ","End":"01:23.134","Text":"that would be equal to 0.25."},{"Start":"01:23.134 ","End":"01:28.000","Text":"That means that the area from minus infinity to t,"},{"Start":"01:28.000 ","End":"01:30.940","Text":"that equals to 0.75."},{"Start":"01:30.940 ","End":"01:34.135","Text":"Now, what\u0027s the definition of percentile?"},{"Start":"01:34.135 ","End":"01:39.690","Text":"That\u0027s the probability of X being less than or equal to x_p."},{"Start":"01:39.690 ","End":"01:44.865","Text":"We want that probability to be p. In our case,"},{"Start":"01:44.865 ","End":"01:52.885","Text":"we\u0027re looking for the probability of X being less than or equal to x_0.75,"},{"Start":"01:52.885 ","End":"01:56.840","Text":"and we want that to be equal to 0.75."},{"Start":"01:56.840 ","End":"02:05.259","Text":"Or basically we want 0.75 to be equal to F at x_0.75,"},{"Start":"02:05.259 ","End":"02:07.860","Text":"and this is what we\u0027re looking for."},{"Start":"02:07.860 ","End":"02:11.600","Text":"Now, all we have to do is we have to equate"},{"Start":"02:11.600 ","End":"02:16.505","Text":"this expression right here to 0.75 and solve for t,"},{"Start":"02:16.505 ","End":"02:22.040","Text":"where t is our percentile x_0.75."},{"Start":"02:22.040 ","End":"02:28.745","Text":"Let\u0027s do that. That\u0027ll be a 1/4 times e^2 times t,"},{"Start":"02:28.745 ","End":"02:32.195","Text":"and that has to equal to 0.75."},{"Start":"02:32.195 ","End":"02:36.005","Text":"Now let\u0027s multiply both sides by 4,"},{"Start":"02:36.005 ","End":"02:41.030","Text":"and that\u0027ll be e^2t, that equals to 3."},{"Start":"02:41.030 ","End":"02:43.790","Text":"Let\u0027s take ln of both sides."},{"Start":"02:43.790 ","End":"02:51.004","Text":"That\u0027ll be ln of e to the power of 2t and that equals to ln of 3."},{"Start":"02:51.004 ","End":"02:59.430","Text":"Now, we can also write this out as 2t ln e and that equals to ln 3."},{"Start":"02:59.600 ","End":"03:03.570","Text":"Ln of e, that\u0027s 1."},{"Start":"03:03.570 ","End":"03:07.635","Text":"We have 2t equals ln 3,"},{"Start":"03:07.635 ","End":"03:14.570","Text":"and that means that t equals to ln 3 over 2."},{"Start":"03:14.570 ","End":"03:17.120","Text":"Now, this is the value,"},{"Start":"03:17.120 ","End":"03:21.140","Text":"this is the value, this is the upper 25th percentile,"},{"Start":"03:21.140 ","End":"03:27.165","Text":"or the value where from minus infinity to this value,"},{"Start":"03:27.165 ","End":"03:30.860","Text":"the area under the density function is 75 percent,"},{"Start":"03:30.860 ","End":"03:37.710","Text":"or the area above this value is 0.25 percent."}],"ID":32260},{"Watched":false,"Name":"Exercise 4 - Parts a-b","Duration":"5m 19s","ChapterTopicVideoID":30200,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In the following question, we\u0027re given a density function of"},{"Start":"00:02.850 ","End":"00:07.600","Text":"a random variable X and we\u0027re asked to calculate the density function."},{"Start":"00:08.150 ","End":"00:11.775","Text":"The density function that\u0027s small f of x,"},{"Start":"00:11.775 ","End":"00:13.260","Text":"that would equal 2."},{"Start":"00:13.260 ","End":"00:17.085","Text":"Now, we\u0027re going to have to split the density function into ranges here."},{"Start":"00:17.085 ","End":"00:20.197","Text":"The first range is between 1 and 2,"},{"Start":"00:20.197 ","End":"00:24.915","Text":"so X here would be greater or equal to 1 and less than or equal to 2."},{"Start":"00:24.915 ","End":"00:26.340","Text":"Now, why equal?"},{"Start":"00:26.340 ","End":"00:29.270","Text":"Well, that\u0027s because of the black dots right here."},{"Start":"00:29.270 ","End":"00:31.830","Text":"The next range is between 2 and 4,"},{"Start":"00:31.830 ","End":"00:35.130","Text":"so X here would be less than or equal to 4,"},{"Start":"00:35.130 ","End":"00:38.195","Text":"but only greater than 2. Again, why is that?"},{"Start":"00:38.195 ","End":"00:43.775","Text":"Because we have a white dot denoting that 2 is not part of this range,"},{"Start":"00:43.775 ","End":"00:45.910","Text":"but part of this range."},{"Start":"00:45.910 ","End":"00:49.160","Text":"We have all the other ranges as well."},{"Start":"00:49.160 ","End":"00:53.930","Text":"Now what\u0027s the value of the density function for each 1 of the ranges?"},{"Start":"00:53.930 ","End":"00:56.015","Text":"Well, for all other ranges,"},{"Start":"00:56.015 ","End":"00:58.565","Text":"the density function equals 0."},{"Start":"00:58.565 ","End":"01:01.910","Text":"Now, what about the first range between 1 and 2?"},{"Start":"01:01.910 ","End":"01:03.290","Text":"Well, we\u0027re given that,"},{"Start":"01:03.290 ","End":"01:05.794","Text":"that equals a 1/4."},{"Start":"01:05.794 ","End":"01:08.690","Text":"Now, what about this value right here,"},{"Start":"01:08.690 ","End":"01:14.050","Text":"the value of the density function in the range between 2 and 4."},{"Start":"01:14.050 ","End":"01:16.675","Text":"Well, let\u0027s call that A."},{"Start":"01:16.675 ","End":"01:21.500","Text":"Now we know that the total area under the density function equals 1,"},{"Start":"01:21.500 ","End":"01:26.649","Text":"so all we have to do is calculate these areas right here,"},{"Start":"01:26.649 ","End":"01:29.420","Text":"the area of these 2 rectangles,"},{"Start":"01:29.420 ","End":"01:31.280","Text":"equate them to 1 and solve for A."},{"Start":"01:31.280 ","End":"01:35.240","Text":"Let\u0027s do that. Now. The 1st rectangle,"},{"Start":"01:35.240 ","End":"01:37.315","Text":"the base is 1,"},{"Start":"01:37.315 ","End":"01:40.182","Text":"and the height is 1/4,"},{"Start":"01:40.182 ","End":"01:43.110","Text":"so the area here is 1/4."},{"Start":"01:43.110 ","End":"01:47.480","Text":"What about the next rectangle?"},{"Start":"01:47.480 ","End":"01:49.565","Text":"Well, the base here is 2."},{"Start":"01:49.565 ","End":"01:51.610","Text":"The height is A."},{"Start":"01:51.610 ","End":"01:54.720","Text":"Now we know then that 1/4,"},{"Start":"01:54.720 ","End":"01:58.520","Text":"the area of this rectangle plus 2A, well,"},{"Start":"01:58.520 ","End":"02:00.980","Text":"that\u0027s the area of the larger rectangle,"},{"Start":"02:00.980 ","End":"02:02.950","Text":"that has to equal to 1."},{"Start":"02:02.950 ","End":"02:11.805","Text":"That means that we have 2A equals 3/4 and A, then equals 3/8."},{"Start":"02:11.805 ","End":"02:14.440","Text":"We\u0027ll just write this down here."},{"Start":"02:15.290 ","End":"02:21.565","Text":"This then is the density function of the random variable X."},{"Start":"02:21.565 ","End":"02:23.620","Text":"In this section, we\u0027re asked to calculate"},{"Start":"02:23.620 ","End":"02:27.260","Text":"the cumulative distribution function of X, so the CDF of X."},{"Start":"02:27.260 ","End":"02:31.630","Text":"That\u0027s the probability of X being less than or equal to some value"},{"Start":"02:31.630 ","End":"02:37.890","Text":"t. We\u0027ll denote that as big F of t. Now that equals to."},{"Start":"02:37.890 ","End":"02:42.755","Text":"Now let\u0027s divide the CDF into 4 parts,"},{"Start":"02:42.755 ","End":"02:46.165","Text":"where t is less than 1,"},{"Start":"02:46.165 ","End":"02:50.655","Text":"where t is between 1 and 2,"},{"Start":"02:50.655 ","End":"02:53.085","Text":"where t is greater than 2,"},{"Start":"02:53.085 ","End":"02:55.170","Text":"but less than or equal to 4,"},{"Start":"02:55.170 ","End":"02:58.140","Text":"and when t is greater than 4."},{"Start":"02:58.140 ","End":"03:01.005","Text":"Now, when t is less than 1,"},{"Start":"03:01.005 ","End":"03:04.868","Text":"that\u0027s this range right here from minus infinity until 1,"},{"Start":"03:04.868 ","End":"03:08.990","Text":"the density function isn\u0027t defined in this range,"},{"Start":"03:08.990 ","End":"03:15.074","Text":"so the area underneath the density function would be 0,"},{"Start":"03:15.074 ","End":"03:19.190","Text":"so the CDF in this range would be 0."},{"Start":"03:19.190 ","End":"03:21.995","Text":"Now let\u0027s take a look at the other end,"},{"Start":"03:21.995 ","End":"03:24.590","Text":"where t is greater than 4,"},{"Start":"03:24.590 ","End":"03:26.870","Text":"that\u0027s in this range right here."},{"Start":"03:26.870 ","End":"03:30.050","Text":"Now for all values of t in this range right here,"},{"Start":"03:30.050 ","End":"03:37.210","Text":"we\u0027ve already accumulated all of the area underneath the density function."},{"Start":"03:37.210 ","End":"03:42.440","Text":"The value of the CDF in this range would be 1."},{"Start":"03:42.440 ","End":"03:47.075","Text":"Now let\u0027s take a look at this range right here between 1 and 2."},{"Start":"03:47.075 ","End":"03:49.115","Text":"Let\u0027s pick a point here,"},{"Start":"03:49.115 ","End":"03:52.935","Text":"we\u0027ll call this t and what we want to do is we want to"},{"Start":"03:52.935 ","End":"03:58.770","Text":"calculate the area of this rectangle right here."},{"Start":"03:58.770 ","End":"04:01.095","Text":"We\u0027ll, that\u0027s the base times the height."},{"Start":"04:01.095 ","End":"04:09.111","Text":"Now the base would be t minus 1 and the height is 1/4 that\u0027s that\u0027s given,"},{"Start":"04:09.111 ","End":"04:12.515","Text":"so this is the expression of the CDF in this range."},{"Start":"04:12.515 ","End":"04:15.520","Text":"Now what about the other range?"},{"Start":"04:15.520 ","End":"04:17.980","Text":"Again, in the next range,"},{"Start":"04:17.980 ","End":"04:19.715","Text":"right, we\u0027ll pick a point right here,"},{"Start":"04:19.715 ","End":"04:23.660","Text":"we\u0027ll call that t. And what we want to do is want to calculate"},{"Start":"04:23.660 ","End":"04:29.180","Text":"the area under the density function from 1 to t. Now,"},{"Start":"04:29.180 ","End":"04:36.610","Text":"we know what the area is of the smaller rectangle right here, that\u0027s 1/4."},{"Start":"04:36.610 ","End":"04:42.060","Text":"All we have to do is to calculate the area of this rectangle right here."},{"Start":"04:42.060 ","End":"04:44.260","Text":"That\u0027s the base times the height."},{"Start":"04:44.260 ","End":"04:49.070","Text":"Well, the base here, that\u0027s t minus 2 times the height."},{"Start":"04:49.070 ","End":"04:50.240","Text":"Now the height here,"},{"Start":"04:50.240 ","End":"04:54.845","Text":"we\u0027ve calculated that in the last section as 3/8."},{"Start":"04:54.845 ","End":"04:57.815","Text":"The height would be 3/8,"},{"Start":"04:57.815 ","End":"04:59.210","Text":"the base times the height."},{"Start":"04:59.210 ","End":"05:05.285","Text":"The total area under the density function from 1 to t would be"},{"Start":"05:05.285 ","End":"05:12.315","Text":"1/4 plus t minus 2 times 3/8."},{"Start":"05:12.315 ","End":"05:19.500","Text":"This then is the CDF or the cumulative distribution function of X."}],"ID":32261},{"Watched":false,"Name":"Exercise 4 - Parts c-e","Duration":"9m 1s","ChapterTopicVideoID":30201,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this section, we\u0027re asked to calculate the median of X."},{"Start":"00:03.450 ","End":"00:05.940","Text":"That means that we\u0027re looking for the probability of"},{"Start":"00:05.940 ","End":"00:09.030","Text":"X being less than or equal to some value,"},{"Start":"00:09.030 ","End":"00:11.220","Text":"x_0.5, we\u0027ll call it."},{"Start":"00:11.220 ","End":"00:13.890","Text":"We want that to be equal to 1/2."},{"Start":"00:13.890 ","End":"00:19.905","Text":"Now, that means that we\u0027re looking for F at x_0.5,"},{"Start":"00:19.905 ","End":"00:21.930","Text":"we want that to be equal to a 1/2."},{"Start":"00:21.930 ","End":"00:25.330","Text":"Now in order to use the cumulative distribution function,"},{"Start":"00:25.330 ","End":"00:28.975","Text":"we need to know in what range does this value lie?"},{"Start":"00:28.975 ","End":"00:35.315","Text":"Now, we know that it doesn\u0027t lie in the range between 1 and 2. And why is that?"},{"Start":"00:35.315 ","End":"00:38.495","Text":"Because the area under the density function here"},{"Start":"00:38.495 ","End":"00:42.155","Text":"is a 1/4 and we need another 1/4 to get to a 1/2."},{"Start":"00:42.155 ","End":"00:45.980","Text":"So it has to be somewhere here in the range between 2 and 4."},{"Start":"00:45.980 ","End":"00:48.440","Text":"That\u0027ll be x_0.5."},{"Start":"00:48.440 ","End":"00:51.890","Text":"So that means that we\u0027ll be using"},{"Start":"00:51.890 ","End":"00:56.675","Text":"this expression for the CDF and then we\u0027ll equate that to a 1/2,"},{"Start":"00:56.675 ","End":"01:02.267","Text":"we\u0027ll solve for t and t will be our median or x_0.5,"},{"Start":"01:02.267 ","End":"01:03.630","Text":"so let\u0027s do that."},{"Start":"01:03.630 ","End":"01:10.325","Text":"That\u0027ll be a 1/4 plus 3/8 times t minus 2,"},{"Start":"01:10.325 ","End":"01:12.455","Text":"and that has to equal to a 1/2."},{"Start":"01:12.455 ","End":"01:17.105","Text":"That means that 3/8 times t minus 2,"},{"Start":"01:17.105 ","End":"01:19.525","Text":"well, that equals to a 1/4."},{"Start":"01:19.525 ","End":"01:29.725","Text":"That means that t minus 2 equals 2/3 and that means that t then equals 2 and 2/3."},{"Start":"01:29.725 ","End":"01:34.820","Text":"This value right here is the value"},{"Start":"01:34.820 ","End":"01:40.740","Text":"where the area under the density function from 1 to 2 and 2/3,"},{"Start":"01:40.740 ","End":"01:42.990","Text":"the area will equal to a 1/2."},{"Start":"01:42.990 ","End":"01:49.320","Text":"That means by definition that 2 and 2/3 is the median of X."},{"Start":"01:49.320 ","End":"01:50.580","Text":"In this section,"},{"Start":"01:50.580 ","End":"01:53.840","Text":"we\u0027re asked to calculate the expectation and variance of X."},{"Start":"01:53.840 ","End":"01:56.660","Text":"Let\u0027s start with the expectation."},{"Start":"01:56.660 ","End":"02:01.190","Text":"The expectation of X is defined as the integral from minus infinity to"},{"Start":"02:01.190 ","End":"02:06.335","Text":"plus infinity of X times the density function of (x) dx."},{"Start":"02:06.335 ","End":"02:15.735","Text":"In our case, that\u0027ll be the integral from 1 to 2 of X times a 1/4 dx"},{"Start":"02:15.735 ","End":"02:25.748","Text":"plus the integral from 2 to 4 of X times 3/8 dx,"},{"Start":"02:25.748 ","End":"02:27.360","Text":"so let\u0027s do that."},{"Start":"02:27.360 ","End":"02:31.860","Text":"The first integral is equal to"},{"Start":"02:31.860 ","End":"02:39.920","Text":"1/4 times X squared divided by 2 in the range between 1 and 2 plus,"},{"Start":"02:39.920 ","End":"02:42.085","Text":"now this integral right here,"},{"Start":"02:42.085 ","End":"02:50.995","Text":"that equals to 3/8 X squared divided by 2 in the range between 2 and 4."},{"Start":"02:50.995 ","End":"02:54.085","Text":"Now let\u0027s just plug in the numbers here."},{"Start":"02:54.085 ","End":"03:03.040","Text":"That\u0027ll be a 1/4 times 2 squared divided by 2 minus a 1/4 times 1 squared divided by"},{"Start":"03:03.040 ","End":"03:09.720","Text":"2 plus 3/8 times 4 squared"},{"Start":"03:09.720 ","End":"03:17.435","Text":"divided by 2 minus 3/8 times 2 squared divided by 2."},{"Start":"03:17.435 ","End":"03:25.460","Text":"And that equals to 2 squared divided by 8 minus 1 squared divided by"},{"Start":"03:25.460 ","End":"03:31.710","Text":"8 plus 3/16 times 4 squared minus 2"},{"Start":"03:31.710 ","End":"03:38.460","Text":"squared and all that comes out to 2.625."},{"Start":"03:38.460 ","End":"03:44.360","Text":"That equals to the expectation of X."},{"Start":"03:44.360 ","End":"03:46.670","Text":"Let\u0027s take a look now at the variance of X."},{"Start":"03:46.670 ","End":"03:51.065","Text":"The variance of X is defined as the integral between"},{"Start":"03:51.065 ","End":"03:56.210","Text":"minus infinity and plus infinity of X squared times F of X;"},{"Start":"03:56.210 ","End":"03:58.445","Text":"the density function of (x)dx,"},{"Start":"03:58.445 ","End":"04:02.615","Text":"minus the expectation squared of X."},{"Start":"04:02.615 ","End":"04:06.020","Text":"Now, since we\u0027ve already calculated the expectation of X,"},{"Start":"04:06.020 ","End":"04:09.395","Text":"let\u0027s just deal with this integral right here."},{"Start":"04:09.395 ","End":"04:10.835","Text":"And only after that,"},{"Start":"04:10.835 ","End":"04:13.715","Text":"we\u0027ll calculate the variance of X."},{"Start":"04:13.715 ","End":"04:15.065","Text":"So in our case,"},{"Start":"04:15.065 ","End":"04:17.765","Text":"what\u0027s this portion right here?"},{"Start":"04:17.765 ","End":"04:24.160","Text":"Well, that would be the integral from 1 to 2 of"},{"Start":"04:24.160 ","End":"04:31.485","Text":"X squared times a 1/4 plus the integral from 2 to 4,"},{"Start":"04:31.485 ","End":"04:37.155","Text":"of X squared times 3/8."},{"Start":"04:37.155 ","End":"04:39.600","Text":"That\u0027s dx right here."},{"Start":"04:39.600 ","End":"04:44.855","Text":"Let\u0027s just work out these integrals right here."},{"Start":"04:44.855 ","End":"04:47.130","Text":"The first integral right here."},{"Start":"04:47.130 ","End":"04:52.005","Text":"Well, that\u0027s X cubed divided by 3 times"},{"Start":"04:52.005 ","End":"04:58.025","Text":"a 1/4 in the range between 1 and 2 plus,"},{"Start":"04:58.025 ","End":"05:07.090","Text":"now that\u0027ll be X cubed divided by 3 times 3/8 in the range between 2 and 4."},{"Start":"05:07.090 ","End":"05:09.630","Text":"Now let\u0027s just plug in the numbers."},{"Start":"05:09.630 ","End":"05:13.170","Text":"That will be 2 cubed divided by 3 times a"},{"Start":"05:13.170 ","End":"05:19.760","Text":"1/4 minus 1 cubed divided by 3 times a 1/4 plus,"},{"Start":"05:19.760 ","End":"05:22.355","Text":"now let\u0027s take a look at this expression."},{"Start":"05:22.355 ","End":"05:27.755","Text":"That\u0027ll be 4 cubed divided by 3 times 3/8"},{"Start":"05:27.755 ","End":"05:34.135","Text":"minus 2 cubed divided by 3 times 3/8."},{"Start":"05:34.135 ","End":"05:42.405","Text":"So that will equal to 1/12,"},{"Start":"05:42.405 ","End":"05:45.690","Text":"2 to the power 3 minus 1 to the power of"},{"Start":"05:45.690 ","End":"05:54.180","Text":"3 plus 1/8 times 4 cubed minus 2 cubed."},{"Start":"05:54.180 ","End":"06:00.380","Text":"Now, this is the expression of only the first integral right here."},{"Start":"06:00.380 ","End":"06:07.040","Text":"We have to take away the expectation squared minus the expectation squared of X."},{"Start":"06:07.040 ","End":"06:08.600","Text":"And what\u0027s the expectation squared?"},{"Start":"06:08.600 ","End":"06:11.840","Text":"Well, that\u0027s 2.625 squared, right?"},{"Start":"06:11.840 ","End":"06:16.325","Text":"So we have to take away 2.625"},{"Start":"06:16.325 ","End":"06:23.790","Text":"squared and that comes out to 0.6927."},{"Start":"06:23.840 ","End":"06:29.630","Text":"So this is the variance of X."},{"Start":"06:29.630 ","End":"06:31.145","Text":"In this section,"},{"Start":"06:31.145 ","End":"06:34.210","Text":"we\u0027re asked to calculate the expectation of X cubed."},{"Start":"06:34.210 ","End":"06:43.660","Text":"So we know that the expectation of X equals the integral of X times F of (x)dx."},{"Start":"06:43.660 ","End":"06:52.210","Text":"And the expectation of X squared equals the integral of X squared times F(x)dx."},{"Start":"06:52.210 ","End":"06:54.895","Text":"So by the same reasoning,"},{"Start":"06:54.895 ","End":"07:02.540","Text":"the expectation of X cubed is equal to the integral of X cubed times F(x)dx."},{"Start":"07:03.020 ","End":"07:06.580","Text":"Let\u0027s just calculate that."},{"Start":"07:06.580 ","End":"07:09.925","Text":"In our case, they\u0027ll be the integral, right?"},{"Start":"07:09.925 ","End":"07:20.000","Text":"In this range right here will be from 1 to 2 of X cubed times a 1/4 dx plus the integral."},{"Start":"07:20.000 ","End":"07:27.965","Text":"Now this range from 2 to 4 of X cubed times 3/8 dx."},{"Start":"07:27.965 ","End":"07:33.420","Text":"Let\u0027s just calculate these integrals."},{"Start":"07:33.420 ","End":"07:41.910","Text":"That will be X^4 divided by 4 times a 1/4 in"},{"Start":"07:41.910 ","End":"07:50.820","Text":"the range between 1 and 2 plus X^4 divided by 4 times 3/8,"},{"Start":"07:50.820 ","End":"07:54.750","Text":"in the range between 2 and 4."},{"Start":"07:54.750 ","End":"07:57.600","Text":"Now let\u0027s just plug in the numbers here."},{"Start":"07:57.600 ","End":"08:02.610","Text":"That\u0027ll be 2^4 divided by"},{"Start":"08:02.610 ","End":"08:08.445","Text":"4 times a 1/4 minus 1^4,"},{"Start":"08:08.445 ","End":"08:13.350","Text":"divided by 4 times a 1/4 plus,"},{"Start":"08:13.350 ","End":"08:14.820","Text":"now let\u0027s plug in these numbers."},{"Start":"08:14.820 ","End":"08:19.185","Text":"That\u0027s 4^4 divided by 4 times"},{"Start":"08:19.185 ","End":"08:26.010","Text":"3/8 minus 2^4 divided by 4 times 3/8."},{"Start":"08:26.010 ","End":"08:31.575","Text":"Now that equals to"},{"Start":"08:31.575 ","End":"08:37.635","Text":"1/16 times 2^4 minus 1^4,"},{"Start":"08:37.635 ","End":"08:47.855","Text":"plus 3 divided by 32 times 4^4 minus 2^4."},{"Start":"08:47.855 ","End":"08:56.825","Text":"That comes out to 23.4375."},{"Start":"08:56.825 ","End":"09:01.390","Text":"So that\u0027s the expectation of X cubed."}],"ID":32262},{"Watched":false,"Name":"Exercise 5","Duration":"7m 38s","ChapterTopicVideoID":30202,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"A factory manufacturers product A."},{"Start":"00:03.240 ","End":"00:07.155","Text":"The manufacturing time in hours has the following density function,"},{"Start":"00:07.155 ","End":"00:10.800","Text":"f of x equals 6x times 1 minus x,"},{"Start":"00:10.800 ","End":"00:13.260","Text":"and x is defined between 0 and 1."},{"Start":"00:13.260 ","End":"00:16.530","Text":"We\u0027re asked, what\u0027s the probability that"},{"Start":"00:16.530 ","End":"00:21.090","Text":"the manufacturing time of a random product A will be less than 20 minutes."},{"Start":"00:21.090 ","End":"00:27.855","Text":"The first thing that I want to do is I want to see what this function looks like."},{"Start":"00:27.855 ","End":"00:29.925","Text":"Let\u0027s draw this function."},{"Start":"00:29.925 ","End":"00:35.100","Text":"We see here the density function."},{"Start":"00:35.100 ","End":"00:44.050","Text":"We can see that this is a parabola and it crosses the x-axis at 0 and at 1."},{"Start":"00:44.050 ","End":"00:47.870","Text":"Where x equals 0, the function equals 0 and when x equals 1,"},{"Start":"00:47.870 ","End":"00:50.120","Text":"the function equals 0 as well."},{"Start":"00:50.120 ","End":"00:53.900","Text":"Now, the second thing I want to do is I want to"},{"Start":"00:53.900 ","End":"00:59.750","Text":"convert everything here to a standard unit of time."},{"Start":"00:59.750 ","End":"01:04.560","Text":"Now, we\u0027re talking about x being in"},{"Start":"01:04.560 ","End":"01:09.545","Text":"hours but we\u0027re asked about the production time x in minutes."},{"Start":"01:09.545 ","End":"01:14.290","Text":"We know that 20 minutes"},{"Start":"01:14.290 ","End":"01:20.370","Text":"equals 20 over 60 and that equals to 1/3 hour."},{"Start":"01:20.370 ","End":"01:26.300","Text":"Again, we know that x is in hours, so what are we asked?"},{"Start":"01:26.300 ","End":"01:34.965","Text":"We\u0027re asked for the probability that x would be less than 1/3."},{"Start":"01:34.965 ","End":"01:36.815","Text":"Now, how do we do that?"},{"Start":"01:36.815 ","End":"01:46.755","Text":"Well, we\u0027re going to have to take the integral of the density function from 0-1/3."},{"Start":"01:46.755 ","End":"01:50.720","Text":"Basically, we\u0027re looking at this point right here and we want to"},{"Start":"01:50.720 ","End":"01:56.520","Text":"know what this area is right here. Let\u0027s do that."},{"Start":"01:56.520 ","End":"02:04.280","Text":"That\u0027ll be the integral from 0-1/3 of this function right here."},{"Start":"02:04.280 ","End":"02:06.680","Text":"Now, let\u0027s just multiply everything right in."},{"Start":"02:06.680 ","End":"02:13.460","Text":"That\u0027ll be 6x minus 6x squared dx."},{"Start":"02:13.460 ","End":"02:20.330","Text":"Now, that equals to 6x squared divided by 2"},{"Start":"02:20.330 ","End":"02:30.045","Text":"minus 6x cubed divided by 3 in the interval between 0 and 1/3."},{"Start":"02:30.045 ","End":"02:36.090","Text":"Now, that equals to 3x"},{"Start":"02:36.090 ","End":"02:42.885","Text":"squared minus 2x cubed in this interval between 0 and 1/3."},{"Start":"02:42.885 ","End":"02:48.210","Text":"That equals now let\u0027s plug this in that\u0027s 3 times"},{"Start":"02:48.210 ","End":"02:55.080","Text":"1/3 squared minus 2"},{"Start":"02:55.080 ","End":"03:00.870","Text":"times 1/3 cubed minus,"},{"Start":"03:00.870 ","End":"03:03.455","Text":"now, if we plug in 0 we\u0027ll get 0."},{"Start":"03:03.455 ","End":"03:09.125","Text":"Now, that comes out to 7 over 27."},{"Start":"03:09.125 ","End":"03:12.305","Text":"That\u0027s the probability that"},{"Start":"03:12.305 ","End":"03:17.870","Text":"the manufacturing time of a random product A would be less than 20 minutes."},{"Start":"03:17.870 ","End":"03:19.880","Text":"In this section, we\u0027re asked,"},{"Start":"03:19.880 ","End":"03:22.580","Text":"what\u0027s the probability that the manufacturing time of"},{"Start":"03:22.580 ","End":"03:26.690","Text":"a random product A would be exactly 30 minutes?"},{"Start":"03:26.690 ","End":"03:29.120","Text":"We\u0027re looking for the probability of x,"},{"Start":"03:29.120 ","End":"03:30.335","Text":"the production time,"},{"Start":"03:30.335 ","End":"03:33.650","Text":"being equal to 1/2 of an hour."},{"Start":"03:33.650 ","End":"03:35.165","Text":"That\u0027s 30 minutes."},{"Start":"03:35.165 ","End":"03:38.915","Text":"Now, this probability equals 0."},{"Start":"03:38.915 ","End":"03:41.390","Text":"Why is that? Well, first of all,"},{"Start":"03:41.390 ","End":"03:44.330","Text":"we\u0027re dealing with a continuous random variable."},{"Start":"03:44.330 ","End":"03:46.700","Text":"Now let\u0027s look at the density function."},{"Start":"03:46.700 ","End":"03:51.335","Text":"We know that the probability of"},{"Start":"03:51.335 ","End":"03:56.854","Text":"something is equal to the area underneath the density function."},{"Start":"03:56.854 ","End":"04:00.020","Text":"Now, what\u0027s the area under"},{"Start":"04:00.020 ","End":"04:04.520","Text":"the density function of a line of a point right here that\u0027s 1/2?"},{"Start":"04:04.520 ","End":"04:06.185","Text":"Well, that equals to 0."},{"Start":"04:06.185 ","End":"04:10.860","Text":"There\u0027s no area right here on this line."},{"Start":"04:10.860 ","End":"04:13.535","Text":"This is an important concept to remember."},{"Start":"04:13.535 ","End":"04:17.750","Text":"If you\u0027re dealing with a continuous random variable and you\u0027re asked"},{"Start":"04:17.750 ","End":"04:22.260","Text":"the probability of that variable equaling exactly some value,"},{"Start":"04:22.260 ","End":"04:24.515","Text":"that will always be 0."},{"Start":"04:24.515 ","End":"04:27.470","Text":"Now, some other characteristics."},{"Start":"04:27.470 ","End":"04:29.450","Text":"If you were asked,"},{"Start":"04:29.450 ","End":"04:32.400","Text":"what\u0027s the probability of x,"},{"Start":"04:32.400 ","End":"04:34.100","Text":"a continuous random variable,"},{"Start":"04:34.100 ","End":"04:36.590","Text":"being greater or equal to some value?"},{"Start":"04:36.590 ","End":"04:42.325","Text":"That equals to the probability of x being just greater than that value."},{"Start":"04:42.325 ","End":"04:44.565","Text":"The same thing for the other side."},{"Start":"04:44.565 ","End":"04:50.540","Text":"If we were asked for the probability of x being less than or equal to a specific value,"},{"Start":"04:50.540 ","End":"04:56.775","Text":"well, that equals to the probability of x being just less than that value."},{"Start":"04:56.775 ","End":"04:59.840","Text":"These are important characteristics to"},{"Start":"04:59.840 ","End":"05:04.640","Text":"remember when we\u0027re dealing with continuous random variables."},{"Start":"05:04.640 ","End":"05:09.348","Text":"In this section, we\u0027re given that 5 random samples of product A are selected."},{"Start":"05:09.348 ","End":"05:12.170","Text":"We\u0027re asked, what\u0027s the expectation of the number of"},{"Start":"05:12.170 ","End":"05:16.025","Text":"products whose manufacturing time is greater than 20 minutes?"},{"Start":"05:16.025 ","End":"05:18.590","Text":"Well, if we recall from section A,"},{"Start":"05:18.590 ","End":"05:26.060","Text":"we\u0027ve calculated the probability that x is less than 1/3 or 20 minutes."},{"Start":"05:26.060 ","End":"05:30.135","Text":"That equals to 7 divided by 27."},{"Start":"05:30.135 ","End":"05:34.925","Text":"If we\u0027re looking for the probability of x being greater than 1/3,"},{"Start":"05:34.925 ","End":"05:37.235","Text":"well that\u0027s 1 minus this."},{"Start":"05:37.235 ","End":"05:44.750","Text":"The probability of x being greater than 1/3 or 20 minutes,"},{"Start":"05:44.750 ","End":"05:48.725","Text":"well, that equals to 1 minus 7 divided by 27."},{"Start":"05:48.725 ","End":"05:51.920","Text":"That equals to 20 divided by 27."},{"Start":"05:51.920 ","End":"05:53.780","Text":"Now, what else are we given here?"},{"Start":"05:53.780 ","End":"06:00.500","Text":"Well, we\u0027re given that we have 5 random samples so n here equals 5."},{"Start":"06:00.500 ","End":"06:05.285","Text":"Let\u0027s look at what is our success."},{"Start":"06:05.285 ","End":"06:12.155","Text":"Well, we\u0027re looking at success as the production time being greater than 20 minutes."},{"Start":"06:12.155 ","End":"06:14.015","Text":"Let\u0027s write this down."},{"Start":"06:14.015 ","End":"06:24.425","Text":"Success equals production time being greater than 20 minutes or greater than 1/3."},{"Start":"06:24.425 ","End":"06:26.390","Text":"We want to count that."},{"Start":"06:26.390 ","End":"06:33.690","Text":"Let\u0027s define a random variable y and that\u0027ll be the number of successes."},{"Start":"06:33.690 ","End":"06:37.825","Text":"Now, the minute we have that,"},{"Start":"06:37.825 ","End":"06:42.730","Text":"then these are the conditions for a binomial distribution."},{"Start":"06:42.730 ","End":"06:49.450","Text":"We know now that y is distributed with a binomial distribution where n"},{"Start":"06:49.450 ","End":"06:56.780","Text":"equals 5 and p equals 20 divided by 27."},{"Start":"06:56.780 ","End":"07:01.719","Text":"Now, since y is distributed with a binomial distribution,"},{"Start":"07:01.719 ","End":"07:04.945","Text":"we\u0027re looking for the expectation of"},{"Start":"07:04.945 ","End":"07:08.635","Text":"the number of products whose manufacturing time is greater than 20 minutes."},{"Start":"07:08.635 ","End":"07:11.620","Text":"Then we\u0027re looking for the expectation of y."},{"Start":"07:11.620 ","End":"07:15.130","Text":"Now, since that has a binomial distribution where"},{"Start":"07:15.130 ","End":"07:19.840","Text":"the expectation of y is n times p. In our case,"},{"Start":"07:19.840 ","End":"07:24.355","Text":"that\u0027ll be 5 times 20 divided by 27."},{"Start":"07:24.355 ","End":"07:30.940","Text":"That equals to 3.704."},{"Start":"07:30.940 ","End":"07:34.190","Text":"That is the expected number of"},{"Start":"07:34.190 ","End":"07:39.150","Text":"products whose manufacturing time is greater than 20 minutes."}],"ID":32263},{"Watched":false,"Name":"Exercise 6 - Parts a-b","Duration":"6m 56s","ChapterTopicVideoID":30204,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In this question, we\u0027re given that the density function of"},{"Start":"00:02.850 ","End":"00:06.855","Text":"a specific random variable is given by the following formula."},{"Start":"00:06.855 ","End":"00:08.203","Text":"This is it right here,"},{"Start":"00:08.203 ","End":"00:10.305","Text":"and we\u0027re asked to calculate b."},{"Start":"00:10.305 ","End":"00:15.880","Text":"Now we can see that the density function split up into 4 ranges."},{"Start":"00:15.880 ","End":"00:18.665","Text":"Before calculating b,"},{"Start":"00:18.665 ","End":"00:21.965","Text":"let\u0027s just see what this function looks like."},{"Start":"00:21.965 ","End":"00:26.820","Text":"The first range is where x is below 4, so that\u0027s 4."},{"Start":"00:26.820 ","End":"00:30.285","Text":"Next range is where x is between 4 and 5,"},{"Start":"00:30.285 ","End":"00:35.055","Text":"and the next range where x is between 5 and 6."},{"Start":"00:35.055 ","End":"00:39.480","Text":"The last range where x is greater than 6."},{"Start":"00:39.480 ","End":"00:43.980","Text":"Now, where x is less than 4 or greater than 6,"},{"Start":"00:43.980 ","End":"00:47.985","Text":"well, the value of f of x is 0."},{"Start":"00:47.985 ","End":"00:51.755","Text":"Let\u0027s see what happens in these ranges right here."},{"Start":"00:51.755 ","End":"00:56.715","Text":"In this range, f of 4,"},{"Start":"00:56.715 ","End":"01:01.425","Text":"well that will equal to b times 4 minus 4 times b,"},{"Start":"01:01.425 ","End":"01:03.285","Text":"that equals to 0."},{"Start":"01:03.285 ","End":"01:07.060","Text":"The value of f at 4 equals 0."},{"Start":"01:07.060 ","End":"01:13.145","Text":"What happens where x equals 5 or f at 5?"},{"Start":"01:13.145 ","End":"01:18.785","Text":"Now, that equals to b times 5 minus 4 times b,"},{"Start":"01:18.785 ","End":"01:20.920","Text":"and that equals to b."},{"Start":"01:20.920 ","End":"01:27.440","Text":"Here, we have this point right here, and that\u0027s b."},{"Start":"01:27.440 ","End":"01:33.300","Text":"Now, that\u0027s the equation of a straight line."},{"Start":"01:33.300 ","End":"01:38.150","Text":"Let\u0027s just draw the straight line here between these 2 points."},{"Start":"01:38.150 ","End":"01:45.985","Text":"The straight line goes from 0 to b in the range between 4 and 5."},{"Start":"01:45.985 ","End":"01:49.820","Text":"Now, what happens in this range between 5 and 6?"},{"Start":"01:49.820 ","End":"01:52.460","Text":"Well that\u0027s just a constant b,"},{"Start":"01:52.460 ","End":"01:55.680","Text":"so it\u0027ll be like this."},{"Start":"01:56.120 ","End":"02:02.120","Text":"Now that we know what this function looks like, let\u0027s calculate b."},{"Start":"02:02.120 ","End":"02:05.930","Text":"Now, the first thing that we need to know is that"},{"Start":"02:05.930 ","End":"02:09.500","Text":"this is a density function and that\u0027s given to us obviously."},{"Start":"02:09.500 ","End":"02:14.180","Text":"The area under the density function has to equal to 1."},{"Start":"02:14.180 ","End":"02:19.715","Text":"Now, if we take a look at the shape of the density function,"},{"Start":"02:19.715 ","End":"02:21.985","Text":"well, this is a trapezoid."},{"Start":"02:21.985 ","End":"02:29.510","Text":"We can use the equation for the area of the trapezoid to calculate b,"},{"Start":"02:29.510 ","End":"02:31.190","Text":"so let\u0027s do that."},{"Start":"02:31.190 ","End":"02:38.015","Text":"The area of a trapezoid is the sum of the 2 bases times the height divided by 2."},{"Start":"02:38.015 ","End":"02:39.335","Text":"Let\u0027s just write this out."},{"Start":"02:39.335 ","End":"02:42.620","Text":"That\u0027s a1 plus a2,"},{"Start":"02:42.620 ","End":"02:48.195","Text":"the sum of the 2 bases times the height divided by 2."},{"Start":"02:48.195 ","End":"02:50.015","Text":"In our case, well,"},{"Start":"02:50.015 ","End":"02:53.690","Text":"the first base is 2, from 46,"},{"Start":"02:53.690 ","End":"02:57.550","Text":"so that\u0027s 2 plus 1,"},{"Start":"02:57.550 ","End":"03:00.590","Text":"that\u0027s the second base, times b,"},{"Start":"03:00.590 ","End":"03:03.770","Text":"that\u0027s the height divided by 2."},{"Start":"03:03.770 ","End":"03:06.710","Text":"That\u0027s the area of the trapezoid and that has to"},{"Start":"03:06.710 ","End":"03:10.135","Text":"equal to 1 because it\u0027s a density function."},{"Start":"03:10.135 ","End":"03:12.774","Text":"Let\u0027s just do a little bit of math here,"},{"Start":"03:12.774 ","End":"03:14.912","Text":"so that\u0027ll be 3b equals 2,"},{"Start":"03:14.912 ","End":"03:20.315","Text":"and that means that b equals 2/3."},{"Start":"03:20.315 ","End":"03:24.950","Text":"In this section, we\u0027re asked to calculate the expectation of x."},{"Start":"03:24.950 ","End":"03:31.550","Text":"The expectation of x of a continuous random variable x that equals"},{"Start":"03:31.550 ","End":"03:35.180","Text":"to the integral from minus infinity to"},{"Start":"03:35.180 ","End":"03:39.845","Text":"plus infinity of x times the density function of x d x."},{"Start":"03:39.845 ","End":"03:44.180","Text":"In our case, let\u0027s just take a look at the density function after we\u0027ve"},{"Start":"03:44.180 ","End":"03:49.605","Text":"calculated the value of b. Here it is right here."},{"Start":"03:49.605 ","End":"03:53.960","Text":"Let\u0027s continue and calculate the expectation."},{"Start":"03:53.960 ","End":"03:56.635","Text":"Well, that\u0027ll be the expectation of x,"},{"Start":"03:56.635 ","End":"03:58.865","Text":"that\u0027ll be equal to the integral."},{"Start":"03:58.865 ","End":"04:04.335","Text":"Now, let\u0027s take this range between 4 and 5 of"},{"Start":"04:04.335 ","End":"04:12.920","Text":"x times 2/3 x minus 8/3 dx plus the integral."},{"Start":"04:12.920 ","End":"04:14.735","Text":"Now let\u0027s take a look at this range."},{"Start":"04:14.735 ","End":"04:22.185","Text":"That\u0027s between 5 and 6 of 2/3 times x dx,"},{"Start":"04:22.185 ","End":"04:25.515","Text":"x times the density function of x."},{"Start":"04:25.515 ","End":"04:29.575","Text":"Now, let\u0027s just multiply this thing and right here,"},{"Start":"04:29.575 ","End":"04:38.910","Text":"that\u0027ll be the integral between 4 and 5 of 2/3 x squared minus 8/3 x,"},{"Start":"04:39.410 ","End":"04:49.300","Text":"dx plus the integral from 5-6 of 2/3 x dx."},{"Start":"04:49.400 ","End":"04:53.080","Text":"Let\u0027s now do the integrals."},{"Start":"04:53.540 ","End":"04:58.045","Text":"The integral of this expression right here, well,"},{"Start":"04:58.045 ","End":"05:04.925","Text":"that\u0027ll be equal to 2/3 x cubed divided by 3."},{"Start":"05:04.925 ","End":"05:07.555","Text":"The integral of this guy right here,"},{"Start":"05:07.555 ","End":"05:17.755","Text":"that\u0027ll be 8/3 x squared divided by 2 in the range from 4-5."},{"Start":"05:17.755 ","End":"05:20.275","Text":"That\u0027s this guy right here."},{"Start":"05:20.275 ","End":"05:23.575","Text":"Plus, now, what\u0027s the integral of this guy?"},{"Start":"05:23.575 ","End":"05:33.650","Text":"Well, that\u0027s 2/3 x squared divided by 2 in the range 5 and 6."},{"Start":"05:33.650 ","End":"05:37.970","Text":"Now, these guys cancel each other out, we\u0027re okay here."},{"Start":"05:37.970 ","End":"05:41.275","Text":"This becomes a 4."},{"Start":"05:41.275 ","End":"05:44.774","Text":"Let\u0027s now plug in the numbers."},{"Start":"05:44.774 ","End":"05:48.780","Text":"Well, when we plug in 5 into this expression,"},{"Start":"05:48.780 ","End":"05:55.440","Text":"we have 2/9 times 5 cubed minus"},{"Start":"05:55.440 ","End":"06:02.580","Text":"4/3 times 5 squared minus,"},{"Start":"06:02.580 ","End":"06:04.485","Text":"now let\u0027s plug in 4 here."},{"Start":"06:04.485 ","End":"06:09.360","Text":"Well, that\u0027s 2/9 times 4"},{"Start":"06:09.360 ","End":"06:18.220","Text":"cubed minus 4/3 times 4 squared."},{"Start":"06:18.950 ","End":"06:23.070","Text":"Plus, now this guy right here."},{"Start":"06:23.070 ","End":"06:28.215","Text":"Well, that will be 1/3 times"},{"Start":"06:28.215 ","End":"06:34.660","Text":"6 squared minus 1/3 times 5 squared."},{"Start":"06:36.440 ","End":"06:42.655","Text":"This seems like a tedious task for the calculator."},{"Start":"06:42.655 ","End":"06:44.740","Text":"But once we do that,"},{"Start":"06:44.740 ","End":"06:53.580","Text":"then we\u0027ll get the of 5.22."},{"Start":"06:53.580 ","End":"06:57.370","Text":"Now, that\u0027s the expectation of x"}],"ID":32264},{"Watched":false,"Name":"Exercise 6 - Part c","Duration":"3m 16s","ChapterTopicVideoID":30203,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.590","Text":"In this section, we\u0027re given that Y is an indicator variable that receives"},{"Start":"00:04.590 ","End":"00:09.981","Text":"the value 1 if X is less than 5 and we\u0027re asked what\u0027s the variance of Y?"},{"Start":"00:09.981 ","End":"00:13.845","Text":"Since Y is an indicator variable,"},{"Start":"00:13.845 ","End":"00:17.385","Text":"but Y can receive only 2 values?"},{"Start":"00:17.385 ","End":"00:20.520","Text":"It can receive the value of 0 or 1,"},{"Start":"00:20.520 ","End":"00:27.210","Text":"and it\u0027ll be equal to 1 only when Y meets a specific condition."},{"Start":"00:27.210 ","End":"00:30.720","Text":"Now, since we\u0027re looking for the variance of Y, well,"},{"Start":"00:30.720 ","End":"00:32.695","Text":"let\u0027s figure out, first of all,"},{"Start":"00:32.695 ","End":"00:34.970","Text":"the probability of Y."},{"Start":"00:34.970 ","End":"00:37.820","Text":"Now, what did we say?"},{"Start":"00:37.820 ","End":"00:42.260","Text":"We said that we want the probability of Y being equal to 1,"},{"Start":"00:42.260 ","End":"00:45.200","Text":"that means the Y has to meet a specific condition."},{"Start":"00:45.200 ","End":"00:49.645","Text":"That means that we\u0027re looking for the probability, what\u0027s the condition?"},{"Start":"00:49.645 ","End":"00:53.955","Text":"Now Y receives the value 1 if X is less than 5,"},{"Start":"00:53.955 ","End":"00:56.880","Text":"that means that X has to be less than 5."},{"Start":"00:56.880 ","End":"01:00.665","Text":"Now, the probability of X being less than 5, well,"},{"Start":"01:00.665 ","End":"01:04.410","Text":"that\u0027s the probability or"},{"Start":"01:04.410 ","End":"01:11.050","Text":"the area under the density function from 4 to 5."},{"Start":"01:11.050 ","End":"01:13.160","Text":"Now we can do an integral here,"},{"Start":"01:13.160 ","End":"01:19.685","Text":"but this is basically just trying to calculate the area of a triangle. Let\u0027s do that."},{"Start":"01:19.685 ","End":"01:21.080","Text":"Area of a triangle well,"},{"Start":"01:21.080 ","End":"01:29.975","Text":"that\u0027s 1/2 times the base times the height that equals to 1/2 times 1 times 2/3,"},{"Start":"01:29.975 ","End":"01:35.485","Text":"and that equals to 1/3."},{"Start":"01:35.485 ","End":"01:43.928","Text":"The probability of X being less than 5 equals the probability of Y being equal to 1,"},{"Start":"01:43.928 ","End":"01:45.920","Text":"and that equals to 1/3,"},{"Start":"01:45.920 ","End":"01:47.495","Text":"let\u0027s write that down here."},{"Start":"01:47.495 ","End":"01:50.000","Text":"Now, if that\u0027s the case,"},{"Start":"01:50.000 ","End":"01:56.755","Text":"then when Y equals 0 then the probability of Y equaling 0, that\u0027s 2/3."},{"Start":"01:56.755 ","End":"01:59.740","Text":"Now that we have this,"},{"Start":"01:59.740 ","End":"02:02.710","Text":"then we can go ahead and calculate the variance."},{"Start":"02:02.710 ","End":"02:03.940","Text":"Now before we do that,"},{"Start":"02:03.940 ","End":"02:07.510","Text":"let\u0027s first calculate the expectation of Y."},{"Start":"02:07.510 ","End":"02:12.745","Text":"Well, that equals to the sum of Y times the probability of Y."},{"Start":"02:12.745 ","End":"02:15.985","Text":"Now, we have to pay attention here,"},{"Start":"02:15.985 ","End":"02:19.240","Text":"we\u0027re looking at a discrete variable here."},{"Start":"02:19.240 ","End":"02:26.165","Text":"Y is a discrete variable that\u0027s based on a continuous variable X."},{"Start":"02:26.165 ","End":"02:29.710","Text":"This is something that we have to pay attention to."},{"Start":"02:29.710 ","End":"02:31.825","Text":"Now, let\u0027s just calculate this."},{"Start":"02:31.825 ","End":"02:37.835","Text":"This is 0 times 2/3 plus 1 times 1/3,"},{"Start":"02:37.835 ","End":"02:40.125","Text":"and that equals to 1/3."},{"Start":"02:40.125 ","End":"02:42.680","Text":"Now what about the variance of Y?"},{"Start":"02:42.680 ","End":"02:47.105","Text":"Well, that equals to the sum of Y squared times"},{"Start":"02:47.105 ","End":"02:52.520","Text":"the probability of Y minus the expectation squared of Y."},{"Start":"02:52.520 ","End":"02:57.950","Text":"Now, that will equal to 0 squared times 2/3"},{"Start":"02:57.950 ","End":"03:05.842","Text":"plus 1 squared times 1/3 minus 1/3 squared,"},{"Start":"03:05.842 ","End":"03:10.430","Text":"and that equals to 1/3 minus 1/9,"},{"Start":"03:10.430 ","End":"03:13.830","Text":"that equals to 2/9."},{"Start":"03:13.830 ","End":"03:17.020","Text":"That\u0027s the variance of Y."}],"ID":32265},{"Watched":false,"Name":"Exercise 7 - Part a","Duration":"1m 54s","ChapterTopicVideoID":30205,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.030","Text":"In this function, the following density function is given,"},{"Start":"00:03.030 ","End":"00:04.260","Text":"that\u0027s this thing right here,"},{"Start":"00:04.260 ","End":"00:07.650","Text":"and we\u0027re asked to calculate the value of k. Now,"},{"Start":"00:07.650 ","End":"00:10.110","Text":"we know that the integral of"},{"Start":"00:10.110 ","End":"00:16.990","Text":"the density function between minus infinity and plus infinity, well that equals to 1."},{"Start":"00:16.990 ","End":"00:20.670","Text":"If we take the integral of this function right here, what\u0027s given to us,"},{"Start":"00:20.670 ","End":"00:21.990","Text":"and we equate that to 1,"},{"Start":"00:21.990 ","End":"00:26.070","Text":"we\u0027ll be able to extract k. Let\u0027s do that."},{"Start":"00:26.070 ","End":"00:31.920","Text":"Well, that\u0027ll be the integral between 1 and 2 of x"},{"Start":"00:31.920 ","End":"00:37.820","Text":"squared divided by 4 dx plus that\u0027s the integral right now,"},{"Start":"00:37.820 ","End":"00:43.520","Text":"that\u0027s this guy right here between 2 and 3 of k times xdx,"},{"Start":"00:43.520 ","End":"00:46.085","Text":"and that\u0027ll be equal to 1."},{"Start":"00:46.085 ","End":"00:49.955","Text":"Now, let\u0027s take the integral of this expression."},{"Start":"00:49.955 ","End":"00:54.980","Text":"That\u0027ll be 1/4 times x cubed divided by 3,"},{"Start":"00:54.980 ","End":"00:57.755","Text":"in the range between 1 and 2,"},{"Start":"00:57.755 ","End":"01:03.470","Text":"plus k times x squared divided by 2,"},{"Start":"01:03.470 ","End":"01:05.840","Text":"in the range between 2 and 3,"},{"Start":"01:05.840 ","End":"01:07.730","Text":"and that equals 1."},{"Start":"01:07.730 ","End":"01:10.160","Text":"Now, let\u0027s just plug in the numbers."},{"Start":"01:10.160 ","End":"01:16.235","Text":"That\u0027ll be 8 divided by 12 minus 1 divided by 12,"},{"Start":"01:16.235 ","End":"01:24.380","Text":"plus k times 9 divided by 2 minus 4 divided by 2,"},{"Start":"01:24.380 ","End":"01:26.360","Text":"and that equals to 1."},{"Start":"01:26.360 ","End":"01:35.110","Text":"What we have here is 7/12 plus k times 5/2,"},{"Start":"01:35.110 ","End":"01:36.890","Text":"and that equals to 1."},{"Start":"01:36.890 ","End":"01:38.840","Text":"Let\u0027s just switch sides."},{"Start":"01:38.840 ","End":"01:43.100","Text":"That\u0027ll be 5 over 2 times k,"},{"Start":"01:43.100 ","End":"01:46.970","Text":"that\u0027ll be equal to 5/12."},{"Start":"01:46.970 ","End":"01:54.720","Text":"That means that k now will be equal to 1/6. So that\u0027s the value of k."}],"ID":32266},{"Watched":false,"Name":"Exercise 7 - Parts b-c","Duration":"7m 7s","ChapterTopicVideoID":30206,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.530","Text":"In this section, we\u0027re asked to calculate the cumulative distribution function of x,"},{"Start":"00:04.530 ","End":"00:07.740","Text":"where the density function of x is this."},{"Start":"00:07.740 ","End":"00:15.810","Text":"We\u0027re looking for the probability of x being less than or equal to t or F at t. Now,"},{"Start":"00:15.810 ","End":"00:17.385","Text":"before we can do that,"},{"Start":"00:17.385 ","End":"00:21.610","Text":"let\u0027s just see what this function looks like."},{"Start":"00:21.680 ","End":"00:26.745","Text":"Here we go. Well, in this range right here,"},{"Start":"00:26.745 ","End":"00:29.565","Text":"the density function is x squared over 4."},{"Start":"00:29.565 ","End":"00:34.080","Text":"Now, this is a parabola and it goes through the origin."},{"Start":"00:34.080 ","End":"00:36.235","Text":"It looks something like this right here."},{"Start":"00:36.235 ","End":"00:37.985","Text":"It goes through the origin."},{"Start":"00:37.985 ","End":"00:42.920","Text":"But x is defined between 1 and 2 only."},{"Start":"00:42.920 ","End":"00:46.195","Text":"Let\u0027s just plug in 1,"},{"Start":"00:46.195 ","End":"00:49.160","Text":"the lower bound of the range into this formula right"},{"Start":"00:49.160 ","End":"00:52.060","Text":"here and see what the value of the density function is."},{"Start":"00:52.060 ","End":"00:55.040","Text":"Well, that\u0027s 1 squared over 4, that\u0027s a quarter."},{"Start":"00:55.040 ","End":"00:56.975","Text":"That means that 1,"},{"Start":"00:56.975 ","End":"00:59.440","Text":"the density function equals a quarter."},{"Start":"00:59.440 ","End":"01:01.000","Text":"What about 2?"},{"Start":"01:01.000 ","End":"01:05.300","Text":"Well, if we plug in 2 into this formula right here, this equation,"},{"Start":"01:05.300 ","End":"01:07.280","Text":"well, that\u0027ll be 2 squared over 4,"},{"Start":"01:07.280 ","End":"01:10.745","Text":"that\u0027s 4 over 4, that equals to 1."},{"Start":"01:10.745 ","End":"01:12.905","Text":"Now since this is a parabola,"},{"Start":"01:12.905 ","End":"01:17.340","Text":"the function would look something like this."},{"Start":"01:17.590 ","End":"01:22.930","Text":"Now let\u0027s take a look at the next range between 2 and 3."},{"Start":"01:22.930 ","End":"01:25.260","Text":"Well, at 2,"},{"Start":"01:25.260 ","End":"01:29.660","Text":"the value of the function in this range equals 1/3,"},{"Start":"01:29.660 ","End":"01:33.050","Text":"6 times 2, that\u0027s 1/3."},{"Start":"01:33.050 ","End":"01:38.255","Text":"But 2 belongs to this range right here so at 2 here,"},{"Start":"01:38.255 ","End":"01:43.190","Text":"we\u0027ll just draw an open point right here. What about at 3?"},{"Start":"01:43.190 ","End":"01:44.694","Text":"Well, at 3, well,"},{"Start":"01:44.694 ","End":"01:46.565","Text":"that\u0027s 1/6 times 3,"},{"Start":"01:46.565 ","End":"01:48.955","Text":"that equals to a 1/2."},{"Start":"01:48.955 ","End":"01:52.215","Text":"Now since this is a straight line,"},{"Start":"01:52.215 ","End":"01:55.865","Text":"the density function would look something like this."},{"Start":"01:55.865 ","End":"01:58.745","Text":"This is our density function."},{"Start":"01:58.745 ","End":"02:04.715","Text":"Let\u0027s now construct that cumulative distribution function,"},{"Start":"02:04.715 ","End":"02:08.050","Text":"that\u0027s F at t and that equals 2."},{"Start":"02:08.050 ","End":"02:15.125","Text":"Now, we\u0027ll divide this into 4 ranges where t is less than 1,"},{"Start":"02:15.125 ","End":"02:19.620","Text":"where t is between 1 and 2,"},{"Start":"02:19.620 ","End":"02:23.880","Text":"where t is between 2 and 3,"},{"Start":"02:23.880 ","End":"02:26.965","Text":"and where t is greater than 3."},{"Start":"02:26.965 ","End":"02:32.899","Text":"Now, let\u0027s take a look at the values of the CDF in each 1 of the ranges."},{"Start":"02:32.899 ","End":"02:36.455","Text":"Well, in this range right here where t is less than 1,"},{"Start":"02:36.455 ","End":"02:39.290","Text":"the value here is 0. Now, why is that?"},{"Start":"02:39.290 ","End":"02:46.280","Text":"Because the density function is defined from 1 onwards until 3."},{"Start":"02:46.280 ","End":"02:53.845","Text":"There\u0027s no area under the density function in the ranges of minus infinity to 1."},{"Start":"02:53.845 ","End":"02:59.600","Text":"We haven\u0027t accumulated any area whatsoever under the density function."},{"Start":"02:59.600 ","End":"03:02.210","Text":"Let\u0027s take a look at this range right here,"},{"Start":"03:02.210 ","End":"03:04.595","Text":"where t is greater than 3."},{"Start":"03:04.595 ","End":"03:07.850","Text":"For any value of x above 3,"},{"Start":"03:07.850 ","End":"03:14.905","Text":"we\u0027ve actually accumulated all of the area underneath the density function."},{"Start":"03:14.905 ","End":"03:19.715","Text":"Here, the value of the CDF equals 1."},{"Start":"03:19.715 ","End":"03:24.170","Text":"Now let\u0027s take a look at the expression of the CDF in"},{"Start":"03:24.170 ","End":"03:29.490","Text":"these ranges right here between 1 and 2 and between 2 and 3."},{"Start":"03:29.660 ","End":"03:32.505","Text":"In the range between 1 and 2,"},{"Start":"03:32.505 ","End":"03:33.936","Text":"let\u0027s just pick a point,"},{"Start":"03:33.936 ","End":"03:37.530","Text":"we\u0027ll call it t. What we want to do is we want to"},{"Start":"03:37.530 ","End":"03:42.645","Text":"find the area underneath the density function."},{"Start":"03:42.645 ","End":"03:51.440","Text":"Well, that\u0027s the integral between 1 and t of this expression right here."},{"Start":"03:51.440 ","End":"03:55.685","Text":"1/4 times x squared dx."},{"Start":"03:55.685 ","End":"04:02.960","Text":"Well, that equals to 1/4 times x cubed divided by 3 in"},{"Start":"04:02.960 ","End":"04:12.090","Text":"the range between 1 and t. That equals to t cubed minus 1 divided by 12."},{"Start":"04:12.090 ","End":"04:17.150","Text":"This is the expression that we\u0027ll put here in this range right here."},{"Start":"04:17.150 ","End":"04:18.620","Text":"That will be again,"},{"Start":"04:18.620 ","End":"04:22.800","Text":"t cubed minus 1 divided by 12."},{"Start":"04:23.000 ","End":"04:27.050","Text":"Let\u0027s take a look now at this range right here,"},{"Start":"04:27.050 ","End":"04:29.030","Text":"between 2 and 3."},{"Start":"04:29.030 ","End":"04:31.580","Text":"In this range right here, again,"},{"Start":"04:31.580 ","End":"04:36.250","Text":"we\u0027ll pick a point t and what we want to do is we want to"},{"Start":"04:36.250 ","End":"04:42.055","Text":"calculate the area under the density function from 1 to t. Now,"},{"Start":"04:42.055 ","End":"04:47.800","Text":"we can do that by calculating the area between 1 and 2 and then from 2 to"},{"Start":"04:47.800 ","End":"04:55.135","Text":"t. Now the area underneath the density function from 1 to 2,"},{"Start":"04:55.135 ","End":"04:57.190","Text":"well, that\u0027s easy."},{"Start":"04:57.190 ","End":"05:01.165","Text":"All we have to do is take the upper bound of this range,"},{"Start":"05:01.165 ","End":"05:04.315","Text":"that\u0027s 2 and plug it into this formula right here."},{"Start":"05:04.315 ","End":"05:07.695","Text":"That\u0027ll be 2 cubed minus 1 over 12,"},{"Start":"05:07.695 ","End":"05:12.630","Text":"that equals to 8 minus 1 over 12, or 7/12."},{"Start":"05:12.630 ","End":"05:17.690","Text":"This area right here is equal to 7 divided by 12,"},{"Start":"05:17.690 ","End":"05:20.390","Text":"this area right here."},{"Start":"05:20.390 ","End":"05:24.455","Text":"Now, what about this area right here?"},{"Start":"05:24.455 ","End":"05:33.050","Text":"Well, let\u0027s take the integral from 2 to t of this expression right here."},{"Start":"05:33.050 ","End":"05:35.915","Text":"That\u0027s 1/6 times x dx."},{"Start":"05:35.915 ","End":"05:44.505","Text":"Well, that equals to 1/6 times x squared divided by 2 between 2 and"},{"Start":"05:44.505 ","End":"05:54.580","Text":"t. That equals to t squared minus 4 divided by 12."},{"Start":"05:54.890 ","End":"05:59.950","Text":"The expression here would be"},{"Start":"06:02.360 ","End":"06:11.985","Text":"7/12 plus t squared minus 4 divided by 12."},{"Start":"06:11.985 ","End":"06:19.025","Text":"This then is the cumulative distribution function of x."},{"Start":"06:19.025 ","End":"06:25.325","Text":"In this section, we\u0027re asked to calculate the probability of x being greater than 2.5."},{"Start":"06:25.325 ","End":"06:30.170","Text":"The probability of x being greater than 2.5,"},{"Start":"06:30.170 ","End":"06:35.285","Text":"that equals to 1 minus the probability of x being less than or equal to"},{"Start":"06:35.285 ","End":"06:41.220","Text":"2.5 or 1 minus F at 2.5."},{"Start":"06:41.220 ","End":"06:46.545","Text":"Now, 2.5 is in this range right here so we\u0027ll use this expression."},{"Start":"06:46.545 ","End":"06:51.525","Text":"That will equal to 1 minus, now this expression,"},{"Start":"06:51.525 ","End":"06:58.229","Text":"7 divided by 12 plus 2.5 squared minus 4"},{"Start":"06:58.229 ","End":"07:06.460","Text":"divided by 12 and that equals to 0.229."}],"ID":32267},{"Watched":false,"Name":"Exercise 8","Duration":"8m 55s","ChapterTopicVideoID":30207,"CourseChapterTopicPlaylistID":296881,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.500","Text":"In this question, we\u0027re given"},{"Start":"00:01.500 ","End":"00:04.920","Text":"a random variable X that has the following density function,"},{"Start":"00:04.920 ","End":"00:08.655","Text":"f(x) equals 1 divided by b minus a,"},{"Start":"00:08.655 ","End":"00:11.790","Text":"where X is defined between a and b."},{"Start":"00:11.790 ","End":"00:16.645","Text":"We\u0027re asked to calculate the cumulative distribution function of X."},{"Start":"00:16.645 ","End":"00:18.650","Text":"Before we do that,"},{"Start":"00:18.650 ","End":"00:25.115","Text":"let\u0027s just take a look at what this function looks like. Here it is."},{"Start":"00:25.115 ","End":"00:28.340","Text":"Now X is defined between a and b."},{"Start":"00:28.340 ","End":"00:32.120","Text":"This must be a and this must be b."},{"Start":"00:32.120 ","End":"00:38.120","Text":"Now, the value of the density function is a constant and it equals to"},{"Start":"00:38.120 ","End":"00:45.400","Text":"1 divided by b minus a for all values of X between a and b."},{"Start":"00:45.400 ","End":"00:50.360","Text":"We\u0027re looking now to calculate the CDF."},{"Start":"00:50.360 ","End":"00:54.320","Text":"That\u0027s the probability of X being less than"},{"Start":"00:54.320 ","End":"00:58.680","Text":"or equal to t. That means that we\u0027re looking at f(t)."},{"Start":"00:58.680 ","End":"01:01.750","Text":"Now, in order to calculate f and t,"},{"Start":"01:01.750 ","End":"01:06.410","Text":"we\u0027re going to have to divide the function into several ranges."},{"Start":"01:06.410 ","End":"01:10.950","Text":"The first range is where t is less than a."},{"Start":"01:11.590 ","End":"01:17.780","Text":"The second is where t is between a and b."},{"Start":"01:17.780 ","End":"01:21.815","Text":"The third is where t is greater than b."},{"Start":"01:21.815 ","End":"01:27.440","Text":"Now, when t is less than a what\u0027s the value of the CDF?"},{"Start":"01:27.440 ","End":"01:29.810","Text":"Well, that\u0027s 0. Why is that?"},{"Start":"01:29.810 ","End":"01:32.480","Text":"Because we haven\u0027t accumulated any area under"},{"Start":"01:32.480 ","End":"01:38.420","Text":"the density function because the density function is defined between a and b."},{"Start":"01:38.420 ","End":"01:42.470","Text":"Now, what happens when t is greater than b?"},{"Start":"01:42.470 ","End":"01:45.095","Text":"That means we\u0027re in this range right here."},{"Start":"01:45.095 ","End":"01:53.360","Text":"Well, that means that we\u0027ve accumulated all of the area underneath the density function."},{"Start":"01:53.360 ","End":"01:57.920","Text":"That means that the cumulative distribution function well that equals 1."},{"Start":"01:57.920 ","End":"02:00.725","Text":"Now, what happens in this range?"},{"Start":"02:00.725 ","End":"02:05.480","Text":"What\u0027s the expression of the cumulative distribution function in this range?"},{"Start":"02:05.480 ","End":"02:08.000","Text":"In order to calculate that,"},{"Start":"02:08.000 ","End":"02:10.475","Text":"let\u0027s just define a point here,"},{"Start":"02:10.475 ","End":"02:14.120","Text":"t. What we want to do is we want to calculate"},{"Start":"02:14.120 ","End":"02:19.835","Text":"the area under the density function from a to t. Well,"},{"Start":"02:19.835 ","End":"02:21.050","Text":"we can use an integral,"},{"Start":"02:21.050 ","End":"02:26.330","Text":"but this is just a rectangle and calculating the area of a rectangle is simple."},{"Start":"02:26.330 ","End":"02:28.565","Text":"That\u0027s just the base times the height."},{"Start":"02:28.565 ","End":"02:31.715","Text":"Well, the base here, t minus a,"},{"Start":"02:31.715 ","End":"02:37.780","Text":"and the height is 1 divided by b minus a."},{"Start":"02:37.780 ","End":"02:41.300","Text":"This expression here, that would be"},{"Start":"02:41.300 ","End":"02:45.530","Text":"the expression for the cumulative distribution function in this range."},{"Start":"02:45.530 ","End":"02:46.760","Text":"Let\u0027s write it down."},{"Start":"02:46.760 ","End":"02:52.675","Text":"That\u0027ll be t minus a divided by b minus a."},{"Start":"02:52.675 ","End":"02:57.300","Text":"Here we have the cumulative distribution function."},{"Start":"02:57.580 ","End":"03:00.290","Text":"In this section, we\u0027re asked to calculate"},{"Start":"03:00.290 ","End":"03:04.570","Text":"the expectation variance of the probability distribution."},{"Start":"03:04.570 ","End":"03:09.230","Text":"The definition of the expectation of a continuous random variable,"},{"Start":"03:09.230 ","End":"03:13.295","Text":"well that\u0027s the integral from minus infinity to plus infinity"},{"Start":"03:13.295 ","End":"03:18.290","Text":"of X times the density function of x dx."},{"Start":"03:18.290 ","End":"03:22.085","Text":"Now, we can use this to calculate the expectation,"},{"Start":"03:22.085 ","End":"03:27.860","Text":"or we can use the characteristic of a symmetrical distribution."},{"Start":"03:27.860 ","End":"03:30.140","Text":"Here we have a symmetrical distribution."},{"Start":"03:30.140 ","End":"03:31.730","Text":"Now, what\u0027s the characteristic?"},{"Start":"03:31.730 ","End":"03:36.860","Text":"We say that the expectation of X,"},{"Start":"03:36.860 ","End":"03:40.340","Text":"in a symmetrical distribution is right in the middle of"},{"Start":"03:40.340 ","End":"03:45.350","Text":"the range where the density function is defined."},{"Start":"03:45.350 ","End":"03:48.470","Text":"In this case, that\u0027ll be right here,"},{"Start":"03:48.470 ","End":"03:51.770","Text":"between, in the middle, between a and b."},{"Start":"03:51.770 ","End":"03:54.420","Text":"Now, what\u0027s this point right here?"},{"Start":"03:54.420 ","End":"03:56.395","Text":"That\u0027ll be the expectation."},{"Start":"03:56.395 ","End":"04:00.525","Text":"That\u0027ll be a plus b divided by 2."},{"Start":"04:00.525 ","End":"04:05.710","Text":"That\u0027s the expectation of x."},{"Start":"04:06.050 ","End":"04:09.530","Text":"Now let\u0027s take a look at the variance of X."},{"Start":"04:09.530 ","End":"04:13.220","Text":"Well, the variance of a continuous random variable X is"},{"Start":"04:13.220 ","End":"04:16.760","Text":"defined as the integral from minus infinity to plus"},{"Start":"04:16.760 ","End":"04:19.670","Text":"infinity of X squared times"},{"Start":"04:19.670 ","End":"04:25.900","Text":"the density function of x dx minus the expectation squared of x."},{"Start":"04:25.900 ","End":"04:31.400","Text":"Let\u0027s, first of all, take a look at this expression right here."},{"Start":"04:31.400 ","End":"04:34.025","Text":"That\u0027s the expectation of X squared."},{"Start":"04:34.025 ","End":"04:39.755","Text":"Now, in our case,"},{"Start":"04:39.755 ","End":"04:46.790","Text":"that would equal to the integral between a and b of X squared times the density function,"},{"Start":"04:46.790 ","End":"04:52.540","Text":"which is 1 minus b minus 1 over b minus a dx."},{"Start":"04:52.540 ","End":"05:01.990","Text":"Now that equals to 1 divided by b minus a times X cubed divided by 3 in"},{"Start":"05:01.990 ","End":"05:06.070","Text":"the range between a and b and that equals to b cubed"},{"Start":"05:06.070 ","End":"05:12.295","Text":"minus a cubed divided by 3 times b minus a."},{"Start":"05:12.295 ","End":"05:17.305","Text":"Now that we know what the expectation of X squared is,"},{"Start":"05:17.305 ","End":"05:19.210","Text":"let\u0027s get back to our variance."},{"Start":"05:19.210 ","End":"05:23.173","Text":"Now the variance of X would be equal to this expression,"},{"Start":"05:23.173 ","End":"05:30.790","Text":"that\u0027s b cubed minus a cubed divided by 3 times b minus a minus the expectation squared."},{"Start":"05:30.790 ","End":"05:39.940","Text":"Well, that\u0027s a plus b divided by 2 squared."},{"Start":"05:39.940 ","End":"05:42.524","Text":"Now, let\u0027s simplify that."},{"Start":"05:42.524 ","End":"05:45.815","Text":"Now, what\u0027s b cubed minus a cubed,"},{"Start":"05:45.815 ","End":"05:54.740","Text":"that\u0027s b minus a times b squared plus ab plus a"},{"Start":"05:54.740 ","End":"06:04.760","Text":"squared divided by 3 times b minus a minus this guy,"},{"Start":"06:04.760 ","End":"06:11.285","Text":"which is a plus b squared divided by 4."},{"Start":"06:11.285 ","End":"06:14.560","Text":"Let\u0027s just multiply things out."},{"Start":"06:14.560 ","End":"06:19.010","Text":"This expression cancels this expression out."},{"Start":"06:19.010 ","End":"06:23.735","Text":"We\u0027ll take a common denominator and we\u0027ll write this out like this."},{"Start":"06:23.735 ","End":"06:33.965","Text":"Now, this equals to 4 times b squared plus 4ab plus"},{"Start":"06:33.965 ","End":"06:38.720","Text":"4a squared minus 3a"},{"Start":"06:38.720 ","End":"06:46.695","Text":"squared minus 6ab minus 3b squared."},{"Start":"06:46.695 ","End":"06:51.295","Text":"All this is divided by 12."},{"Start":"06:51.295 ","End":"06:54.020","Text":"Now, let\u0027s simplify this."},{"Start":"06:54.020 ","End":"06:59.480","Text":"That will equal to b squared minus"},{"Start":"06:59.480 ","End":"07:06.225","Text":"2ab plus a squared divided by 12."},{"Start":"07:06.225 ","End":"07:15.790","Text":"This equals to a minus b squared divided by 12."},{"Start":"07:15.790 ","End":"07:24.740","Text":"Class, we\u0027ve gone to the end of our calculation and this then is the variance of X."},{"Start":"07:24.740 ","End":"07:29.375","Text":"In this section, we\u0027re asked to find the expectation of 1 over X."},{"Start":"07:29.375 ","End":"07:36.170","Text":"Now if you recall the expectation of any function of x that equals to the"},{"Start":"07:36.170 ","End":"07:40.069","Text":"integral from minus infinity to plus infinity"},{"Start":"07:40.069 ","End":"07:45.005","Text":"of that function times the density function of x dx."},{"Start":"07:45.005 ","End":"07:47.150","Text":"Now, in our case,"},{"Start":"07:47.150 ","End":"07:52.310","Text":"we want the expectation of 1 over X and that equals to the integral from"},{"Start":"07:52.310 ","End":"08:00.035","Text":"a to b of1 over X times 1 divided by b minus a dx,"},{"Start":"08:00.035 ","End":"08:04.010","Text":"and that equals to 1 divided by b minus a."},{"Start":"08:04.010 ","End":"08:11.525","Text":"Now the integral of 1 over X is lone X in the range between a and b."},{"Start":"08:11.525 ","End":"08:18.030","Text":"That will equal to ln b"},{"Start":"08:18.030 ","End":"08:24.570","Text":"minus ln a divided by b minus a."},{"Start":"08:24.570 ","End":"08:29.160","Text":"Now, we can write this out as ln of"},{"Start":"08:29.160 ","End":"08:36.485","Text":"b divided by a over b minus a."},{"Start":"08:36.485 ","End":"08:41.555","Text":"Now, the thing that we have to remember to add is that a,"},{"Start":"08:41.555 ","End":"08:45.695","Text":"both a and b both have to be greater than 0."},{"Start":"08:45.695 ","End":"08:49.940","Text":"Why is that? Because their arguments for the ln operator"},{"Start":"08:49.940 ","End":"08:55.380","Text":"and the ln operator takes only positive values."}],"ID":32268}],"Thumbnail":null,"ID":296881},{"Name":"Normal Probability","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"8m 1s","ChapterTopicVideoID":13266,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"In this chapter we\u0027ll be talking about special continuous probability,"},{"Start":"00:03.240 ","End":"00:05.730","Text":"specifically the normal distribution."},{"Start":"00:05.730 ","End":"00:08.340","Text":"When we\u0027re dealing with a normal distribution,"},{"Start":"00:08.340 ","End":"00:11.040","Text":"we\u0027re dealing with continuous random variables."},{"Start":"00:11.040 ","End":"00:12.765","Text":"Just as a reminder,"},{"Start":"00:12.765 ","End":"00:14.790","Text":"continuous random variable can have"},{"Start":"00:14.790 ","End":"00:18.150","Text":"an infinite number of values within a specified range."},{"Start":"00:18.150 ","End":"00:21.975","Text":"For example, if a random variable is, let\u0027s say,"},{"Start":"00:21.975 ","End":"00:29.280","Text":"a height of a person and we\u0027re dealing with a range between 810 and 181 centimeter."},{"Start":"00:29.280 ","End":"00:35.980","Text":"Within that 1 centimeter range we can have an infinite number of values."},{"Start":"00:35.980 ","End":"00:37.640","Text":"Now, as we said,"},{"Start":"00:37.640 ","End":"00:41.150","Text":"this is a common distribution."},{"Start":"00:41.150 ","End":"00:45.592","Text":"The variables that have a normal distribution can describe many things,"},{"Start":"00:45.592 ","End":"00:48.845","Text":"production times, babies weight,"},{"Start":"00:48.845 ","End":"00:52.290","Text":"height, and so on and so forth."},{"Start":"00:52.460 ","End":"00:57.845","Text":"The density function of the normal probability looks like a bell."},{"Start":"00:57.845 ","End":"01:00.095","Text":"It looks like this right here."},{"Start":"01:00.095 ","End":"01:05.945","Text":"Now, this curve then may be called either a bell curve or a Gauss curve."},{"Start":"01:05.945 ","End":"01:08.030","Text":"I\u0027ll be using the term bell curve."},{"Start":"01:08.030 ","End":"01:12.800","Text":"These curves are distinguished from each other by their averages and standard deviation."},{"Start":"01:12.800 ","End":"01:16.030","Text":"These are the parameters that define the distribution."},{"Start":"01:16.030 ","End":"01:17.300","Text":"As we can see here,"},{"Start":"01:17.300 ","End":"01:20.660","Text":"if we have our random variable X and that\u0027s"},{"Start":"01:20.660 ","End":"01:24.455","Text":"the X-axis and we can see here various bell curves."},{"Start":"01:24.455 ","End":"01:26.495","Text":"We have tall thin ones,"},{"Start":"01:26.495 ","End":"01:28.205","Text":"we have short fat ones,"},{"Start":"01:28.205 ","End":"01:33.255","Text":"and they\u0027re all defined by their averages Mu,"},{"Start":"01:33.255 ","End":"01:36.360","Text":"and standard deviation, Sigma."},{"Start":"01:36.360 ","End":"01:40.010","Text":"If we say that we have a random variable X,"},{"Start":"01:40.010 ","End":"01:41.720","Text":"we say that it\u0027s distributed with"},{"Start":"01:41.720 ","End":"01:46.070","Text":"a normal distribution with Mu and Sigma squared, where again,"},{"Start":"01:46.070 ","End":"01:52.640","Text":"Mu is the average and Sigma squared is the standard deviation squared or the variance."},{"Start":"01:52.640 ","End":"01:56.690","Text":"Now presented here also is the equation of the density function."},{"Start":"01:56.690 ","End":"01:59.825","Text":"We won\u0027t go into that right now in this chapter."},{"Start":"01:59.825 ","End":"02:03.650","Text":"In order to calculate probabilities with the normal distribution,"},{"Start":"02:03.650 ","End":"02:07.085","Text":"it\u0027s necessary to calculate the relevant areas under the curve."},{"Start":"02:07.085 ","End":"02:08.495","Text":"Now we\u0027ve seen this in"},{"Start":"02:08.495 ","End":"02:12.380","Text":"previous distributions where we wanted to calculate the probability"},{"Start":"02:12.380 ","End":"02:18.065","Text":"that some random variable X is less than or equal to a specific value a."},{"Start":"02:18.065 ","End":"02:19.800","Text":"Now, how did we do that?"},{"Start":"02:19.800 ","End":"02:25.790","Text":"Well, we took the density function of X and we\u0027ve calculated the area"},{"Start":"02:25.790 ","End":"02:32.515","Text":"under the density function from minus infinity till this value a."},{"Start":"02:32.515 ","End":"02:34.670","Text":"We\u0027ll do the same thing here."},{"Start":"02:34.670 ","End":"02:37.310","Text":"Now in order to calculate these areas,"},{"Start":"02:37.310 ","End":"02:40.100","Text":"we need to transform any normal distribution to"},{"Start":"02:40.100 ","End":"02:43.495","Text":"a standard normal distribution. Why is that?"},{"Start":"02:43.495 ","End":"02:49.970","Text":"Again, we\u0027ve seen that the bell curves can come in many shapes and sizes,"},{"Start":"02:49.970 ","End":"02:53.810","Text":"tall thin ones, short fat ones and so on and so forth."},{"Start":"02:53.810 ","End":"02:59.720","Text":"It\u0027s very complicated to calculate the areas under these types of curves."},{"Start":"02:59.720 ","End":"03:06.920","Text":"Now if we transform these density functions into a standardized density function,"},{"Start":"03:06.920 ","End":"03:10.790","Text":"then it\u0027ll be very easy for us to calculate the area."},{"Start":"03:10.790 ","End":"03:18.680","Text":"Now, this transformation, or this process is known as standardization or normalization."},{"Start":"03:18.680 ","End":"03:21.185","Text":"These 2 terms are interchangeable."},{"Start":"03:21.185 ","End":"03:26.240","Text":"Now, the standard normal distribution after transformation is"},{"Start":"03:26.240 ","End":"03:29.000","Text":"just a regular normal distribution with"},{"Start":"03:29.000 ","End":"03:35.315","Text":"the exceptions that the average is 0 and the standard deviation is 1."},{"Start":"03:35.315 ","End":"03:37.475","Text":"Now it\u0027ll be denoted as Z."},{"Start":"03:37.475 ","End":"03:39.740","Text":"That means that if we take X,"},{"Start":"03:39.740 ","End":"03:44.405","Text":"and after the transformation or the standardization,"},{"Start":"03:44.405 ","End":"03:46.190","Text":"it will become Z."},{"Start":"03:46.190 ","End":"03:50.270","Text":"Z will be distributed with"},{"Start":"03:50.270 ","End":"03:57.790","Text":"a normal distribution where Mu equals 0 and Sigma squared equals 1 squared."},{"Start":"03:57.790 ","End":"04:00.800","Text":"We talked about the standardization process,"},{"Start":"04:00.800 ","End":"04:02.915","Text":"but how do we actually do it?"},{"Start":"04:02.915 ","End":"04:07.775","Text":"Well, the standardization process is carried out using the following formula."},{"Start":"04:07.775 ","End":"04:12.725","Text":"We have X, a random variable, we subtract Mu,"},{"Start":"04:12.725 ","End":"04:16.340","Text":"our average, and we divide by Sigma,"},{"Start":"04:16.340 ","End":"04:18.905","Text":"that\u0027s the standard deviation."},{"Start":"04:18.905 ","End":"04:21.740","Text":"This transformation we\u0027ll call that Z."},{"Start":"04:21.740 ","End":"04:30.680","Text":"That some schools will teach you that Z equals to X minus X bar divided by S,"},{"Start":"04:30.680 ","End":"04:36.740","Text":"where X-bar is comparable to Mu and S is comparable to Sigma,"},{"Start":"04:36.740 ","End":"04:39.155","Text":"the average and standard deviation."},{"Start":"04:39.155 ","End":"04:41.300","Text":"They\u0027re comparable."},{"Start":"04:41.300 ","End":"04:44.300","Text":"You can choose to use either 1."},{"Start":"04:44.300 ","End":"04:47.695","Text":"I prefer to use Mu and Sigma."},{"Start":"04:47.695 ","End":"04:53.800","Text":"After standardization we get a value called the standard score."},{"Start":"04:53.800 ","End":"04:57.785","Text":"Once Z receives a value after this transformation,"},{"Start":"04:57.785 ","End":"05:00.755","Text":"that\u0027s called the standard score. We\u0027ll call that Z."},{"Start":"05:00.755 ","End":"05:05.150","Text":"Now the standard score refers to the number of standard deviations"},{"Start":"05:05.150 ","End":"05:10.910","Text":"the value X deviates from the average, from Mu."},{"Start":"05:11.060 ","End":"05:16.820","Text":"After calculating the standard score of a given value,"},{"Start":"05:16.820 ","End":"05:22.565","Text":"we use a table of the standard normal probability to calculate the desired area."},{"Start":"05:22.565 ","End":"05:26.570","Text":"Basically, this makes things a whole lot easier. Why is that?"},{"Start":"05:26.570 ","End":"05:30.650","Text":"Because once I\u0027ve transformed X that has"},{"Start":"05:30.650 ","End":"05:33.930","Text":"any normal distribution with any Mu and any"},{"Start":"05:33.930 ","End":"05:37.550","Text":"Sigma into a standardized normal distribution,"},{"Start":"05:37.550 ","End":"05:44.234","Text":"where the normal distribution has Mu equaling 0 and Sigma equaling 1,"},{"Start":"05:44.234 ","End":"05:48.110","Text":"then we have a table that already calculates for"},{"Start":"05:48.110 ","End":"05:52.765","Text":"us the area underneath the density function."},{"Start":"05:52.765 ","End":"05:54.725","Text":"Why not use that?"},{"Start":"05:54.725 ","End":"05:56.820","Text":"It\u0027ll make things a whole lot easier."},{"Start":"05:56.820 ","End":"06:01.595","Text":"Let\u0027s just take a peak at what this table looks like."},{"Start":"06:01.595 ","End":"06:05.055","Text":"Here we can see the actual table."},{"Start":"06:05.055 ","End":"06:08.170","Text":"But let\u0027s first look at this density function."},{"Start":"06:08.170 ","End":"06:09.730","Text":"We have here the density function,"},{"Start":"06:09.730 ","End":"06:14.035","Text":"the normal standard normal probability distribution or density function."},{"Start":"06:14.035 ","End":"06:18.075","Text":"We have here our standard score, that\u0027s Z."},{"Start":"06:18.075 ","End":"06:22.230","Text":"Here, we have 0 that\u0027s our Mu."},{"Start":"06:22.230 ","End":"06:25.230","Text":"The grayed-out area right here,"},{"Start":"06:25.230 ","End":"06:27.624","Text":"that\u0027s the area underneath"},{"Start":"06:27.624 ","End":"06:33.280","Text":"the density function from minus infinity until z or standard score."},{"Start":"06:33.280 ","End":"06:36.025","Text":"This grayed-out area right here,"},{"Start":"06:36.025 ","End":"06:41.415","Text":"that\u0027s exactly what\u0027s calculated in this table right here."},{"Start":"06:41.415 ","End":"06:45.805","Text":"This is a table that we\u0027ll be using shortly when we do the exercises."},{"Start":"06:45.805 ","End":"06:51.800","Text":"This is the table and we\u0027ll use it in a bit."},{"Start":"06:51.800 ","End":"06:56.825","Text":"Now, all this process might seem a little bit complicated, really isn\u0027t."},{"Start":"06:56.825 ","End":"07:00.095","Text":"But let\u0027s just review what we\u0027ve learned to now."},{"Start":"07:00.095 ","End":"07:05.150","Text":"In general, we describe the standardization process in the following schematic diagram."},{"Start":"07:05.150 ","End":"07:08.180","Text":"First of all, we have X that\u0027s distributed with"},{"Start":"07:08.180 ","End":"07:12.035","Text":"a normal distribution with Mu and Sigma squared."},{"Start":"07:12.035 ","End":"07:14.510","Text":"The next thing that we want to do,"},{"Start":"07:14.510 ","End":"07:17.785","Text":"we want to transform this,"},{"Start":"07:17.785 ","End":"07:21.990","Text":"we want to standardize X into Z and how do we do that?"},{"Start":"07:21.990 ","End":"07:25.535","Text":"We have X, we subtract Mu,"},{"Start":"07:25.535 ","End":"07:28.790","Text":"and we divide by Sigma and getting Z."},{"Start":"07:28.790 ","End":"07:33.320","Text":"Now Z is distributed with a normal distribution,"},{"Start":"07:33.320 ","End":"07:37.910","Text":"but now with Mu equaling 0 and Sigma squared equaling 1 squared."},{"Start":"07:37.910 ","End":"07:42.679","Text":"Now, once we have that and we get the standard score,"},{"Start":"07:42.679 ","End":"07:47.690","Text":"then we can go to the standard normal table and we can"},{"Start":"07:47.690 ","End":"07:54.140","Text":"look up the value of the area underneath the density function."},{"Start":"07:54.140 ","End":"07:56.915","Text":"Well, that\u0027s our probability."},{"Start":"07:56.915 ","End":"08:01.500","Text":"Enough theory, let\u0027s get to the exercise."}],"ID":13754},{"Watched":false,"Name":"Example - Part a","Duration":"8m 12s","ChapterTopicVideoID":13267,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"Okay. Let\u0027s do an example to see how things work."},{"Start":"00:03.300 ","End":"00:07.020","Text":"Now, the weight of a chocolate bar produced by a company has"},{"Start":"00:07.020 ","End":"00:09.570","Text":"a normal probability distribution with an average of"},{"Start":"00:09.570 ","End":"00:12.410","Text":"100 grams and a standard deviation of 8 grams,"},{"Start":"00:12.410 ","End":"00:18.090","Text":"and we\u0027re asked what proportion of chocolate bars weigh less than 110 grams?"},{"Start":"00:18.090 ","End":"00:23.100","Text":"Well, the first thing that we need to know or to make sure is that we\u0027re dealing"},{"Start":"00:23.100 ","End":"00:27.750","Text":"with a normal probability distribution and we are. Why is that?"},{"Start":"00:27.750 ","End":"00:30.375","Text":"Because it\u0027s given to us right here."},{"Start":"00:30.375 ","End":"00:32.700","Text":"If it\u0027s not given to us,"},{"Start":"00:32.700 ","End":"00:36.419","Text":"we can\u0027t assume that we\u0027re dealing with a normal probability distribution."},{"Start":"00:36.419 ","End":"00:38.220","Text":"We don\u0027t know what we\u0027re dealing with,"},{"Start":"00:38.220 ","End":"00:43.355","Text":"but here we are given that this is a normal probability distribution."},{"Start":"00:43.355 ","End":"00:48.260","Text":"The next thing that we need to do is to define a random variable."},{"Start":"00:48.260 ","End":"00:52.340","Text":"Well, let\u0027s call a random variable X and that\u0027ll be"},{"Start":"00:52.340 ","End":"00:58.500","Text":"defined as the weight of the chocolate bar."},{"Start":"00:59.480 ","End":"01:04.820","Text":"We\u0027re given that that\u0027s distributed with a normal distribution,"},{"Start":"01:04.820 ","End":"01:07.955","Text":"where new equals 100,"},{"Start":"01:07.955 ","End":"01:09.380","Text":"that\u0027s 100 grams right here,"},{"Start":"01:09.380 ","End":"01:12.450","Text":"that\u0027s the average, and Sigma,"},{"Start":"01:12.450 ","End":"01:16.350","Text":"the standard deviation, is 8,"},{"Start":"01:16.350 ","End":"01:19.035","Text":"that\u0027s given right here."},{"Start":"01:19.035 ","End":"01:22.730","Text":"This is information that we\u0027re given and we\u0027re asked what proportion"},{"Start":"01:22.730 ","End":"01:25.970","Text":"of chocolate bars weigh less than 110 grams."},{"Start":"01:25.970 ","End":"01:30.620","Text":"The next step that we need to do is we need to draw"},{"Start":"01:30.620 ","End":"01:36.250","Text":"the density function and map this information onto the density function."},{"Start":"01:36.250 ","End":"01:44.450","Text":"Let\u0027s do that. This is our density function right here and in the middle right here,"},{"Start":"01:44.450 ","End":"01:45.860","Text":"well, that\u0027s our average."},{"Start":"01:45.860 ","End":"01:51.240","Text":"That\u0027s new, so let\u0027s write this new then would be equal to 100."},{"Start":"01:51.240 ","End":"01:56.660","Text":"We\u0027re looking for the proportion of chocolate bars that weigh less than 110 grams."},{"Start":"01:56.660 ","End":"02:01.450","Text":"Let\u0027s write here, 110 grams."},{"Start":"02:01.450 ","End":"02:03.500","Text":"What does that mean?"},{"Start":"02:03.500 ","End":"02:06.620","Text":"What proportion of chocolate bars weigh less than 110 grams?"},{"Start":"02:06.620 ","End":"02:15.940","Text":"We\u0027re looking for the area underneath the density function from minus infinity to 110."},{"Start":"02:15.940 ","End":"02:20.790","Text":"That would be defined as the proportion."},{"Start":"02:20.790 ","End":"02:23.625","Text":"Let\u0027s just draw this,"},{"Start":"02:23.625 ","End":"02:27.200","Text":"so we\u0027re looking for the proportion of"},{"Start":"02:27.200 ","End":"02:31.220","Text":"the yellow area in relation to the total area under the curve,"},{"Start":"02:31.220 ","End":"02:35.585","Text":"which equals to 1 because it\u0027s a density function."},{"Start":"02:35.585 ","End":"02:40.905","Text":"Let\u0027s write this out. We\u0027re looking for the probability of"},{"Start":"02:40.905 ","End":"02:46.620","Text":"X being less than 110 grams."},{"Start":"02:46.620 ","End":"02:49.580","Text":"Now if we remember the steps that we had to take,"},{"Start":"02:49.580 ","End":"02:56.015","Text":"we had X and we needed to standardize it into Z,"},{"Start":"02:56.015 ","End":"03:01.715","Text":"and only then could we figure out or calculate the probability."},{"Start":"03:01.715 ","End":"03:04.040","Text":"Now Z, if we recall,"},{"Start":"03:04.040 ","End":"03:10.100","Text":"that\u0027s X minus Mu divided by Sigma, that\u0027s the transformation."},{"Start":"03:10.100 ","End":"03:13.105","Text":"When we\u0027re dealing with this probability,"},{"Start":"03:13.105 ","End":"03:19.460","Text":"we\u0027re looking for the probability of X minus Mu divided by Sigma,"},{"Start":"03:19.460 ","End":"03:26.160","Text":"that has to be less than a 110 minus Mu divided by Sigma."},{"Start":"03:26.300 ","End":"03:32.923","Text":"That equals to the probability now X minus Mu divided by Sigma that\u0027s Z."},{"Start":"03:32.923 ","End":"03:34.975","Text":"That\u0027ll be Z,"},{"Start":"03:34.975 ","End":"03:37.960","Text":"which is less than, now, Mu,"},{"Start":"03:37.960 ","End":"03:41.300","Text":"is a 100 and sigma is 8,"},{"Start":"03:41.300 ","End":"03:48.335","Text":"so that\u0027ll be 110 minus 100 divided by 8."},{"Start":"03:48.335 ","End":"03:57.485","Text":"That means that we\u0027re looking for the probability of Z being less than 1.25."},{"Start":"03:57.485 ","End":"03:59.300","Text":"Now, as we can see,"},{"Start":"03:59.300 ","End":"04:05.405","Text":"the probability of X being less than 110 is comparable,"},{"Start":"04:05.405 ","End":"04:09.970","Text":"basically equal to the probability of Z now."},{"Start":"04:09.970 ","End":"04:14.615","Text":"The transformation being less than 1.25."},{"Start":"04:14.615 ","End":"04:16.760","Text":"Now, what does that mean?"},{"Start":"04:16.760 ","End":"04:20.930","Text":"How does this look on this density function?"},{"Start":"04:20.930 ","End":"04:23.430","Text":"Let\u0027s try this out."},{"Start":"04:24.620 ","End":"04:27.630","Text":"This then is the z-axis,"},{"Start":"04:27.630 ","End":"04:33.240","Text":"which is comparable to the x-axis and we can see in the middle of"},{"Start":"04:33.240 ","End":"04:40.790","Text":"the z-axis we have 0 which is comparable to the 100 on the x-axis."},{"Start":"04:40.790 ","End":"04:46.520","Text":"We\u0027ve calculated that Z being equal to"},{"Start":"04:46.520 ","End":"04:55.010","Text":"1.25 and that\u0027s comparable to the 110 on the x-axis."},{"Start":"04:55.010 ","End":"04:57.050","Text":"What does this mean?"},{"Start":"04:57.050 ","End":"05:02.455","Text":"All this means is that the value 110 on the x-axis is"},{"Start":"05:02.455 ","End":"05:10.145","Text":"1.25 standard deviations away from the average of 100."},{"Start":"05:10.145 ","End":"05:15.355","Text":"Now, once we\u0027ve calculated this, the Z value,"},{"Start":"05:15.355 ","End":"05:25.230","Text":"only then can we go to the standard normal distribution table and figure"},{"Start":"05:25.230 ","End":"05:35.840","Text":"out what the area is under the density function from minus infinity until Z."},{"Start":"05:35.840 ","End":"05:38.880","Text":"Let\u0027s do that."},{"Start":"05:39.230 ","End":"05:43.340","Text":"This then is the table that helps us to calculate"},{"Start":"05:43.340 ","End":"05:47.030","Text":"the area under a standard normal density function."},{"Start":"05:47.030 ","End":"05:51.965","Text":"Now, let\u0027s look at the structure of this table before we can use this."},{"Start":"05:51.965 ","End":"05:54.265","Text":"We have here,"},{"Start":"05:54.265 ","End":"05:59.325","Text":"as row names and column names are Z values."},{"Start":"05:59.325 ","End":"06:05.270","Text":"In the middle, we have what\u0027s called Phi of Z. Phi is"},{"Start":"06:05.270 ","End":"06:11.345","Text":"the area under the density function that we\u0027re looking for."},{"Start":"06:11.345 ","End":"06:13.145","Text":"In our case, Z,"},{"Start":"06:13.145 ","End":"06:17.730","Text":"our standard score, was equal to 1.25."},{"Start":"06:17.740 ","End":"06:21.990","Text":"Let\u0027s try to find this value right here,"},{"Start":"06:21.990 ","End":"06:29.430","Text":"1.25 means that we\u0027re looking for Z in the rows here and we have 1.2."},{"Start":"06:29.430 ","End":"06:33.435","Text":"Now, we need that extra 0.05,"},{"Start":"06:33.435 ","End":"06:37.410","Text":"because we\u0027re at 1.25 and that\u0027s right here,"},{"Start":"06:37.410 ","End":"06:39.900","Text":"that\u0027s this column right here."},{"Start":"06:39.900 ","End":"06:46.790","Text":"Let\u0027s take a look now at the intersection of this column and this row,"},{"Start":"06:46.790 ","End":"06:50.000","Text":"that\u0027s this value right here."},{"Start":"06:50.000 ","End":"06:56.610","Text":"That means that the area under the normal density function,"},{"Start":"06:56.610 ","End":"07:02.240","Text":"the standard normal density function from minus infinity until Z,"},{"Start":"07:02.240 ","End":"07:05.870","Text":"our standard score, that equals to 1.25,"},{"Start":"07:05.870 ","End":"07:10.260","Text":"that area equals to 0.8944."},{"Start":"07:13.820 ","End":"07:17.210","Text":"Excellent. Now we know how to use this table."},{"Start":"07:17.210 ","End":"07:20.670","Text":"Let\u0027s get back to our question."},{"Start":"07:22.670 ","End":"07:27.410","Text":"We last left off right here where we\u0027re looking for"},{"Start":"07:27.410 ","End":"07:32.090","Text":"the probability of Z being less than 1.25."},{"Start":"07:32.090 ","End":"07:37.140","Text":"Now we know that that equals to Phi of Z,"},{"Start":"07:37.140 ","End":"07:44.530","Text":"which is 1.25 and we know that that equals to 0.8944."},{"Start":"07:45.470 ","End":"07:51.835","Text":"That is the proportion of this yellow area,"},{"Start":"07:51.835 ","End":"07:56.285","Text":"in comparison with the total area under the curve."},{"Start":"07:56.285 ","End":"08:02.660","Text":"We\u0027re looking at the proportion of chocolate bars that weigh less than 110 grams,"},{"Start":"08:02.660 ","End":"08:12.360","Text":"that\u0027s 89.44 percent and this is the answer for this question."}],"ID":13755},{"Watched":false,"Name":"Example - Parts b-c","Duration":"6m 27s","ChapterTopicVideoID":13269,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.040","Text":"In this section, we\u0027re asked what proportion of"},{"Start":"00:02.040 ","End":"00:05.745","Text":"the chocolate bars weigh more than 110``` grams."},{"Start":"00:05.745 ","End":"00:07.980","Text":"Now, if you remember from the last section,"},{"Start":"00:07.980 ","End":"00:09.615","Text":"we\u0027re asked the same question,"},{"Start":"00:09.615 ","End":"00:14.895","Text":"but we\u0027re looking for the proportion of chocolate bars that weigh less than 110 grams."},{"Start":"00:14.895 ","End":"00:19.980","Text":"Here we\u0027re asked for the proportion of the bars that weigh more than 110 grams,"},{"Start":"00:19.980 ","End":"00:26.250","Text":"that means that we\u0027re looking for the probability of x being greater than 110."},{"Start":"00:26.250 ","End":"00:35.055","Text":"Well, that\u0027s comparable, that\u0027s equal to the probability of Z being greater than 1.25."},{"Start":"00:35.055 ","End":"00:38.925","Text":"Now, that equals to what?"},{"Start":"00:38.925 ","End":"00:44.120","Text":"Well, if we take a look at the density function, right now,"},{"Start":"00:44.120 ","End":"00:47.435","Text":"we\u0027re not interested in the yellow shaded area,"},{"Start":"00:47.435 ","End":"00:49.985","Text":"but in the area in white."},{"Start":"00:49.985 ","End":"00:53.930","Text":"Now we know that the total area under the density function equals 1."},{"Start":"00:53.930 ","End":"00:59.690","Text":"We\u0027re looking at 1 minus the area shaded in yellow."},{"Start":"00:59.690 ","End":"01:06.380","Text":"The area shaded in yellow is the probability of Z being less than 1.25."},{"Start":"01:06.380 ","End":"01:11.045","Text":"As we said, we\u0027ve calculated that in the last section that equals to 1"},{"Start":"01:11.045 ","End":"01:20.780","Text":"minus 0.8944 and that equals to 0.1056."},{"Start":"01:20.780 ","End":"01:25.080","Text":"So this is the answer, for this question."},{"Start":"01:25.080 ","End":"01:32.575","Text":"This is basically the proportion of chocolate bars that weigh more than 110 grams."},{"Start":"01:32.575 ","End":"01:35.180","Text":"Let\u0027s just sum this point up."},{"Start":"01:35.180 ","End":"01:36.680","Text":"Let\u0027s first of all,"},{"Start":"01:36.680 ","End":"01:42.575","Text":"draw the standardized normal density function. Here it is."},{"Start":"01:42.575 ","End":"01:48.785","Text":"Now, this is our z-axis and we\u0027re looking at a value of Z,"},{"Start":"01:48.785 ","End":"01:51.740","Text":"let\u0027s say a, a positive number."},{"Start":"01:51.740 ","End":"01:56.720","Text":"We\u0027re interested in the area to the left of a."},{"Start":"01:56.720 ","End":"02:01.730","Text":"That means that we\u0027re looking for the area from minus infinity to a."},{"Start":"02:01.730 ","End":"02:04.910","Text":"That\u0027s the area in yellow right here."},{"Start":"02:04.910 ","End":"02:11.750","Text":"Now this area is Phi of a."},{"Start":"02:11.750 ","End":"02:15.500","Text":"Now since the total area under the density function equals 1,"},{"Start":"02:15.500 ","End":"02:21.065","Text":"then the area in white then is 1 minus Phi of a."},{"Start":"02:21.065 ","End":"02:26.840","Text":"Let\u0027s just take a look at the table to make sure that we\u0027re right."},{"Start":"02:26.850 ","End":"02:33.395","Text":"Here we can see the standard normal density function."},{"Start":"02:33.395 ","End":"02:36.020","Text":"Here we have our standard score Z,"},{"Start":"02:36.020 ","End":"02:37.625","Text":"and here we have Phi of Z,"},{"Start":"02:37.625 ","End":"02:39.455","Text":"the grayed-out area right here,"},{"Start":"02:39.455 ","End":"02:43.940","Text":"which is the area that\u0027s given to us by the table,"},{"Start":"02:43.940 ","End":"02:46.845","Text":"these values within the table right here."},{"Start":"02:46.845 ","End":"02:53.320","Text":"So this then is the summation to the left of a,"},{"Start":"02:53.320 ","End":"02:56.245","Text":"what\u0027s given in the table that\u0027s Phi of a."},{"Start":"02:56.245 ","End":"02:59.470","Text":"If we\u0027re interested in the area to the right of a,"},{"Start":"02:59.470 ","End":"03:02.230","Text":"well that\u0027s 1 minus Phi of a."},{"Start":"03:02.230 ","End":"03:07.945","Text":"In this section we\u0027re asked what proportion of chocolate bars weigh less than 92 grams."},{"Start":"03:07.945 ","End":"03:12.880","Text":"The first thing we want to do is want to draw the density function."},{"Start":"03:12.880 ","End":"03:15.785","Text":"Here\u0027s our density function."},{"Start":"03:15.785 ","End":"03:19.449","Text":"We have here a value of 100,"},{"Start":"03:19.449 ","End":"03:21.430","Text":"or that\u0027s our average."},{"Start":"03:21.430 ","End":"03:28.750","Text":"We\u0027re looking for the proportion of chocolate bars that weigh less than 92 grams."},{"Start":"03:29.090 ","End":"03:33.650","Text":"We\u0027re looking to calculate the area colored in red right here."},{"Start":"03:33.650 ","End":"03:40.384","Text":"So that\u0027s the probability of x being less than 92 grams."},{"Start":"03:40.384 ","End":"03:42.185","Text":"Now for recall the process,"},{"Start":"03:42.185 ","End":"03:45.780","Text":"we have x, we have to standardize this into to z."},{"Start":"03:45.780 ","End":"03:54.320","Text":"Only then can we calculate the probability that z is x minus Mu divided by Sigma."},{"Start":"03:54.320 ","End":"03:55.880","Text":"Let\u0027s do that right here."},{"Start":"03:55.880 ","End":"04:01.640","Text":"That\u0027s the probability of X minus mu divided by sigma."},{"Start":"04:01.640 ","End":"04:05.585","Text":"Well, that has to be less than 92 minus now,"},{"Start":"04:05.585 ","End":"04:08.375","Text":"Mu is a 100,"},{"Start":"04:08.375 ","End":"04:10.560","Text":"and sigma is a."},{"Start":"04:11.330 ","End":"04:17.745","Text":"That equals to the probability of Z being less than minus 1."},{"Start":"04:17.745 ","End":"04:22.600","Text":"Well, that\u0027s Phi of minus 1."},{"Start":"04:22.610 ","End":"04:28.520","Text":"Now we can go to the standard table and see if we can"},{"Start":"04:28.520 ","End":"04:35.200","Text":"calculate the area under the curve where our standard score is minus 1."},{"Start":"04:35.200 ","End":"04:37.050","Text":"Let\u0027s sum up this point."},{"Start":"04:37.050 ","End":"04:41.150","Text":"Now, assume you have a standard normal density function,"},{"Start":"04:41.150 ","End":"04:47.170","Text":"that means that the axis here is the z-axis."},{"Start":"04:47.170 ","End":"04:49.460","Text":"When we\u0027re talking about the z-axis,"},{"Start":"04:49.460 ","End":"04:53.150","Text":"and that means that the average here is 0."},{"Start":"04:53.150 ","End":"04:57.920","Text":"After standardization, you encounter a negative value for z,"},{"Start":"04:57.920 ","End":"04:59.780","Text":"for example, minus a."},{"Start":"04:59.780 ","End":"05:02.900","Text":"That means that you want to calculate the area under"},{"Start":"05:02.900 ","End":"05:07.355","Text":"the density function from minus infinity till minus a."},{"Start":"05:07.355 ","End":"05:10.630","Text":"That\u0027s this red area right here."},{"Start":"05:10.630 ","End":"05:16.160","Text":"So you want to calculate Phi of minus a,"},{"Start":"05:16.160 ","End":"05:17.870","Text":"so you go to the standard table,"},{"Start":"05:17.870 ","End":"05:23.565","Text":"but you can\u0027t find minus a because a table supports only positive values of Z."},{"Start":"05:23.565 ","End":"05:27.070","Text":"Now, in order to resolve this issue,"},{"Start":"05:27.070 ","End":"05:31.420","Text":"you\u0027re going to have to use 1 of the characteristics of the density function and that"},{"Start":"05:31.420 ","End":"05:36.340","Text":"is that the density function is symmetrical around the average."},{"Start":"05:36.340 ","End":"05:43.120","Text":"What does that mean? That means that this area right here from minus infinity to"},{"Start":"05:43.120 ","End":"05:50.860","Text":"minus a is equal to the area under the density function from a to plus infinity."},{"Start":"05:50.860 ","End":"05:54.160","Text":"That\u0027s this area right here in red."},{"Start":"05:54.160 ","End":"05:57.130","Text":"Now we know how to calculate this area."},{"Start":"05:57.130 ","End":"05:59.485","Text":"We\u0027ve done this before in the past section,"},{"Start":"05:59.485 ","End":"06:05.925","Text":"that equals to 1 minus Phi of a."},{"Start":"06:05.925 ","End":"06:10.760","Text":"This white area equals Phi of a."},{"Start":"06:10.760 ","End":"06:20.320","Text":"We know that 5 minus a then equals 1 minus Phi of a."},{"Start":"06:20.540 ","End":"06:27.540","Text":"That also means that this white area also equals Phi of a."}],"ID":13756},{"Watched":false,"Name":"Example - Part d","Duration":"6m 52s","ChapterTopicVideoID":13268,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"In this section, 90 percent of the chocolate bars from"},{"Start":"00:03.390 ","End":"00:07.750","Text":"the production line weigh less than what weight?"},{"Start":"00:08.360 ","End":"00:12.075","Text":"If you recall from the previous sections,"},{"Start":"00:12.075 ","End":"00:13.575","Text":"in the previous sections,"},{"Start":"00:13.575 ","End":"00:20.385","Text":"we were given the weight and we wanted to calculate the probability, or the proportion."},{"Start":"00:20.385 ","End":"00:23.790","Text":"Now, we have to work backwards. Why is that?"},{"Start":"00:23.790 ","End":"00:31.425","Text":"Because we\u0027re given the proportion and we want to calculate the actual weight."},{"Start":"00:31.425 ","End":"00:33.750","Text":"Now, what is this weight?"},{"Start":"00:33.750 ","End":"00:35.610","Text":"Well, 90 percent of the chocolate bars from"},{"Start":"00:35.610 ","End":"00:37.925","Text":"the production line weigh less than what weight?"},{"Start":"00:37.925 ","End":"00:44.010","Text":"That means that we\u0027re looking at x_0.9,"},{"Start":"00:44.010 ","End":"00:49.150","Text":"and that is the 90th percentile."},{"Start":"00:50.500 ","End":"00:58.350","Text":"Or we can call this the top 10 percentile."},{"Start":"01:00.610 ","End":"01:04.115","Text":"Again, in the previous sections,"},{"Start":"01:04.115 ","End":"01:12.245","Text":"we were given x and we had to calculate z through a process of standardization."},{"Start":"01:12.245 ","End":"01:14.675","Text":"Then we had to go to the standard table,"},{"Start":"01:14.675 ","End":"01:19.790","Text":"and calculate or find the probability, or the proportion."},{"Start":"01:19.790 ","End":"01:22.340","Text":"Now here, we\u0027re not doing that."},{"Start":"01:22.340 ","End":"01:24.020","Text":"Here we have to work backwards."},{"Start":"01:24.020 ","End":"01:26.825","Text":"Here, we\u0027re given the proportion."},{"Start":"01:26.825 ","End":"01:33.465","Text":"We have to go to the table and find the standard score z,"},{"Start":"01:33.465 ","End":"01:38.465","Text":"and from there we have to calculate x."},{"Start":"01:38.465 ","End":"01:46.375","Text":"How do we do that? Well, we know that when we\u0027re dealing with the x-axis,"},{"Start":"01:46.375 ","End":"01:54.645","Text":"our average was 100 and we\u0027re trying to find out x_0.9, the 90th percentile."},{"Start":"01:54.645 ","End":"01:58.125","Text":"Again, this, the 90th percentile,"},{"Start":"01:58.125 ","End":"01:59.740","Text":"is the value of x,"},{"Start":"01:59.740 ","End":"02:05.255","Text":"where 90 percent of the area underneath the density function."},{"Start":"02:05.255 ","End":"02:11.450","Text":"We can find that from minus infinity to x_0.9."},{"Start":"02:11.450 ","End":"02:15.575","Text":"That\u0027s this shaded area in yellow."},{"Start":"02:15.575 ","End":"02:21.445","Text":"Now, when we go to the z-axis,"},{"Start":"02:21.445 ","End":"02:27.540","Text":"we know that zero is comparable to 100 on the x-axis."},{"Start":"02:27.540 ","End":"02:31.100","Text":"What we\u0027re trying to find out from the table is,"},{"Start":"02:31.100 ","End":"02:32.960","Text":"what\u0027s the value of z?"},{"Start":"02:32.960 ","End":"02:35.910","Text":"What\u0027s the standard score here?"},{"Start":"02:35.910 ","End":"02:40.040","Text":"We\u0027ll call this z_0.9."},{"Start":"02:40.040 ","End":"02:45.430","Text":"What\u0027s this value right here that\u0027s comparable to x_0.9."},{"Start":"02:45.430 ","End":"02:48.020","Text":"Let\u0027s go to the standard table,"},{"Start":"02:48.020 ","End":"02:54.330","Text":"and see how we can find this value right here for z."},{"Start":"02:54.950 ","End":"03:00.350","Text":"Here we see our standard normal probability distribution table."},{"Start":"03:00.350 ","End":"03:03.080","Text":"Now, we were given, if we recall,"},{"Start":"03:03.080 ","End":"03:05.000","Text":"Phi of z,"},{"Start":"03:05.000 ","End":"03:08.180","Text":"and that was 0.9."},{"Start":"03:08.180 ","End":"03:10.550","Text":"That means that the area under"},{"Start":"03:10.550 ","End":"03:20.225","Text":"the standard density function from minus infinity to a specific value is 90 percent."},{"Start":"03:20.225 ","End":"03:23.630","Text":"Now, where can we find that value?"},{"Start":"03:23.630 ","End":"03:26.720","Text":"Well, this value, 90 percent or 0.9,"},{"Start":"03:26.720 ","End":"03:30.410","Text":"should be found right here within the table itself."},{"Start":"03:30.410 ","End":"03:35.345","Text":"Now, we have to find a value that\u0027s closest to 0.9."},{"Start":"03:35.345 ","End":"03:36.770","Text":"Where can we find that?"},{"Start":"03:36.770 ","End":"03:39.245","Text":"Well, we find it right here."},{"Start":"03:39.245 ","End":"03:43.430","Text":"This is the value 0.8997."},{"Start":"03:43.430 ","End":"03:47.030","Text":"That\u0027s the closest we can come to 0.9."},{"Start":"03:47.030 ","End":"03:51.305","Text":"What\u0027s the corresponding z-value, or standard score?"},{"Start":"03:51.305 ","End":"03:58.010","Text":"Well, it\u0027s right here, because this value,"},{"Start":"03:58.010 ","End":"04:06.580","Text":"0.8997 is in the intersection of this row right here and this column."},{"Start":"04:07.970 ","End":"04:13.440","Text":"Z_0.9, that would be equal to 1.28."},{"Start":"04:13.440 ","End":"04:18.635","Text":"Again, here\u0027s 1.2 and here\u0027s the extra 0.8 right here."},{"Start":"04:18.635 ","End":"04:21.710","Text":"Now, some of you may have"},{"Start":"04:21.710 ","End":"04:26.885","Text":"a more detailed table specifying"},{"Start":"04:26.885 ","End":"04:32.890","Text":"specific values of Phi and their corresponding standard scores."},{"Start":"04:32.890 ","End":"04:36.695","Text":"Here we can see this type of table right here."},{"Start":"04:36.695 ","End":"04:41.800","Text":"Now for a specific Phi value, let\u0027s say of 0.9,"},{"Start":"04:41.800 ","End":"04:47.845","Text":"we have a corresponding z-value of 1.282,"},{"Start":"04:47.845 ","End":"04:53.615","Text":"which is a little bit more exactly what we found out within the table."},{"Start":"04:53.615 ","End":"04:56.555","Text":"Let\u0027s use this value right here."},{"Start":"04:56.555 ","End":"05:00.065","Text":"We\u0027ll just add this onto here."},{"Start":"05:00.065 ","End":"05:03.410","Text":"Again, z_0.9,"},{"Start":"05:03.410 ","End":"05:07.715","Text":"that\u0027s equal to 1.282."},{"Start":"05:07.715 ","End":"05:11.855","Text":"Again, let\u0027s just write this out."},{"Start":"05:11.855 ","End":"05:15.840","Text":"Z_0.9, that\u0027s equal to 1.282."},{"Start":"05:16.610 ","End":"05:21.390","Text":"We\u0027ll put that right here on the z-axis as well."},{"Start":"05:21.390 ","End":"05:24.820","Text":"That equals to 1.282."},{"Start":"05:24.820 ","End":"05:32.090","Text":"Now, what\u0027s the corresponding x-value to z_0.9?"},{"Start":"05:32.090 ","End":"05:34.595","Text":"Well, if we recall,"},{"Start":"05:34.595 ","End":"05:40.650","Text":"z equals to x minus Mu divided by Sigma."},{"Start":"05:40.650 ","End":"05:47.205","Text":"In our case, z_0.9, that\u0027s 1.282,"},{"Start":"05:47.205 ","End":"05:51.780","Text":"and that equals to x_0.9 minus,"},{"Start":"05:51.780 ","End":"05:55.320","Text":"now Mu is 100 divided by 8."},{"Start":"05:55.320 ","End":"06:01.415","Text":"That\u0027s our Sigma. Now we have an equation with 1 variable."},{"Start":"06:01.415 ","End":"06:02.870","Text":"It\u0027s very easy to solve."},{"Start":"06:02.870 ","End":"06:04.640","Text":"Basic math. Let\u0027s do it."},{"Start":"06:04.640 ","End":"06:11.280","Text":"We\u0027ll divide both sides by 8 and we\u0027ll get 10.256."},{"Start":"06:11.280 ","End":"06:15.605","Text":"That equals to x_0.9 minus a 100."},{"Start":"06:15.605 ","End":"06:18.920","Text":"That means that x_0.9, well,"},{"Start":"06:18.920 ","End":"06:24.835","Text":"that equals to 110.256."},{"Start":"06:24.835 ","End":"06:26.850","Text":"Let\u0027s write this up right here,"},{"Start":"06:26.850 ","End":"06:31.905","Text":"that equals to 110.256."},{"Start":"06:31.905 ","End":"06:34.250","Text":"This value right here,"},{"Start":"06:34.250 ","End":"06:36.695","Text":"which is in grams,"},{"Start":"06:36.695 ","End":"06:43.025","Text":"is the 90th percentile of the weight of the chocolate bars."},{"Start":"06:43.025 ","End":"06:52.260","Text":"Meaning that 90 percent of the chocolate bars weigh less than 110.256 grams."}],"ID":13757},{"Watched":false,"Name":"Exercise 1 - Part a","Duration":"6m 4s","ChapterTopicVideoID":13270,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"In this question, we\u0027re given that the height of people in a given population has"},{"Start":"00:04.500 ","End":"00:09.000","Text":"a normal probability distribution with an average of 107 centimeters,"},{"Start":"00:09.000 ","End":"00:11.520","Text":"and a standard deviation of 10 centimeters."},{"Start":"00:11.520 ","End":"00:17.900","Text":"We\u0027re asked what\u0027s the proportion of people who are shorter than 182.4 centimeters?"},{"Start":"00:17.900 ","End":"00:21.005","Text":"Well, the first thing that we have to make sure is that"},{"Start":"00:21.005 ","End":"00:24.245","Text":"we\u0027re dealing with a normal probability distribution,"},{"Start":"00:24.245 ","End":"00:27.200","Text":"and we are, that\u0027s given to us right here."},{"Start":"00:27.200 ","End":"00:28.595","Text":"If it wasn\u0027t given,"},{"Start":"00:28.595 ","End":"00:32.465","Text":"then we wouldn\u0027t know what distribution we have."},{"Start":"00:32.465 ","End":"00:37.175","Text":"The second thing that we need to do is define a random variable."},{"Start":"00:37.175 ","End":"00:39.380","Text":"Now let\u0027s call that x,"},{"Start":"00:39.380 ","End":"00:47.390","Text":"and x is defined as the height of people in centimeters,"},{"Start":"00:47.390 ","End":"00:54.655","Text":"and we\u0027re given that that\u0027s distributed with a normal distribution where Mu,"},{"Start":"00:54.655 ","End":"00:58.603","Text":"the average is 170 centimeters,"},{"Start":"00:58.603 ","End":"01:00.890","Text":"and the standard deviation, well,"},{"Start":"01:00.890 ","End":"01:04.255","Text":"that equals to 10 centimeters."},{"Start":"01:04.255 ","End":"01:09.005","Text":"Now, since this is a normal probability distribution,"},{"Start":"01:09.005 ","End":"01:11.860","Text":"let\u0027s just draw the density function."},{"Start":"01:11.860 ","End":"01:14.630","Text":"Here we have the density functions."},{"Start":"01:14.630 ","End":"01:21.625","Text":"Let\u0027s just put the data onto the chart."},{"Start":"01:21.625 ","End":"01:23.260","Text":"Mu is 170,"},{"Start":"01:23.260 ","End":"01:25.730","Text":"so right here that\u0027s the average."},{"Start":"01:25.730 ","End":"01:29.210","Text":"That\u0027s 170 centimeters."},{"Start":"01:29.210 ","End":"01:35.615","Text":"Now, we\u0027re looking at the proportion of people who are shorter than 182.4 centimeters."},{"Start":"01:35.615 ","End":"01:39.980","Text":"That\u0027s larger than the average so we\u0027ll put that here,"},{"Start":"01:39.980 ","End":"01:44.660","Text":"and that\u0027ll be 182.4 centimeters,"},{"Start":"01:44.660 ","End":"01:50.630","Text":"that\u0027s right here and we\u0027re looking also for the area under"},{"Start":"01:50.630 ","End":"02:00.190","Text":"the density function from minus infinity to this value right here, 182.4 centimeters."},{"Start":"02:00.190 ","End":"02:03.660","Text":"That\u0027s this area in yellow right here."},{"Start":"02:03.660 ","End":"02:09.545","Text":"What are the steps that we have to go through in order to figure out this area?"},{"Start":"02:09.545 ","End":"02:12.095","Text":"Well, first of all,"},{"Start":"02:12.095 ","End":"02:14.300","Text":"we had x, if we remember,"},{"Start":"02:14.300 ","End":"02:19.310","Text":"then we needed to standardize x into z,"},{"Start":"02:19.310 ","End":"02:22.235","Text":"and then go to the standard normal table,"},{"Start":"02:22.235 ","End":"02:25.850","Text":"and then find or calculate what p was,"},{"Start":"02:25.850 ","End":"02:27.965","Text":"what the proportion was."},{"Start":"02:27.965 ","End":"02:36.980","Text":"Well, we know that x is 182.4 centimeters. Now what\u0027s that?"},{"Start":"02:36.980 ","End":"02:46.595","Text":"We\u0027re looking for the probability of x being less than 182.4 centimeters."},{"Start":"02:46.595 ","End":"02:48.985","Text":"Now, that equals,"},{"Start":"02:48.985 ","End":"02:54.606","Text":"remember that z is x minus Mu divided by Sigma."},{"Start":"02:54.606 ","End":"03:00.970","Text":"We\u0027re looking here for the probability of x minus Mu divided by"},{"Start":"03:00.970 ","End":"03:08.550","Text":"Sigma that has to be less than 182.4 minus, now what\u0027s Mu?"},{"Start":"03:08.550 ","End":"03:15.630","Text":"That\u0027s 170, 170 divided by Sigma, Sigma is 10."},{"Start":"03:15.630 ","End":"03:21.220","Text":"We\u0027re looking for the probability then of z,"},{"Start":"03:21.220 ","End":"03:24.130","Text":"x minus Mu divided by Sigma, that\u0027s right here,"},{"Start":"03:24.130 ","End":"03:28.575","Text":"that\u0027s z, has to be less than this expression right here."},{"Start":"03:28.575 ","End":"03:31.445","Text":"That\u0027s 182.4 minus 170."},{"Start":"03:31.445 ","End":"03:35.140","Text":"That\u0027s 12.4 divided by 10."},{"Start":"03:35.140 ","End":"03:38.495","Text":"So that\u0027s 1.24."},{"Start":"03:38.495 ","End":"03:44.210","Text":"When we\u0027re dealing with the z axis right now,"},{"Start":"03:44.210 ","End":"03:48.190","Text":"we know that 0 is comparable to 170."},{"Start":"03:48.190 ","End":"03:58.640","Text":"So 182.4, that\u0027s comparable to this value right here, which is 1.24."},{"Start":"03:58.640 ","End":"04:01.220","Text":"Now, what does this value mean?"},{"Start":"04:01.220 ","End":"04:04.730","Text":"All it means is that 182.4 centimeters,"},{"Start":"04:04.730 ","End":"04:12.180","Text":"that\u0027s 1.24 standard deviations away from 170."},{"Start":"04:13.070 ","End":"04:24.450","Text":"Now we have to go to the standard table and we have to look up the value of z,"},{"Start":"04:24.450 ","End":"04:32.940","Text":"1.24 and find out what Phi of 1.24 is."},{"Start":"04:32.940 ","End":"04:35.025","Text":"Let\u0027s go to the table."},{"Start":"04:35.025 ","End":"04:38.105","Text":"Here we are and in this table,"},{"Start":"04:38.105 ","End":"04:43.720","Text":"the standard normal table and we\u0027re looking for z,"},{"Start":"04:43.720 ","End":"04:49.245","Text":"the standard score that equals 1.24."},{"Start":"04:49.245 ","End":"04:55.770","Text":"Let\u0027s just look right here on this side."},{"Start":"04:55.770 ","End":"05:01.685","Text":"That\u0027s 1.2, that\u0027s this line right here and we need 1.24."},{"Start":"05:01.685 ","End":"05:04.340","Text":"That means that we\u0027re looking at this column."},{"Start":"05:04.340 ","End":"05:09.500","Text":"The intersection of this row and this column,"},{"Start":"05:09.500 ","End":"05:15.320","Text":"that equals to 0.8925."},{"Start":"05:15.320 ","End":"05:19.640","Text":"So Phi of 1.24,"},{"Start":"05:19.640 ","End":"05:25.300","Text":"that equals to 0.8925."},{"Start":"05:25.300 ","End":"05:31.245","Text":"Let\u0027s now go back to our question. Here we go."},{"Start":"05:31.245 ","End":"05:35.810","Text":"The probability of z being less than 1.24,"},{"Start":"05:35.810 ","End":"05:40.745","Text":"that equals to Phi of 1.24,"},{"Start":"05:40.745 ","End":"05:45.000","Text":"and that equals to 0.8925."},{"Start":"05:47.500 ","End":"05:51.260","Text":"That\u0027s our answer."},{"Start":"05:51.260 ","End":"05:54.380","Text":"The proportion of people who are shorter than"},{"Start":"05:54.380 ","End":"05:58.085","Text":"182.4 centimeters or if you want it in percent,"},{"Start":"05:58.085 ","End":"06:04.440","Text":"that equals to 89.25 percent."}],"ID":13758},{"Watched":false,"Name":"Exercise 1 - Parts b-c","Duration":"5m 59s","ChapterTopicVideoID":13271,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.560","Text":"In this section, we\u0027re asked,"},{"Start":"00:01.560 ","End":"00:06.255","Text":"What\u0027s the proportion of people who are taller than 190 centimeters?"},{"Start":"00:06.255 ","End":"00:13.785","Text":"We\u0027re looking at the probability of x being greater than 190."},{"Start":"00:13.785 ","End":"00:19.230","Text":"Let\u0027s just draw this out on a density function."},{"Start":"00:19.230 ","End":"00:21.690","Text":"Here we have our density function,"},{"Start":"00:21.690 ","End":"00:23.880","Text":"that\u0027s our x axis."},{"Start":"00:23.880 ","End":"00:28.485","Text":"The average is 170,"},{"Start":"00:28.485 ","End":"00:33.370","Text":"and we\u0027re looking at a 190 centimeters."},{"Start":"00:33.370 ","End":"00:36.920","Text":"We want this area right here."},{"Start":"00:36.920 ","End":"00:41.930","Text":"We want the proportion of people who are taller than 190,"},{"Start":"00:41.930 ","End":"00:45.020","Text":"or that x is greater than 190."},{"Start":"00:45.020 ","End":"00:49.805","Text":"Now, if we recall the process, we have our x,"},{"Start":"00:49.805 ","End":"00:53.330","Text":"we have to standardize x into our standard score,"},{"Start":"00:53.330 ","End":"00:59.630","Text":"that\u0027ll be z, and then we have to go to the table and see what the proportion is."},{"Start":"00:59.630 ","End":"01:04.539","Text":"Well here, our x is 190,"},{"Start":"01:04.539 ","End":"01:10.560","Text":"the transformation of z is x minus Mu divided by Sigma."},{"Start":"01:10.560 ","End":"01:17.353","Text":"That equals then to the probability of x minus mu,"},{"Start":"01:17.353 ","End":"01:23.870","Text":"divided by sigma that has to be greater than 190 minus 170."},{"Start":"01:23.870 ","End":"01:28.340","Text":"That\u0027s our mu divided by 10. That\u0027s our Sigma."},{"Start":"01:28.340 ","End":"01:34.325","Text":"That equals to the probability of z being greater,"},{"Start":"01:34.325 ","End":"01:36.590","Text":"now, what\u0027s this expression?"},{"Start":"01:36.590 ","End":"01:38.630","Text":"Well, it\u0027s 190 minus 170,"},{"Start":"01:38.630 ","End":"01:42.055","Text":"that\u0027s 20 divided by 10, that\u0027s 2."},{"Start":"01:42.055 ","End":"01:44.240","Text":"But if we recall,"},{"Start":"01:44.240 ","End":"01:49.355","Text":"whenever we have the probability of z being greater than 2,"},{"Start":"01:49.355 ","End":"01:58.280","Text":"the standardized table doesn\u0027t give us the proportion of z greater than our value."},{"Start":"01:58.280 ","End":"02:06.410","Text":"It gives us the proportion of the area where z is less than a value."},{"Start":"02:06.410 ","End":"02:10.790","Text":"That means that we have to do this,"},{"Start":"02:10.790 ","End":"02:18.040","Text":"that equals to 1 minus the probability of z being less than 2."},{"Start":"02:18.040 ","End":"02:24.725","Text":"That means that this equals to 1 minus Phi of 2."},{"Start":"02:24.725 ","End":"02:29.960","Text":"Excellent. Now, we\u0027re ready to go to our table."},{"Start":"02:29.960 ","End":"02:33.125","Text":"Here\u0027s our table."},{"Start":"02:33.125 ","End":"02:40.700","Text":"We\u0027re looking for z being equal to 2,"},{"Start":"02:40.700 ","End":"02:45.505","Text":"and we want to find out what Phi of 2 is."},{"Start":"02:45.505 ","End":"02:51.910","Text":"Well, let\u0027s just go down this column right here until we hit 2."},{"Start":"02:51.910 ","End":"02:54.060","Text":"Now that\u0027s 2.0,"},{"Start":"02:54.060 ","End":"02:56.645","Text":"so we\u0027re dealing with this column right here."},{"Start":"02:56.645 ","End":"03:01.430","Text":"Phi of 2 is this value right here,"},{"Start":"03:01.430 ","End":"03:06.700","Text":"that equals to 0.9772."},{"Start":"03:06.700 ","End":"03:09.970","Text":"Let\u0027s get back to our question."},{"Start":"03:09.980 ","End":"03:18.420","Text":"This then equals 1 minus 0.9772,"},{"Start":"03:18.420 ","End":"03:22.390","Text":"which equals to 0.0228."},{"Start":"03:24.800 ","End":"03:31.085","Text":"That\u0027s our answer for the proportion of people who are taller than a 190 centimeters,"},{"Start":"03:31.085 ","End":"03:33.770","Text":"or if you want this in percentages,"},{"Start":"03:33.770 ","End":"03:38.245","Text":"well, that\u0027s 2.28 percent."},{"Start":"03:38.245 ","End":"03:40.145","Text":"In this section, we\u0027re asked,"},{"Start":"03:40.145 ","End":"03:44.950","Text":"What\u0027s the proportion of people who are exactly a 173.6 centimeters tall."},{"Start":"03:44.950 ","End":"03:53.650","Text":"We\u0027re looking for the probability of x being equal to 173.6 centimeters."},{"Start":"03:53.650 ","End":"03:55.610","Text":"Now, this is a trick question."},{"Start":"03:55.610 ","End":"03:57.710","Text":"There\u0027s nothing to calculate here."},{"Start":"03:57.710 ","End":"04:01.345","Text":"Why is that? Well, the answer is 0."},{"Start":"04:01.345 ","End":"04:05.720","Text":"Again, let\u0027s just take a look at our density function."},{"Start":"04:05.720 ","End":"04:09.125","Text":"This is a continuous density function."},{"Start":"04:09.125 ","End":"04:13.310","Text":"Now, when we\u0027re asked about the probability of something,"},{"Start":"04:13.310 ","End":"04:18.439","Text":"then we\u0027re looking for the area under the density function."},{"Start":"04:18.439 ","End":"04:21.050","Text":"Now, let\u0027s take a look at the x-axis."},{"Start":"04:21.050 ","End":"04:23.608","Text":"Well, the average is a 170,"},{"Start":"04:23.608 ","End":"04:26.840","Text":"and we\u0027re looking at a 173.6."},{"Start":"04:26.840 ","End":"04:28.490","Text":"Now, if I would have asked,"},{"Start":"04:28.490 ","End":"04:33.860","Text":"What\u0027s the probability of x being less than a 173.6 centimeters?"},{"Start":"04:33.860 ","End":"04:37.220","Text":"Then I would have asked about this area right here,"},{"Start":"04:37.220 ","End":"04:40.175","Text":"from minus infinity to this value right here."},{"Start":"04:40.175 ","End":"04:43.430","Text":"Now, if I would have asked for the probability of x being"},{"Start":"04:43.430 ","End":"04:46.880","Text":"greater than a 173.6 centimeters,"},{"Start":"04:46.880 ","End":"04:49.880","Text":"well, that\u0027s this area right here, but I didn\u0027t."},{"Start":"04:49.880 ","End":"04:56.930","Text":"I asked for the probability of x being equal to a 173.6 centimeters."},{"Start":"04:56.930 ","End":"05:03.170","Text":"That means I\u0027m looking for the area under this line right here, and that equals to 0."},{"Start":"05:03.170 ","End":"05:06.950","Text":"Now, this is true for all continuous distributions,"},{"Start":"05:06.950 ","End":"05:09.545","Text":"not only for the normal distribution."},{"Start":"05:09.545 ","End":"05:12.860","Text":"Now, you might say, well,"},{"Start":"05:12.860 ","End":"05:17.165","Text":"there are lots of people who are a 173.6 centimeters tall."},{"Start":"05:17.165 ","End":"05:19.415","Text":"But in actual fact,"},{"Start":"05:19.415 ","End":"05:23.900","Text":"what you mean to say is that there are around a 173.6,"},{"Start":"05:23.900 ","End":"05:28.940","Text":"they\u0027re not exactly a 173.6 centimeters tall."},{"Start":"05:28.940 ","End":"05:34.835","Text":"There\u0027s no person on earth that\u0027s exactly a 173.6 centimeters tall."},{"Start":"05:34.835 ","End":"05:36.710","Text":"They\u0027re around a 170,"},{"Start":"05:36.710 ","End":"05:40.940","Text":"between maybe 173, and 174 centimeters."},{"Start":"05:40.940 ","End":"05:42.275","Text":"Now, if that was the case,"},{"Start":"05:42.275 ","End":"05:46.820","Text":"then there would be some area under the curve that we can calculate."},{"Start":"05:46.820 ","End":"05:54.260","Text":"But, for x to be equal exactly to a 173.6 centimeters,"},{"Start":"05:54.260 ","End":"05:56.450","Text":"then the probability of that being true,"},{"Start":"05:56.450 ","End":"05:59.040","Text":"that equals to 0."}],"ID":13759},{"Watched":false,"Name":"Exercise 1 - Parts d-e","Duration":"4m 13s","ChapterTopicVideoID":13272,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.350","Text":"In this section we\u0027re asked,"},{"Start":"00:01.350 ","End":"00:04.950","Text":"what\u0027s the proportion of people who are shorter than 170 centimeters?"},{"Start":"00:04.950 ","End":"00:10.995","Text":"We\u0027re looking at the probability of x being less than 170 centimeters."},{"Start":"00:10.995 ","End":"00:15.150","Text":"Now, let\u0027s take a look at the density function."},{"Start":"00:15.150 ","End":"00:24.270","Text":"Here it is, and we can see that 170 is actually the average of the density function."},{"Start":"00:24.270 ","End":"00:30.825","Text":"If that\u0027s the case, we\u0027re looking at the probability of x being less than the average."},{"Start":"00:30.825 ","End":"00:36.060","Text":"We can easily see that this equals to 1/2."},{"Start":"00:36.060 ","End":"00:39.730","Text":"Now if we really wanted to be formal about this,"},{"Start":"00:39.730 ","End":"00:42.035","Text":"then we would go through the steps."},{"Start":"00:42.035 ","End":"00:46.360","Text":"Well, we have x, we have to standardize x into z,"},{"Start":"00:46.360 ","End":"00:51.200","Text":"then we have to go to the table to find the proportion."},{"Start":"00:51.200 ","End":"00:54.630","Text":"X here is 170."},{"Start":"00:54.630 ","End":"01:00.680","Text":"Now, z, we know that that\u0027s x minus Mu divided by Sigma,"},{"Start":"01:00.680 ","End":"01:02.645","Text":"that\u0027s a transformation,"},{"Start":"01:02.645 ","End":"01:07.605","Text":"so the probability of x being less than 170,"},{"Start":"01:07.605 ","End":"01:12.935","Text":"that\u0027s the probability of x minus Mu divided by Sigma,"},{"Start":"01:12.935 ","End":"01:17.270","Text":"that has to equal to 170 minus Mu."},{"Start":"01:17.270 ","End":"01:23.570","Text":"Now Mu is 170 divided by 10."},{"Start":"01:23.570 ","End":"01:27.560","Text":"That means that we\u0027re looking for the probability of z,"},{"Start":"01:27.560 ","End":"01:29.720","Text":"x minus Mu divided by Sigma,"},{"Start":"01:29.720 ","End":"01:35.850","Text":"that\u0027s z, and it has to be less than 0."},{"Start":"01:35.850 ","End":"01:41.745","Text":"You know why 0? Well, that\u0027s 170 minus 170 divided by 10, that\u0027s 0."},{"Start":"01:41.745 ","End":"01:46.065","Text":"We\u0027re looking at Phi of 0."},{"Start":"01:46.065 ","End":"01:50.880","Text":"Let\u0027s go now to the standard table to see what Phi of 0 is."},{"Start":"01:50.880 ","End":"01:55.460","Text":"Here\u0027s the table and we can see that when z equals 0,"},{"Start":"01:55.460 ","End":"02:01.010","Text":"that means that we\u0027re at this line and this column."},{"Start":"02:01.010 ","End":"02:07.720","Text":"That means that Phi of 0 is 0.5."},{"Start":"02:07.730 ","End":"02:11.970","Text":"Again, that equals to 0.5."},{"Start":"02:11.970 ","End":"02:16.730","Text":"We solve this using 2 methods."},{"Start":"02:16.730 ","End":"02:22.910","Text":"1 method is the formal method where we had x and then we standardized it,"},{"Start":"02:22.910 ","End":"02:27.665","Text":"we went to the table and looked up Phi of 0,"},{"Start":"02:27.665 ","End":"02:35.470","Text":"or we used the characteristic of symmetry around the average of the normal distribution."},{"Start":"02:35.470 ","End":"02:40.900","Text":"Since we\u0027re looking for the probability of x being less than the average,"},{"Start":"02:40.900 ","End":"02:45.640","Text":"it was easily seen that that was 0.5."},{"Start":"02:45.640 ","End":"02:47.840","Text":"In this section, we\u0027re asked,"},{"Start":"02:47.840 ","End":"02:51.815","Text":"what\u0027s the proportion of people who are at most 170 centimeters tall?"},{"Start":"02:51.815 ","End":"02:57.905","Text":"We\u0027re looking for the probability of x being less than or equal to 170."},{"Start":"02:57.905 ","End":"03:00.440","Text":"Now if we recall from the last section,"},{"Start":"03:00.440 ","End":"03:06.500","Text":"we were asked what\u0027s the probability of x being just less than 170?"},{"Start":"03:06.500 ","End":"03:10.265","Text":"Now, because we\u0027re dealing with a continuous random variable,"},{"Start":"03:10.265 ","End":"03:13.700","Text":"then these 2 probabilities are the same,"},{"Start":"03:13.700 ","End":"03:15.380","Text":"they\u0027re equal. Why is that?"},{"Start":"03:15.380 ","End":"03:17.780","Text":"Because when we\u0027re dealing with probabilities,"},{"Start":"03:17.780 ","End":"03:21.470","Text":"we\u0027re looking at areas under the density function,"},{"Start":"03:21.470 ","End":"03:28.680","Text":"and the area under the value 170 is 0."},{"Start":"03:28.680 ","End":"03:31.560","Text":"We\u0027ve talked about this previously."},{"Start":"03:31.560 ","End":"03:36.170","Text":"That means that these 2 probabilities are the same with or"},{"Start":"03:36.170 ","End":"03:40.580","Text":"without the inclusion of 170 centimeters,"},{"Start":"03:40.580 ","End":"03:41.825","Text":"this value right here."},{"Start":"03:41.825 ","End":"03:44.810","Text":"Now if we were talking about discrete random variables,"},{"Start":"03:44.810 ","End":"03:48.440","Text":"for example, the number of cars a family has,"},{"Start":"03:48.440 ","End":"03:53.280","Text":"or the number of pieces of candy on our kitchen table,"},{"Start":"03:53.280 ","End":"03:56.255","Text":"or whatnot, then that would be something else."},{"Start":"03:56.255 ","End":"04:00.605","Text":"But no, we\u0027re dealing with a continuous random variable and as such,"},{"Start":"04:00.605 ","End":"04:08.240","Text":"the area under a specific value is equal to 0."},{"Start":"04:08.240 ","End":"04:13.890","Text":"That means that these probabilities then must be the same."}],"ID":13760},{"Watched":false,"Name":"Exercise 2 - Part a","Duration":"5m 8s","ChapterTopicVideoID":13273,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.960","Text":"In this question we assume that the time it takes for a certain medication to take"},{"Start":"00:03.960 ","End":"00:08.468","Text":"effect has a normal probability distribution with an average of 30 minutes,"},{"Start":"00:08.468 ","End":"00:10.980","Text":"and a variance of 9 minutes and we\u0027re asked,"},{"Start":"00:10.980 ","End":"00:16.755","Text":"what\u0027s the proportion of cases where the medication takes longer than an hour to work."},{"Start":"00:16.755 ","End":"00:20.760","Text":"The first thing that we need to do is define a random variable."},{"Start":"00:20.760 ","End":"00:22.260","Text":"Well, that\u0027ll be x,"},{"Start":"00:22.260 ","End":"00:29.730","Text":"and x would be defined as the time it takes for the medication to take effect."},{"Start":"00:29.730 ","End":"00:37.780","Text":"Let\u0027s just write this out as the time to take effect."},{"Start":"00:38.780 ","End":"00:46.520","Text":"That has a normal distribution where Mu equals to 30 minutes,"},{"Start":"00:46.520 ","End":"00:49.995","Text":"and what about Sigma squared?"},{"Start":"00:49.995 ","End":"00:52.230","Text":"Well, Sigma squared is the variance."},{"Start":"00:52.230 ","End":"00:57.165","Text":"Sigma squared, that equals to 9 minutes."},{"Start":"00:57.165 ","End":"01:04.085","Text":"Now Sigma then equals to the square root of 9, which equals to 3."},{"Start":"01:04.085 ","End":"01:06.860","Text":"Now it\u0027s accepted to write this out like this."},{"Start":"01:06.860 ","End":"01:10.060","Text":"Sigma squared will be 3 squared."},{"Start":"01:10.060 ","End":"01:12.620","Text":"Now that we have all the data,"},{"Start":"01:12.620 ","End":"01:14.690","Text":"we\u0027re asked to find the proportion of cases where"},{"Start":"01:14.690 ","End":"01:17.270","Text":"the medication takes longer than an hour to work."},{"Start":"01:17.270 ","End":"01:20.150","Text":"Let\u0027s just draw the density function and plug"},{"Start":"01:20.150 ","End":"01:23.890","Text":"in the numbers on the under density function."},{"Start":"01:23.890 ","End":"01:26.292","Text":"Here\u0027s our density function,"},{"Start":"01:26.292 ","End":"01:29.150","Text":"now the average Mu equals 30,"},{"Start":"01:29.150 ","End":"01:31.745","Text":"that\u0027s this thing right here."},{"Start":"01:31.745 ","End":"01:34.760","Text":"That\u0027s 30 on the x-axis."},{"Start":"01:34.760 ","End":"01:37.670","Text":"We\u0027re looking for the proportion of"},{"Start":"01:37.670 ","End":"01:40.990","Text":"cases where the medication takes longer than an hour to work."},{"Start":"01:40.990 ","End":"01:42.950","Text":"1 hour is 60 minutes."},{"Start":"01:42.950 ","End":"01:45.320","Text":"Let\u0027s write this thing right here,"},{"Start":"01:45.320 ","End":"01:46.400","Text":"that\u0027s 60 minutes,"},{"Start":"01:46.400 ","End":"01:53.720","Text":"and we\u0027re looking for the probability where x is greater than 60 minutes."},{"Start":"01:53.720 ","End":"01:56.790","Text":"That\u0027s this area right here."},{"Start":"01:57.620 ","End":"02:00.050","Text":"How do we go about doing this?"},{"Start":"02:00.050 ","End":"02:01.955","Text":"Well, we have x,"},{"Start":"02:01.955 ","End":"02:06.290","Text":"we have to standardize it into the standard score that said,"},{"Start":"02:06.290 ","End":"02:11.040","Text":"and then we have to go to the table and find the proportion."},{"Start":"02:11.140 ","End":"02:17.025","Text":"X here, that equals to 60 and"},{"Start":"02:17.025 ","End":"02:22.575","Text":"z that\u0027s x minus Mu divided by Sigma that\u0027s our transformation."},{"Start":"02:22.575 ","End":"02:24.270","Text":"Let\u0027s do that here."},{"Start":"02:24.270 ","End":"02:27.290","Text":"The probability of x being greater than 60,"},{"Start":"02:27.290 ","End":"02:33.130","Text":"well that equals to the probability of x minus Mu divided by Sigma,"},{"Start":"02:33.130 ","End":"02:36.290","Text":"well that\u0027s greater than 60 minus,"},{"Start":"02:36.290 ","End":"02:40.370","Text":"now Mu equals 30 divided by Sigma."},{"Start":"02:40.370 ","End":"02:42.990","Text":"Well Sigma equals 3."},{"Start":"02:43.310 ","End":"02:48.325","Text":"Now that equals to the probability of z,"},{"Start":"02:48.325 ","End":"02:51.205","Text":"x minus Mu divided by Sigma that\u0027s z,"},{"Start":"02:51.205 ","End":"02:54.430","Text":"has to be greater than 60 minus 30,"},{"Start":"02:54.430 ","End":"02:57.620","Text":"well, that 30 divided by 3, that\u0027s 10."},{"Start":"02:57.620 ","End":"03:01.620","Text":"On the z axis right here,"},{"Start":"03:01.620 ","End":"03:08.860","Text":"60 on the x-axis would be comparable to 10 and the z axis."},{"Start":"03:08.860 ","End":"03:18.710","Text":"Since our standard table shows us the probability of z being less than a specific value,"},{"Start":"03:18.710 ","End":"03:22.510","Text":"then let\u0027s just convert this probability and that\u0027d be equal to 1"},{"Start":"03:22.510 ","End":"03:28.785","Text":"minus the probability of z being less than 10."},{"Start":"03:28.785 ","End":"03:36.000","Text":"That is equal to 1 minus Phi of 10."},{"Start":"03:36.000 ","End":"03:41.365","Text":"Now we\u0027re ready to look at our standard table."},{"Start":"03:41.365 ","End":"03:47.625","Text":"This is our table, and we\u0027re looking for z equaling 10."},{"Start":"03:47.625 ","End":"03:50.025","Text":"Let\u0027s look at this column right here,"},{"Start":"03:50.025 ","End":"03:54.280","Text":"and we\u0027ll look for z, that equals to 10."},{"Start":"03:55.910 ","End":"04:00.915","Text":"Now here the highest value of z is 3.4,"},{"Start":"04:00.915 ","End":"04:02.730","Text":"and in the whole of the table,"},{"Start":"04:02.730 ","End":"04:08.620","Text":"that\u0027s 3.49, that\u0027s this value right here."},{"Start":"04:08.690 ","End":"04:12.870","Text":"Anything above this value,"},{"Start":"04:12.870 ","End":"04:15.360","Text":"3.49 or 3.5,"},{"Start":"04:15.360 ","End":"04:18.180","Text":"well, it\u0027s so close to 1,"},{"Start":"04:18.180 ","End":"04:22.400","Text":"it\u0027s insignificant, so that means that if we\u0027re looking for"},{"Start":"04:22.400 ","End":"04:27.410","Text":"Phi of 10 which is way higher than 3.5,"},{"Start":"04:27.410 ","End":"04:31.710","Text":"well, that equals almost to 1."},{"Start":"04:31.710 ","End":"04:34.725","Text":"Now that we have Phi of 10,"},{"Start":"04:34.725 ","End":"04:39.304","Text":"let\u0027s go back to our question. Here we go."},{"Start":"04:39.304 ","End":"04:45.150","Text":"This then equals to 1 minus Phi of 10,"},{"Start":"04:45.150 ","End":"04:48.190","Text":"as we said is equal to 1."},{"Start":"04:48.260 ","End":"04:52.775","Text":"This is almost equal to 1 minus 1,"},{"Start":"04:52.775 ","End":"04:55.385","Text":"that equals to 0."},{"Start":"04:55.385 ","End":"04:58.520","Text":"What we\u0027re looking at is"},{"Start":"04:58.520 ","End":"05:02.744","Text":"the proportion of cases where the medication takes longer than an hour to work,"},{"Start":"05:02.744 ","End":"05:06.380","Text":"well, that\u0027s 0 or 0 percent."}],"ID":13761},{"Watched":false,"Name":"Exercise 2 - Parts b-c","Duration":"6m 2s","ChapterTopicVideoID":13275,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.350","Text":"In this section, we\u0027re asked,"},{"Start":"00:01.350 ","End":"00:03.840","Text":"what\u0027s the proportion of cases with the medication takes"},{"Start":"00:03.840 ","End":"00:06.990","Text":"between 35 and 37 minutes to work?"},{"Start":"00:06.990 ","End":"00:17.205","Text":"We\u0027re looking for the probability where x is between 35 and 37 minutes."},{"Start":"00:17.205 ","End":"00:23.240","Text":"Let\u0027s just draw our density function and plug in these numbers."},{"Start":"00:23.240 ","End":"00:26.010","Text":"Here we have our density function."},{"Start":"00:26.010 ","End":"00:27.615","Text":"The average is 30,"},{"Start":"00:27.615 ","End":"00:31.695","Text":"and we\u0027re interested in this range right here,"},{"Start":"00:31.695 ","End":"00:36.630","Text":"between 35 and 37."},{"Start":"00:36.630 ","End":"00:39.510","Text":"That\u0027s this range right here."},{"Start":"00:39.510 ","End":"00:42.175","Text":"How do we go about doing that?"},{"Start":"00:42.175 ","End":"00:43.640","Text":"Well, as we said,"},{"Start":"00:43.640 ","End":"00:45.440","Text":"we have to take our x,"},{"Start":"00:45.440 ","End":"00:48.830","Text":"we have to standardize it into standard scores,"},{"Start":"00:48.830 ","End":"00:56.917","Text":"that\u0027s z, and then we have to go to the standard table to calculate the proportions."},{"Start":"00:56.917 ","End":"01:00.515","Text":"We have to standardize x for both values,"},{"Start":"01:00.515 ","End":"01:03.755","Text":"for 35 and 37. Let\u0027s do that."},{"Start":"01:03.755 ","End":"01:07.550","Text":"So z for x equals 35,"},{"Start":"01:07.550 ","End":"01:15.025","Text":"that equals to 35 minus 30 divided by 3."},{"Start":"01:15.025 ","End":"01:18.575","Text":"X minus Mu divided by Sigma."},{"Start":"01:18.575 ","End":"01:21.980","Text":"That equals to 5 divided by 3."},{"Start":"01:21.980 ","End":"01:25.580","Text":"That\u0027s 1.67 approximately."},{"Start":"01:25.580 ","End":"01:30.005","Text":"Now z for 37."},{"Start":"01:30.005 ","End":"01:35.105","Text":"That equals to 37 minus 30 divided by 3."},{"Start":"01:35.105 ","End":"01:44.225","Text":"Again, that\u0027s 7 divided by 3, that\u0027s 2.33."},{"Start":"01:44.225 ","End":"01:47.089","Text":"What\u0027s the correlation here,"},{"Start":"01:47.089 ","End":"01:49.370","Text":"or what\u0027s comparable to these numbers?"},{"Start":"01:49.370 ","End":"01:53.210","Text":"Well,135 again, that\u0027s 1.67,"},{"Start":"01:53.210 ","End":"01:57.265","Text":"and here that\u0027ll be 2.33."},{"Start":"01:57.265 ","End":"02:00.890","Text":"Now how do we calculate this area right here?"},{"Start":"02:00.890 ","End":"02:09.770","Text":"We\u0027re going to have to take the area from minus infinity till 2.33,"},{"Start":"02:09.770 ","End":"02:17.000","Text":"and we have to subtract the area from minus infinity till 1.67."},{"Start":"02:17.000 ","End":"02:20.480","Text":"That\u0027ll give us this area right here."},{"Start":"02:20.480 ","End":"02:22.010","Text":"Let\u0027s do that."},{"Start":"02:22.010 ","End":"02:26.990","Text":"We have probability of 35,"},{"Start":"02:26.990 ","End":"02:29.960","Text":"being less than x,"},{"Start":"02:29.960 ","End":"02:32.915","Text":"or x being between 35 and 37,"},{"Start":"02:32.915 ","End":"02:36.860","Text":"that equals to the probability of"},{"Start":"02:36.860 ","End":"02:44.630","Text":"z being between 1.67 and 2.33."},{"Start":"02:44.630 ","End":"02:53.195","Text":"As we said, that equals to Phi of 2.33."},{"Start":"02:53.195 ","End":"02:58.340","Text":"That\u0027s this area right here from minus infinity till 2.33,"},{"Start":"02:58.340 ","End":"03:05.270","Text":"and we have to subtract Phi of 1.67."},{"Start":"03:05.270 ","End":"03:08.855","Text":"That\u0027s this area from minus infinity to 1.67."},{"Start":"03:08.855 ","End":"03:13.310","Text":"What\u0027s left is this area right here."},{"Start":"03:13.310 ","End":"03:16.650","Text":"Now we\u0027re ready to go to our table."},{"Start":"03:17.050 ","End":"03:27.435","Text":"For z equaling 1.67"},{"Start":"03:27.435 ","End":"03:31.980","Text":"or Phi 1.67 equals,"},{"Start":"03:31.980 ","End":"03:37.865","Text":"well, here\u0027s 1.6 and here\u0027s the 7."},{"Start":"03:37.865 ","End":"03:42.260","Text":"So the intersection right here is this number right here."},{"Start":"03:42.260 ","End":"03:47.640","Text":"That equals to 0.9525."},{"Start":"03:47.920 ","End":"03:53.435","Text":"What about z equaling 2.33?"},{"Start":"03:53.435 ","End":"03:56.930","Text":"Phi of 2.33,"},{"Start":"03:56.930 ","End":"04:05.610","Text":"that equals, let\u0027s just scroll down here, 2.33."},{"Start":"04:05.610 ","End":"04:07.545","Text":"That\u0027s 2.3 right here,"},{"Start":"04:07.545 ","End":"04:09.060","Text":"that\u0027s 0, 1, 2,"},{"Start":"04:09.060 ","End":"04:12.875","Text":"3, that\u0027s this value right here."},{"Start":"04:12.875 ","End":"04:17.540","Text":"That equals to 0.9901."},{"Start":"04:17.540 ","End":"04:22.235","Text":"Let\u0027s now go back to our question. Here we go."},{"Start":"04:22.235 ","End":"04:25.850","Text":"Phi of 2.33, as we said,"},{"Start":"04:25.850 ","End":"04:33.345","Text":"that\u0027s 0.9901 minus Phi of 1.67."},{"Start":"04:33.345 ","End":"04:36.945","Text":"That\u0027s 0.9525."},{"Start":"04:36.945 ","End":"04:42.770","Text":"That equals to 0.0376."},{"Start":"04:42.770 ","End":"04:45.830","Text":"So the proportion of cases where the medication takes"},{"Start":"04:45.830 ","End":"04:56.395","Text":"between 35 and 37 minutes is 0.0376 or 3.76%."},{"Start":"04:56.395 ","End":"04:58.340","Text":"In this section we\u0027re asked,"},{"Start":"04:58.340 ","End":"05:02.330","Text":"what are the chances that the medication will help after exactly 36 minutes?"},{"Start":"05:02.330 ","End":"05:08.675","Text":"We\u0027re looking for the probability of x being equal to 36."},{"Start":"05:08.675 ","End":"05:12.050","Text":"We don\u0027t have to do a lot of calculations here."},{"Start":"05:12.050 ","End":"05:14.240","Text":"We know that since this is"},{"Start":"05:14.240 ","End":"05:18.185","Text":"a continuous variable and we\u0027re dealing with a continuous distribution,"},{"Start":"05:18.185 ","End":"05:21.485","Text":"the probability of x being equal to something,"},{"Start":"05:21.485 ","End":"05:24.575","Text":"that\u0027s always equals to 0."},{"Start":"05:24.575 ","End":"05:27.560","Text":"Again, let\u0027s just take a look why."},{"Start":"05:27.560 ","End":"05:30.230","Text":"We have our density function here."},{"Start":"05:30.230 ","End":"05:32.795","Text":"This is our average 30,"},{"Start":"05:32.795 ","End":"05:36.380","Text":"and we\u0027re looking at 36."},{"Start":"05:36.380 ","End":"05:39.680","Text":"What\u0027s the probability of x being equaling 36?"},{"Start":"05:39.680 ","End":"05:45.200","Text":"What\u0027s the area under the density function at exactly 36, the point 36?"},{"Start":"05:45.200 ","End":"05:46.820","Text":"That equals to 0."},{"Start":"05:46.820 ","End":"05:48.920","Text":"Again, let\u0027s remember,"},{"Start":"05:48.920 ","End":"05:52.760","Text":"whenever we have a continuous random variable and we\u0027re asked for"},{"Start":"05:52.760 ","End":"05:57.035","Text":"the probability that that variable equals something,"},{"Start":"05:57.035 ","End":"05:58.910","Text":"that always equals to 0,"},{"Start":"05:58.910 ","End":"06:02.430","Text":"the probability always equals 0."}],"ID":13762},{"Watched":false,"Name":"Exercise 2 - Part d","Duration":"5m 46s","ChapterTopicVideoID":13274,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.660","Text":"In this section we\u0027re asked what\u0027s the proportion of cases where the time"},{"Start":"00:03.660 ","End":"00:07.380","Text":"taken by the medication to work deviates from 30 minutes."},{"Start":"00:07.380 ","End":"00:10.950","Text":"That\u0027s our average by less than 3 minutes."},{"Start":"00:10.950 ","End":"00:13.500","Text":"Now, in order to make sense of this question,"},{"Start":"00:13.500 ","End":"00:16.665","Text":"let\u0027s draw the density function."},{"Start":"00:16.665 ","End":"00:19.140","Text":"Here\u0027s our density function,"},{"Start":"00:19.140 ","End":"00:25.410","Text":"and on the x-axis we have our average that\u0027s 30,"},{"Start":"00:25.410 ","End":"00:28.130","Text":"and what are we looking for?"},{"Start":"00:28.130 ","End":"00:31.520","Text":"We\u0027re looking for the proportion of cases,"},{"Start":"00:31.520 ","End":"00:33.875","Text":"that\u0027s the area under the graph."},{"Start":"00:33.875 ","End":"00:36.650","Text":"Where the time taken by the medication to work deviates"},{"Start":"00:36.650 ","End":"00:39.110","Text":"from the average by less than 3 minutes."},{"Start":"00:39.110 ","End":"00:44.119","Text":"That means that we\u0027re looking for 30 plus minus 3 minutes,"},{"Start":"00:44.119 ","End":"00:46.390","Text":"so that\u0027s 27 on this side,"},{"Start":"00:46.390 ","End":"00:48.870","Text":"and 33 on this side."},{"Start":"00:48.870 ","End":"00:54.870","Text":"These are the boundaries of the range that we\u0027re interested in."},{"Start":"00:55.480 ","End":"01:02.180","Text":"We\u0027re looking then for the probability where X is"},{"Start":"01:02.180 ","End":"01:08.059","Text":"between 27 and 33 minutes,"},{"Start":"01:08.059 ","End":"01:11.470","Text":"that\u0027s the proportion that we\u0027re looking for."},{"Start":"01:11.470 ","End":"01:14.855","Text":"Now, how do we isolate this area right here?"},{"Start":"01:14.855 ","End":"01:16.400","Text":"But that\u0027s pretty simple."},{"Start":"01:16.400 ","End":"01:24.718","Text":"All we have to do is we have to calculate the area from minus infinity to 33,"},{"Start":"01:24.718 ","End":"01:32.080","Text":"and subtract from that the area from minus infinity to 27,"},{"Start":"01:32.080 ","End":"01:36.395","Text":"and what\u0027s left is the area in yellow right here."},{"Start":"01:36.395 ","End":"01:41.060","Text":"Having done that, how do we calculate this proportion?"},{"Start":"01:41.060 ","End":"01:44.750","Text":"Well, we have x."},{"Start":"01:44.750 ","End":"01:51.365","Text":"These are the steps that we have to take to standardize the values of x, we have x,"},{"Start":"01:51.365 ","End":"01:54.650","Text":"we have to standardize them into standard scores,"},{"Start":"01:54.650 ","End":"01:59.090","Text":"then we have to go to our standard table to find the proportions."},{"Start":"01:59.090 ","End":"02:03.045","Text":"Now here, X we have 2 values of x."},{"Start":"02:03.045 ","End":"02:05.130","Text":"We have 27 and 33,"},{"Start":"02:05.130 ","End":"02:09.980","Text":"and Z is our standardization transformation."},{"Start":"02:09.980 ","End":"02:14.270","Text":"That\u0027s X minus mu divided by Sigma."},{"Start":"02:14.270 ","End":"02:18.350","Text":"Let\u0027s do that. Let\u0027s standardize our X."},{"Start":"02:18.350 ","End":"02:25.825","Text":"Now, that means that we have the probability here of X minus mu divided by Sigma,"},{"Start":"02:25.825 ","End":"02:30.900","Text":"and that has to be between 33 minus mu."},{"Start":"02:30.900 ","End":"02:34.640","Text":"Mu is 30 divided by sigma,"},{"Start":"02:34.640 ","End":"02:36.740","Text":"where Sigma is 3,"},{"Start":"02:36.740 ","End":"02:42.875","Text":"and on the other side that will be 27 minus 30 divided by 3,"},{"Start":"02:42.875 ","End":"02:50.095","Text":"and that equals to the probability of X minus mu divided by Sigma, that\u0027s Z."},{"Start":"02:50.095 ","End":"02:54.905","Text":"Z then has to be between 33 minus 30."},{"Start":"02:54.905 ","End":"02:57.335","Text":"Well, that\u0027s 3 divided by 3, that\u0027s 1,"},{"Start":"02:57.335 ","End":"03:00.275","Text":"and 27 minus 30 divided by 3,"},{"Start":"03:00.275 ","End":"03:02.815","Text":"well, that\u0027s minus 1."},{"Start":"03:02.815 ","End":"03:07.310","Text":"These values are comparable to 27,"},{"Start":"03:07.310 ","End":"03:09.485","Text":"and 33 on the z scale."},{"Start":"03:09.485 ","End":"03:12.028","Text":"For 27 we have minus 1,"},{"Start":"03:12.028 ","End":"03:15.030","Text":"and for 33 we have plus 1."},{"Start":"03:15.320 ","End":"03:21.845","Text":"As we said, in order to isolate this area right here that we\u0027re interested in,"},{"Start":"03:21.845 ","End":"03:28.378","Text":"we\u0027re going to have to subtract the areas from minus infinity to 1,"},{"Start":"03:28.378 ","End":"03:31.250","Text":"and the Z scale, and we have to subtract from that,"},{"Start":"03:31.250 ","End":"03:34.385","Text":"the area from minus infinity to minus 1,"},{"Start":"03:34.385 ","End":"03:42.720","Text":"and that means that we\u0027re looking at Phi of 1 minus Phi of minus 1."},{"Start":"03:42.720 ","End":"03:48.170","Text":"Great. Now we\u0027re ready to look at are standard table."},{"Start":"03:48.170 ","End":"03:50.680","Text":"Here\u0027s our standard table,"},{"Start":"03:50.680 ","End":"03:55.820","Text":"and we\u0027re interested in are standard score where Z equals 1,"},{"Start":"03:55.820 ","End":"03:58.988","Text":"and we\u0027re looking for Phi of 1."},{"Start":"03:58.988 ","End":"04:02.053","Text":"Well, let\u0027s look at Z,"},{"Start":"04:02.053 ","End":"04:05.420","Text":"and try to identify where Z equals 1."},{"Start":"04:05.420 ","End":"04:06.965","Text":"Well, it\u0027s right here."},{"Start":"04:06.965 ","End":"04:10.310","Text":"This is a row, and it\u0027s 1.0,"},{"Start":"04:10.310 ","End":"04:12.710","Text":"that means that this is a column right here,"},{"Start":"04:12.710 ","End":"04:15.830","Text":"and the intersection is this value right here."},{"Start":"04:15.830 ","End":"04:21.220","Text":"That means that 5 1 equals 0.8413."},{"Start":"04:22.790 ","End":"04:25.070","Text":"Now that we have that,"},{"Start":"04:25.070 ","End":"04:27.185","Text":"let\u0027s go back to our question."},{"Start":"04:27.185 ","End":"04:37.115","Text":"Here we go, so 5 1 is we said that equals to 0.8413 minus."},{"Start":"04:37.115 ","End":"04:39.425","Text":"Now, what\u0027s 5 minus 1?"},{"Start":"04:39.425 ","End":"04:43.100","Text":"Well, we learned previously that for any value,"},{"Start":"04:43.100 ","End":"04:45.065","Text":"for any negative value,"},{"Start":"04:45.065 ","End":"04:47.720","Text":"Phi, let\u0027s say of minus a."},{"Start":"04:47.720 ","End":"04:51.725","Text":"Well that equals to 1 minus Phi of a."},{"Start":"04:51.725 ","End":"04:57.215","Text":"Now here, minus a is equal to minus 1, a equals 1."},{"Start":"04:57.215 ","End":"05:00.650","Text":"That means that 5 minus 1,"},{"Start":"05:00.650 ","End":"05:06.340","Text":"that\u0027s 1 minus Phi of 1."},{"Start":"05:06.340 ","End":"05:09.680","Text":"Now, let\u0027s just calculate that."},{"Start":"05:09.680 ","End":"05:19.490","Text":"That will be equal to 0.8413 minus 1 minus 0.8413,"},{"Start":"05:19.490 ","End":"05:23.670","Text":"and that equals to"},{"Start":"05:23.670 ","End":"05:30.665","Text":"0.6826 or in percentages,"},{"Start":"05:30.665 ","End":"05:36.720","Text":"that\u0027ll be equal to 68.26 percent."},{"Start":"05:36.720 ","End":"05:41.210","Text":"Now, that is the proportion of cases where the time taken by"},{"Start":"05:41.210 ","End":"05:47.460","Text":"the medication to work deviates from the average by less than 3 minutes."}],"ID":13763},{"Watched":false,"Name":"Exercise 3 - Part a","Duration":"6m 17s","ChapterTopicVideoID":13276,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"The weight of people in a given population has normal probability distribution"},{"Start":"00:04.290 ","End":"00:09.150","Text":"with an average of 60 kilograms and a standard deviation of 8 kilograms."},{"Start":"00:09.150 ","End":"00:15.450","Text":"Then we\u0027re asked, what\u0027s the proportion of people who weigh less than 55 kilograms?"},{"Start":"00:15.450 ","End":"00:18.990","Text":"The first thing that we need to know is that we\u0027re dealing with"},{"Start":"00:18.990 ","End":"00:23.685","Text":"a normal probability distribution and that\u0027s right here."},{"Start":"00:23.685 ","End":"00:26.760","Text":"That\u0027s given to us."},{"Start":"00:26.760 ","End":"00:32.055","Text":"The next thing that we need to do is to define our random variable."},{"Start":"00:32.055 ","End":"00:38.990","Text":"Well, we\u0027ll call that x and that\u0027ll be the weight of the people in kilograms."},{"Start":"00:38.990 ","End":"00:45.365","Text":"Let\u0027s just write that down in kilograms and that has, as we said,"},{"Start":"00:45.365 ","End":"00:51.125","Text":"a normal distribution where Mu equals 60,"},{"Start":"00:51.125 ","End":"00:56.775","Text":"it\u0027s right here, and a standard deviation of 8 kilograms."},{"Start":"00:56.775 ","End":"00:59.090","Text":"That means that Sigma squared,"},{"Start":"00:59.090 ","End":"01:02.430","Text":"that equals to 8 squared."},{"Start":"01:02.960 ","End":"01:05.030","Text":"What are we looking for?"},{"Start":"01:05.030 ","End":"01:08.890","Text":"The proportion of people who weigh less than 55 kilograms."},{"Start":"01:08.890 ","End":"01:16.480","Text":"We\u0027re looking for the probability that x is less than 55."},{"Start":"01:16.550 ","End":"01:19.290","Text":"Now we have all this data,"},{"Start":"01:19.290 ","End":"01:26.350","Text":"let\u0027s just draw the density function and put all this data on the graph."},{"Start":"01:26.350 ","End":"01:29.010","Text":"Here is our density function."},{"Start":"01:29.010 ","End":"01:33.535","Text":"We have both the x-axis and the z-axis."},{"Start":"01:33.535 ","End":"01:35.345","Text":"Now on the x-axis,"},{"Start":"01:35.345 ","End":"01:37.250","Text":"our average is 60."},{"Start":"01:37.250 ","End":"01:43.670","Text":"Well that\u0027s this line right here so we\u0027ll put 60 here and what are we looking for?"},{"Start":"01:43.670 ","End":"01:48.800","Text":"We\u0027re looking for the probability of x being less than 55."},{"Start":"01:48.800 ","End":"01:57.355","Text":"Let\u0027s just put 55 here again on the x-axis and we\u0027re looking for this area right here."},{"Start":"01:57.355 ","End":"01:59.950","Text":"Now as we can see,"},{"Start":"01:59.950 ","End":"02:02.990","Text":"this area right here is less than 50 percent."},{"Start":"02:02.990 ","End":"02:06.575","Text":"Why is that? Because from minus infinity to 60,"},{"Start":"02:06.575 ","End":"02:08.635","Text":"well, that\u0027s 50 percent."},{"Start":"02:08.635 ","End":"02:12.830","Text":"That\u0027s half of the area under the density function so"},{"Start":"02:12.830 ","End":"02:17.375","Text":"for looking at the area from minus infinity to 55,"},{"Start":"02:17.375 ","End":"02:20.760","Text":"well, that\u0027s less than 50 percent."},{"Start":"02:21.560 ","End":"02:24.200","Text":"How do we get this area?"},{"Start":"02:24.200 ","End":"02:26.345","Text":"How do we find this proportion?"},{"Start":"02:26.345 ","End":"02:28.310","Text":"Well, what are our steps?"},{"Start":"02:28.310 ","End":"02:34.130","Text":"We have x, we have to standardize it into Z our standard score,"},{"Start":"02:34.130 ","End":"02:39.770","Text":"and then we have to go to the standard table to find the proportion."},{"Start":"02:39.770 ","End":"02:47.570","Text":"In our case, x is 55 and Z or the transformation of x,"},{"Start":"02:47.570 ","End":"02:51.691","Text":"that\u0027s x minus Mu divided by Sigma,"},{"Start":"02:51.691 ","End":"02:54.100","Text":"so having said that,"},{"Start":"02:54.100 ","End":"02:57.030","Text":"let\u0027s see what the proportion is,"},{"Start":"02:57.030 ","End":"03:00.120","Text":"what\u0027s the probability of x being less than 55 is."},{"Start":"03:00.120 ","End":"03:06.665","Text":"Well, that equals to the probability of x minus Mu divided by Sigma,"},{"Start":"03:06.665 ","End":"03:12.210","Text":"that being less than now, 55 minus Mu."},{"Start":"03:12.210 ","End":"03:16.995","Text":"Now, Mu here is 60 divided by Sigma,"},{"Start":"03:16.995 ","End":"03:19.775","Text":"and Sigma is 8 divided by 8."},{"Start":"03:19.775 ","End":"03:23.240","Text":"Well, that equals to the probability."},{"Start":"03:23.240 ","End":"03:26.360","Text":"Now x minus Mu divided by Sigma well that\u0027s Z."},{"Start":"03:26.360 ","End":"03:32.555","Text":"That Z has to be less than 55 minus 60 divided by 8,"},{"Start":"03:32.555 ","End":"03:38.660","Text":"that\u0027s minus 5 divided by 8 that equals to 0.625."},{"Start":"03:38.660 ","End":"03:45.176","Text":"Let\u0027s just round that up to 0.63 and that\u0027s minus,"},{"Start":"03:45.176 ","End":"03:55.220","Text":"so we\u0027re looking for Phi of minus 0.63."},{"Start":"03:55.220 ","End":"04:01.710","Text":"Now, if we recall Phi of minus a,"},{"Start":"04:01.710 ","End":"04:08.160","Text":"any value that equals to 1 minus Phi of a."},{"Start":"04:08.160 ","End":"04:09.755","Text":"Why is that? Again,"},{"Start":"04:09.755 ","End":"04:18.115","Text":"because the normal density function is symmetrical around the average."},{"Start":"04:18.115 ","End":"04:25.320","Text":"This area right here is equal to this area right here where"},{"Start":"04:25.320 ","End":"04:32.850","Text":"here it\u0027s 65 and what are the comparable values on the z-axis?"},{"Start":"04:32.850 ","End":"04:37.680","Text":"Well, at 55, it was minus 0.63."},{"Start":"04:37.680 ","End":"04:43.305","Text":"Here it would be plus 0.63."},{"Start":"04:43.305 ","End":"04:50.325","Text":"Now we\u0027re ready to go to our standard table."},{"Start":"04:50.325 ","End":"04:55.800","Text":"That\u0027ll be 1 minus Phi of 0.63."},{"Start":"04:55.800 ","End":"05:00.700","Text":"Let\u0027s see what Phi 0.63 is."},{"Start":"05:01.100 ","End":"05:08.615","Text":"Here is our table and we\u0027re looking for Phi of 0.63."},{"Start":"05:08.615 ","End":"05:12.890","Text":"Well, here is 0.6,"},{"Start":"05:12.890 ","End":"05:14.710","Text":"that\u0027s this row right here,"},{"Start":"05:14.710 ","End":"05:16.915","Text":"and here is that 3."},{"Start":"05:16.915 ","End":"05:21.965","Text":"We\u0027re looking at 0.63 and what\u0027s the intersection right here?"},{"Start":"05:21.965 ","End":"05:24.425","Text":"That\u0027s this value right here."},{"Start":"05:24.425 ","End":"05:30.540","Text":"That equals to 0.7357."},{"Start":"05:30.740 ","End":"05:36.605","Text":"Let\u0027s get back to our question now. Here we go."},{"Start":"05:36.605 ","End":"05:40.025","Text":"That means that this then equals 1 minus"},{"Start":"05:40.025 ","End":"05:47.765","Text":"0.7357 and"},{"Start":"05:47.765 ","End":"05:55.800","Text":"that equals to 0.2643."},{"Start":"05:55.800 ","End":"05:59.600","Text":"If we\u0027re looking to show this in percentages,"},{"Start":"05:59.600 ","End":"06:05.030","Text":"well that equals to 26.43 percent."},{"Start":"06:05.030 ","End":"06:08.510","Text":"Now, this then is the proportion of people who weigh"},{"Start":"06:08.510 ","End":"06:12.080","Text":"less than 55 kilos and as we can see,"},{"Start":"06:12.080 ","End":"06:17.190","Text":"this is less than 50 percent as we said previously here."}],"ID":13764},{"Watched":false,"Name":"Exercise 3 - Parts b-c","Duration":"7m 10s","ChapterTopicVideoID":13277,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In this section, we\u0027re asked what\u0027s the proportion of people in"},{"Start":"00:02.850 ","End":"00:06.600","Text":"the population who weigh at least 50 kilograms."},{"Start":"00:06.600 ","End":"00:11.265","Text":"Let\u0027s just look at our density function, see where we\u0027re at."},{"Start":"00:11.265 ","End":"00:19.500","Text":"We\u0027re looking at the probability of X being greater than 50 or 50 kilograms."},{"Start":"00:19.500 ","End":"00:24.880","Text":"That\u0027s the proportion that\u0027s colored in yellow right here."},{"Start":"00:24.880 ","End":"00:26.450","Text":"Now, how do we do that?"},{"Start":"00:26.450 ","End":"00:28.699","Text":"How do we calculate this proportion?"},{"Start":"00:28.699 ","End":"00:32.390","Text":"Well, the process is that we have our X and we have"},{"Start":"00:32.390 ","End":"00:36.395","Text":"to standardize it into our standard score, that\u0027s Z."},{"Start":"00:36.395 ","End":"00:41.720","Text":"Then we have to go to the standard table to find the proportion."},{"Start":"00:41.720 ","End":"00:44.840","Text":"Here X is 50."},{"Start":"00:44.840 ","End":"00:52.580","Text":"Now, the standardization transformation is x minus Mu divided by Sigma."},{"Start":"00:52.580 ","End":"00:55.790","Text":"Let\u0027s do this standardization right now."},{"Start":"00:55.790 ","End":"01:01.695","Text":"That\u0027s the probability of X minus Mu divided by Sigma,"},{"Start":"01:01.695 ","End":"01:04.675","Text":"that has to be greater than 50."},{"Start":"01:04.675 ","End":"01:07.115","Text":"Now, Mu is 60,"},{"Start":"01:07.115 ","End":"01:10.740","Text":"divided by Sigma, that\u0027s 8."},{"Start":"01:10.740 ","End":"01:15.625","Text":"Now, that equals to the probability of"},{"Start":"01:15.625 ","End":"01:24.560","Text":"Z being greater than minus 10 divided by 8, that\u0027s minus 1.25."},{"Start":"01:24.980 ","End":"01:28.780","Text":"We have this probability right here."},{"Start":"01:28.780 ","End":"01:30.580","Text":"Now, how do we deal with this?"},{"Start":"01:30.580 ","End":"01:35.755","Text":"Well, whenever we have Z that\u0027s greater than a negative number,"},{"Start":"01:35.755 ","End":"01:38.515","Text":"well, how do we calculate this?"},{"Start":"01:38.515 ","End":"01:43.540","Text":"Well, we have to remember 1 of the characteristics of the normal distribution is that"},{"Start":"01:43.540 ","End":"01:49.380","Text":"it\u0027s symmetrical around the average. What does that mean?"},{"Start":"01:49.380 ","End":"01:55.495","Text":"Well, let\u0027s take a look at these 2 identical distributions right here."},{"Start":"01:55.495 ","End":"02:01.360","Text":"Now, assuming that this right here is minus a, well,"},{"Start":"02:01.360 ","End":"02:05.800","Text":"the area from minus infinity to minus a, well,"},{"Start":"02:05.800 ","End":"02:12.340","Text":"that equals to the area from a here to plus infinity."},{"Start":"02:12.340 ","End":"02:16.285","Text":"Both areas in red are identical."},{"Start":"02:16.285 ","End":"02:22.280","Text":"That means then that these areas in green are also identical."},{"Start":"02:22.580 ","End":"02:28.510","Text":"If we have a to be equal to 1.25,"},{"Start":"02:28.510 ","End":"02:32.350","Text":"so this thing right here will be minus 1.25,"},{"Start":"02:32.350 ","End":"02:43.230","Text":"and here that\u0027ll be Phi of minus 1.25."},{"Start":"02:43.230 ","End":"02:50.210","Text":"This area right here is actually the same as this area right here."},{"Start":"02:50.330 ","End":"02:56.235","Text":"1 minus 1.25,"},{"Start":"02:56.235 ","End":"03:03.400","Text":"that equals to Phi of 1.25."},{"Start":"03:06.020 ","End":"03:09.010","Text":"This probability right here,"},{"Start":"03:09.010 ","End":"03:15.190","Text":"actually equals to Phi of 1.25."},{"Start":"03:15.650 ","End":"03:23.815","Text":"Now we\u0027re ready to go to our standard table to see what the proportion is."},{"Start":"03:23.815 ","End":"03:30.200","Text":"Here\u0027s our table and we\u0027re looking for Phi of 1.25."},{"Start":"03:30.200 ","End":"03:34.070","Text":"That means that we\u0027re going down here until we get"},{"Start":"03:34.070 ","End":"03:39.390","Text":"to this row and this column right here."},{"Start":"03:39.390 ","End":"03:47.760","Text":"Phi of 1.25 was Z equals 1.25 and that\u0027s 1.25, that\u0027s right here."},{"Start":"03:47.760 ","End":"03:52.040","Text":"The intersection here is this value right here,"},{"Start":"03:52.040 ","End":"03:57.160","Text":"that equals to 0.8944."},{"Start":"03:57.160 ","End":"04:00.240","Text":"Let\u0027s get back to our question now."},{"Start":"04:00.240 ","End":"04:06.690","Text":"Phi of 1.25, well that equals, as we saw,"},{"Start":"04:06.690 ","End":"04:09.220","Text":"to 0.8944,"},{"Start":"04:11.440 ","End":"04:19.100","Text":"or in percentages that will be equal to 89.44 percent."},{"Start":"04:19.100 ","End":"04:21.290","Text":"Now, this is the proportion of the people in"},{"Start":"04:21.290 ","End":"04:26.915","Text":"the population who weigh at least 50 kilograms."},{"Start":"04:26.915 ","End":"04:30.950","Text":"In this section, we\u0027re asked what\u0027s the relative frequency of the people in"},{"Start":"04:30.950 ","End":"04:34.775","Text":"the population who weigh between 60 and 70 kilograms."},{"Start":"04:34.775 ","End":"04:36.830","Text":"When we\u0027re talking about relative frequency,"},{"Start":"04:36.830 ","End":"04:41.090","Text":"we\u0027re looking at the proportion or the probability, it\u0027s the same thing."},{"Start":"04:41.090 ","End":"04:45.305","Text":"Let\u0027s just draw a density function and see where we\u0027re at."},{"Start":"04:45.305 ","End":"04:49.879","Text":"We\u0027re looking to calculate this area right here in green."},{"Start":"04:49.879 ","End":"04:58.770","Text":"Now, that\u0027s the probability of X being between 60 and 70 kilograms,"},{"Start":"04:58.770 ","End":"05:00.800","Text":"that\u0027s what we want to find out."},{"Start":"05:00.800 ","End":"05:04.235","Text":"Now, how do we isolate this area?"},{"Start":"05:04.235 ","End":"05:09.350","Text":"Well, all we have to do is to calculate the area from minus infinity to"},{"Start":"05:09.350 ","End":"05:16.030","Text":"70 and subtract from that the area from minus infinity to 60."},{"Start":"05:16.030 ","End":"05:20.765","Text":"Now on the z scale,60 is comparable to 0."},{"Start":"05:20.765 ","End":"05:26.200","Text":"Now, what does 70 compare to on the z-scale?"},{"Start":"05:26.200 ","End":"05:28.795","Text":"Well, let\u0027s standardize 70."},{"Start":"05:28.795 ","End":"05:35.525","Text":"Z is equal to 70 minus 60 divided by 8,"},{"Start":"05:35.525 ","End":"05:38.405","Text":"that equals to 10 divided by 8,"},{"Start":"05:38.405 ","End":"05:41.820","Text":"that equals to 1.25."},{"Start":"05:41.990 ","End":"05:52.365","Text":"So 1.25, we know what Phi of 1.25 is from the last section,"},{"Start":"05:52.365 ","End":"05:57.390","Text":"that equals to 0.8944."},{"Start":"05:57.390 ","End":"05:58.815","Text":"Now why do we need that?"},{"Start":"05:58.815 ","End":"06:04.600","Text":"Well, that\u0027s 1.25 here."},{"Start":"06:04.600 ","End":"06:07.070","Text":"In order to calculate this area,"},{"Start":"06:07.070 ","End":"06:12.120","Text":"this is comparable to saying Phi of"},{"Start":"06:12.120 ","End":"06:18.765","Text":"1.25 minus Phi of 0."},{"Start":"06:18.765 ","End":"06:21.620","Text":"Now, Phi of 1.25,"},{"Start":"06:21.620 ","End":"06:23.975","Text":"we know, as we said from the last section,"},{"Start":"06:23.975 ","End":"06:30.340","Text":"that\u0027s 0.8944 minus the Phi of 0."},{"Start":"06:30.340 ","End":"06:33.560","Text":"Well, we don\u0027t have to go to the table for that we know that"},{"Start":"06:33.560 ","End":"06:36.785","Text":"this is 0.5. How do we know that?"},{"Start":"06:36.785 ","End":"06:45.635","Text":"Well, we know that half of the area lies between minus infinity and 0 on the Z scale."},{"Start":"06:45.635 ","End":"06:55.795","Text":"Here, that equals to 0.3944,"},{"Start":"06:55.795 ","End":"07:01.980","Text":"or in percentages that would be 39.44 percent."},{"Start":"07:01.980 ","End":"07:04.670","Text":"That\u0027s the relative frequency of the people in"},{"Start":"07:04.670 ","End":"07:10.020","Text":"the population who weigh between 60 and 70 kilos."}],"ID":13765},{"Watched":false,"Name":"Exercise 3 - Parts d-e","Duration":"5m 59s","ChapterTopicVideoID":13278,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.100","Text":"In this section, we\u0027re asked,"},{"Start":"00:02.100 ","End":"00:04.260","Text":"what\u0027s the proportion of the population whose weight"},{"Start":"00:04.260 ","End":"00:07.560","Text":"deviates from the average by no more than 4 kilograms."},{"Start":"00:07.560 ","End":"00:14.100","Text":"We\u0027re looking at the probability of x being between 60,"},{"Start":"00:14.100 ","End":"00:16.620","Text":"that\u0027s our average minus 4 kilograms,"},{"Start":"00:16.620 ","End":"00:20.175","Text":"and 60 plus 4 kilograms."},{"Start":"00:20.175 ","End":"00:27.240","Text":"Or the probability of x being between 56 and 64."},{"Start":"00:27.240 ","End":"00:31.320","Text":"Let\u0027s just draw our density function and see where we\u0027re at."},{"Start":"00:31.320 ","End":"00:35.280","Text":"Here we have our density function and we can see"},{"Start":"00:35.280 ","End":"00:39.555","Text":"the boundaries of the rates that we\u0027re interested in 56 and 64."},{"Start":"00:39.555 ","End":"00:43.880","Text":"Now, how do we isolate this area right here?"},{"Start":"00:43.880 ","End":"00:49.280","Text":"Well, we have to calculate the area from minus infinity to 64 on"},{"Start":"00:49.280 ","End":"00:56.235","Text":"the x axis and subtract from that the area from minus infinity to 56,"},{"Start":"00:56.235 ","End":"00:59.030","Text":"and that\u0027ll isolate this area right here."},{"Start":"00:59.030 ","End":"01:04.970","Text":"Now, let\u0027s just standardize the values of x into z."},{"Start":"01:04.970 ","End":"01:07.830","Text":"Let\u0027s take, for example 64,"},{"Start":"01:07.830 ","End":"01:12.710","Text":"so z equals to x minus Mu divided by Sigma."},{"Start":"01:12.710 ","End":"01:18.710","Text":"That means that we\u0027re looking at 64 minus 60 divided by 8."},{"Start":"01:18.710 ","End":"01:20.210","Text":"That\u0027s our standard deviation."},{"Start":"01:20.210 ","End":"01:25.710","Text":"That means that we\u0027re looking at 4 divided by 8 and that is 1.5."},{"Start":"01:25.710 ","End":"01:29.025","Text":"It\u0027s very easy to see,"},{"Start":"01:29.025 ","End":"01:34.100","Text":"we could have calculated it in our heads because 4,"},{"Start":"01:34.100 ","End":"01:38.340","Text":"this is half of our standard deviation 8."},{"Start":"01:38.890 ","End":"01:44.960","Text":"We\u0027re looking then at the probability of z now being"},{"Start":"01:44.960 ","End":"01:51.715","Text":"between 0.5 and minus 0.5."},{"Start":"01:51.715 ","End":"01:57.170","Text":"This right here is 0.5 and here is minus 0.5."},{"Start":"01:57.170 ","End":"01:58.265","Text":"Now how do I know that?"},{"Start":"01:58.265 ","End":"02:04.530","Text":"Well, because the density function is symmetrical around the average."},{"Start":"02:05.720 ","End":"02:08.510","Text":"How do we calculate this?"},{"Start":"02:08.510 ","End":"02:15.827","Text":"This is Phi of 0.5 minus,"},{"Start":"02:15.827 ","End":"02:17.730","Text":"Phi of 0.5,"},{"Start":"02:17.730 ","End":"02:20.300","Text":"that\u0027s this area right here,"},{"Start":"02:20.300 ","End":"02:24.395","Text":"from minus infinity to 0.5 on the z scale,"},{"Start":"02:24.395 ","End":"02:29.600","Text":"minus now Phi of minus 0.5."},{"Start":"02:29.600 ","End":"02:35.095","Text":"Well, that\u0027s minus infinity to minus 0.5 on the z scale."},{"Start":"02:35.095 ","End":"02:41.390","Text":"Now, if we go to our standard table,"},{"Start":"02:41.390 ","End":"02:47.660","Text":"we\u0027ll see that this value right here equals to 0.6915."},{"Start":"02:47.660 ","End":"02:50.315","Text":"I invite you to check this,"},{"Start":"02:50.315 ","End":"02:54.650","Text":"and 5 minus 0.5, well,"},{"Start":"02:54.650 ","End":"03:00.365","Text":"that equals to 1 minus Phi of 0.5,"},{"Start":"03:00.365 ","End":"03:03.800","Text":"and that equals to"},{"Start":"03:03.800 ","End":"03:12.730","Text":"0.6915 minus 1 minus 0.6915,"},{"Start":"03:12.730 ","End":"03:22.825","Text":"and that comes out to 0.383 or 38.3 percent."},{"Start":"03:22.825 ","End":"03:28.010","Text":"Now, that\u0027s the proportion right here of the population whose weight"},{"Start":"03:28.010 ","End":"03:33.350","Text":"deviates from the average by no more than 4 kilograms."},{"Start":"03:33.350 ","End":"03:34.505","Text":"In this section we\u0027re asked,"},{"Start":"03:34.505 ","End":"03:39.440","Text":"what are the chances of a randomly selected person weighing less than 140 kilograms?"},{"Start":"03:39.440 ","End":"03:47.595","Text":"We\u0027re looking then at the probability of x being less than 140."},{"Start":"03:47.595 ","End":"03:51.880","Text":"Let\u0027s just look at our density function."},{"Start":"03:52.280 ","End":"03:54.920","Text":"Here\u0027s our density function,"},{"Start":"03:54.920 ","End":"04:00.710","Text":"we\u0027re looking to calculate the probability or the proportion of the area in yellow here,"},{"Start":"04:00.710 ","End":"04:04.130","Text":"where x is below 140."},{"Start":"04:04.130 ","End":"04:08.575","Text":"In order to do that, let\u0027s standardize x at 140."},{"Start":"04:08.575 ","End":"04:14.075","Text":"That means that z then equals to x minus Mu divided by Sigma."},{"Start":"04:14.075 ","End":"04:20.450","Text":"That means that we\u0027re looking at 140 minus 60 divided by 8,"},{"Start":"04:20.450 ","End":"04:21.739","Text":"our standard deviation,"},{"Start":"04:21.739 ","End":"04:23.555","Text":"and that equals to 10."},{"Start":"04:23.555 ","End":"04:33.115","Text":"That means that 10 is the comparable value on the z scale of x at 140."},{"Start":"04:33.115 ","End":"04:40.145","Text":"That means that we\u0027re looking at probability then of z being less than 10."},{"Start":"04:40.145 ","End":"04:44.875","Text":"Now that equals to Phi at 10."},{"Start":"04:44.875 ","End":"04:52.925","Text":"Now let\u0027s go to our standard table to see what this value is."},{"Start":"04:52.925 ","End":"04:57.545","Text":"Here\u0027s this stable, and we\u0027re looking at the values of z."},{"Start":"04:57.545 ","End":"04:59.795","Text":"Now if we scroll down the table,"},{"Start":"04:59.795 ","End":"05:06.320","Text":"we see that the highest value for z is 3.49."},{"Start":"05:06.320 ","End":"05:10.510","Text":"That means that we\u0027re looking at this value right here."},{"Start":"05:10.510 ","End":"05:17.945","Text":"At 3.49, that means Phi of 3.49,"},{"Start":"05:17.945 ","End":"05:23.525","Text":"that equals to 0.9998,"},{"Start":"05:23.525 ","End":"05:26.880","Text":"which is almost equal to 1,"},{"Start":"05:26.880 ","End":"05:31.870","Text":"for any value above 3.49,"},{"Start":"05:31.870 ","End":"05:35.310","Text":"we\u0027ll consider that to be equal to 1."},{"Start":"05:36.290 ","End":"05:41.025","Text":"That means that Phi of 10 then,"},{"Start":"05:41.025 ","End":"05:43.920","Text":"that almost equals to 1."},{"Start":"05:43.920 ","End":"05:47.390","Text":"That\u0027s our answer, and what does this mean?"},{"Start":"05:47.390 ","End":"05:51.085","Text":"It means that if we pick a person randomly,"},{"Start":"05:51.085 ","End":"05:56.198","Text":"then the chances of that person weighing less than 140 kilograms,"},{"Start":"05:56.198 ","End":"05:59.160","Text":"well, that\u0027s almost a certainty."}],"ID":13766},{"Watched":false,"Name":"Exercise 4 - Part a","Duration":"6m 20s","ChapterTopicVideoID":13279,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.270","Text":"In this question we\u0027re given that the weight of babies at birth has"},{"Start":"00:03.270 ","End":"00:06.360","Text":"a normal probability distribution with an average of"},{"Start":"00:06.360 ","End":"00:10.095","Text":"3,300 grams and a standard deviation of 400 grams."},{"Start":"00:10.095 ","End":"00:13.690","Text":"We\u0027re asked to find the upper 10th percentile."},{"Start":"00:14.000 ","End":"00:20.490","Text":"First of all, we\u0027re given that we\u0027re dealing with a normal probability distribution."},{"Start":"00:20.490 ","End":"00:24.530","Text":"Secondly, let\u0027s define a random variable."},{"Start":"00:24.530 ","End":"00:25.760","Text":"Let\u0027s call that x."},{"Start":"00:25.760 ","End":"00:32.010","Text":"That\u0027ll be the weight of the babies at birth."},{"Start":"00:32.010 ","End":"00:35.820","Text":"We\u0027re dealing with grams here."},{"Start":"00:35.820 ","End":"00:40.445","Text":"Now, that has a normal distribution where"},{"Start":"00:40.445 ","End":"00:46.445","Text":"Mu equals to 3,300 grams and our standard deviation,"},{"Start":"00:46.445 ","End":"00:50.290","Text":"well that equals to 400 grams."},{"Start":"00:50.290 ","End":"00:57.250","Text":"Let\u0027s just look at our density function and see how we can solve this problem."},{"Start":"00:57.250 ","End":"00:59.925","Text":"Here\u0027s our density function."},{"Start":"00:59.925 ","End":"01:03.195","Text":"We\u0027re asked to find the upper 10th percentile."},{"Start":"01:03.195 ","End":"01:07.160","Text":"That means that we\u0027re looking for some value of x."},{"Start":"01:07.160 ","End":"01:10.460","Text":"Let\u0027s call this x sub 0.9,"},{"Start":"01:10.460 ","End":"01:16.010","Text":"where 10 percent of the area under the curve lies above this value"},{"Start":"01:16.010 ","End":"01:23.475","Text":"and 90 percent of the area under the curve falls below this value."},{"Start":"01:23.475 ","End":"01:27.340","Text":"That\u0027s the meaning of the upper 10th percentile."},{"Start":"01:27.340 ","End":"01:30.170","Text":"Now, in previous questions,"},{"Start":"01:30.170 ","End":"01:32.735","Text":"we were given this value right here."},{"Start":"01:32.735 ","End":"01:36.125","Text":"We had to standardize it into our standard score."},{"Start":"01:36.125 ","End":"01:41.165","Text":"Then we had to go to our standard table to find the proportion."},{"Start":"01:41.165 ","End":"01:44.194","Text":"Now here, we have to work backwards."},{"Start":"01:44.194 ","End":"01:46.595","Text":"We\u0027re given the proportion."},{"Start":"01:46.595 ","End":"01:51.930","Text":"We have to go and calculate our z or the standard score,"},{"Start":"01:51.930 ","End":"01:53.575","Text":"and from there,"},{"Start":"01:53.575 ","End":"01:56.870","Text":"we have to calculate our x."},{"Start":"01:56.870 ","End":"01:59.360","Text":"This is what we need to do."},{"Start":"01:59.360 ","End":"02:00.830","Text":"Let\u0027s go ahead and do this."},{"Start":"02:00.830 ","End":"02:04.779","Text":"We\u0027re looking for then some value."},{"Start":"02:04.779 ","End":"02:09.770","Text":"Let\u0027s say Phi of z sub 0.9,"},{"Start":"02:09.770 ","End":"02:13.265","Text":"and that has to equal to 0.9."},{"Start":"02:13.265 ","End":"02:17.025","Text":"Now remember this z sub 0.9"},{"Start":"02:17.025 ","End":"02:23.000","Text":"is the comparable value of what we\u0027re looking for on the x-axis,"},{"Start":"02:23.000 ","End":"02:25.895","Text":"that\u0027s x sub 0.9."},{"Start":"02:25.895 ","End":"02:31.795","Text":"Let\u0027s go to our table and see what this value is."},{"Start":"02:31.795 ","End":"02:34.470","Text":"Here\u0027s our standard table."},{"Start":"02:34.470 ","End":"02:38.000","Text":"We\u0027re going to have to look in the table right here to"},{"Start":"02:38.000 ","End":"02:42.085","Text":"find the value that\u0027s closest to 0.9."},{"Start":"02:42.085 ","End":"02:47.885","Text":"Remember, we want Phi of some value z sub 0.9."},{"Start":"02:47.885 ","End":"02:51.740","Text":"We want that to be equal to 0.9."},{"Start":"02:51.740 ","End":"02:56.230","Text":"We\u0027re looking for this value right here, z sub 0.9."},{"Start":"02:56.230 ","End":"03:02.240","Text":"What is the value right here that\u0027s closest to 0.9?"},{"Start":"03:02.240 ","End":"03:04.775","Text":"Well, that\u0027s right here,"},{"Start":"03:04.775 ","End":"03:08.445","Text":"between this and this number right here."},{"Start":"03:08.445 ","End":"03:16.400","Text":"We\u0027re looking at this row and somewhere between this and this column."},{"Start":"03:16.400 ","End":"03:19.735","Text":"Let\u0027s just decide that it\u0027s right here."},{"Start":"03:19.735 ","End":"03:22.965","Text":"That will be 0.8997."},{"Start":"03:22.965 ","End":"03:28.760","Text":"We\u0027re looking at z sub 0.9,"},{"Start":"03:28.760 ","End":"03:35.170","Text":"that equals to 1.28."},{"Start":"03:35.170 ","End":"03:38.165","Text":"Now, in some cases,"},{"Start":"03:38.165 ","End":"03:41.275","Text":"we can be just a little bit more exact."},{"Start":"03:41.275 ","End":"03:45.315","Text":"Why is that? Well, we can scroll down."},{"Start":"03:45.315 ","End":"03:52.620","Text":"We see here that at Phi of z being equal to 0.9,"},{"Start":"03:52.620 ","End":"03:57.945","Text":"then z will be equal to 1.282."},{"Start":"03:57.945 ","End":"04:05.330","Text":"Let\u0027s just use that as the value for z. Here we go."},{"Start":"04:05.330 ","End":"04:11.535","Text":"Here we found then the value of z sub 0.9,"},{"Start":"04:11.535 ","End":"04:14.350","Text":"that equals to 1.282."},{"Start":"04:17.300 ","End":"04:19.895","Text":"Now that we found z,"},{"Start":"04:19.895 ","End":"04:22.325","Text":"we have to calculate x,"},{"Start":"04:22.325 ","End":"04:26.015","Text":"the comparable value on the x-axis now."},{"Start":"04:26.015 ","End":"04:28.885","Text":"Remember, we\u0027re working backwards."},{"Start":"04:28.885 ","End":"04:32.335","Text":"How did we get from x to z?"},{"Start":"04:32.335 ","End":"04:33.920","Text":"How did we standardize?"},{"Start":"04:33.920 ","End":"04:38.795","Text":"Well, we took x, z was equal to x,"},{"Start":"04:38.795 ","End":"04:44.780","Text":"we subtracted the average and we divided by the standard deviation."},{"Start":"04:44.780 ","End":"04:51.680","Text":"Here, we\u0027ll just plug in the numbers and work backwards. Let\u0027s do that."},{"Start":"04:52.070 ","End":"04:59.870","Text":"We know that z equals to 1.282 and then has to equal to"},{"Start":"04:59.870 ","End":"05:08.940","Text":"x sub 0.9 minus 3,300 divided by 400."},{"Start":"05:08.940 ","End":"05:11.420","Text":"This is now basic math."},{"Start":"05:11.420 ","End":"05:16.440","Text":"Let\u0027s just multiply both sides by 400."},{"Start":"05:16.570 ","End":"05:27.020","Text":"That equals to 512.8 and that equals to x minus 3,300."},{"Start":"05:27.020 ","End":"05:31.680","Text":"Therefore, x sub 0.9,"},{"Start":"05:31.680 ","End":"05:40.980","Text":"that equals to 3,812.8 grams."},{"Start":"05:40.980 ","End":"05:43.660","Text":"This value right here,"},{"Start":"05:43.880 ","End":"05:51.030","Text":"that\u0027s right here, 3,812.8."},{"Start":"05:51.030 ","End":"06:01.050","Text":"That\u0027s the comparable value on the x-axis for z sub 0.9 being equal to 1.282."},{"Start":"06:02.270 ","End":"06:10.160","Text":"This then is the 90th percentile or the top 10th percentile where 90 percent of"},{"Start":"06:10.160 ","End":"06:14.570","Text":"the babies are way less than"},{"Start":"06:14.570 ","End":"06:20.640","Text":"this value and only 10 percent of the babies weigh more than this value."}],"ID":13767},{"Watched":false,"Name":"Exercise 4 - Parts b-c","Duration":"8m 18s","ChapterTopicVideoID":13280,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.855","Text":"In this section, we\u0027re asked to find the 95th percentile."},{"Start":"00:03.855 ","End":"00:09.278","Text":"Let\u0027s draw our density function. Here it is."},{"Start":"00:09.278 ","End":"00:13.170","Text":"We\u0027re asked to find the 95th percentile, what does that mean?"},{"Start":"00:13.170 ","End":"00:15.780","Text":"We\u0027re looking for some value of x."},{"Start":"00:15.780 ","End":"00:19.065","Text":"Let\u0027s call this x_0.95,"},{"Start":"00:19.065 ","End":"00:24.180","Text":"where 95 percent of the area under the curve falls below"},{"Start":"00:24.180 ","End":"00:33.430","Text":"this value and only 5 percent of the area under the curve falls above this value."},{"Start":"00:34.400 ","End":"00:36.960","Text":"How are we going to do that?"},{"Start":"00:36.960 ","End":"00:38.790","Text":"Well, in previous questions,"},{"Start":"00:38.790 ","End":"00:43.820","Text":"we were given x and we were asked to find the proportion."},{"Start":"00:43.820 ","End":"00:49.345","Text":"That means that we had to standardize x into our standard score,"},{"Start":"00:49.345 ","End":"00:53.180","Text":"and then go to the table to find the proportion."},{"Start":"00:53.180 ","End":"00:56.749","Text":"Now here, we\u0027re given the proportion,"},{"Start":"00:56.749 ","End":"01:00.995","Text":"so what we need to do is we need to go to the table and find z,"},{"Start":"01:00.995 ","End":"01:03.095","Text":"and having found z,"},{"Start":"01:03.095 ","End":"01:05.435","Text":"we need to calculate what x is."},{"Start":"01:05.435 ","End":"01:07.580","Text":"We\u0027ll just be working backwards."},{"Start":"01:07.580 ","End":"01:09.350","Text":"That means that right now,"},{"Start":"01:09.350 ","End":"01:13.790","Text":"our first step is to find a specific z."},{"Start":"01:13.790 ","End":"01:16.895","Text":"We\u0027ll call this z_0.95,"},{"Start":"01:16.895 ","End":"01:25.470","Text":"which is the comparable value of x_0.95 on the z scale."},{"Start":"01:25.470 ","End":"01:32.420","Text":"We\u0027re looking then for Phi of z_0.95,"},{"Start":"01:32.420 ","End":"01:37.220","Text":"and we want that to be equal to 0.95."},{"Start":"01:37.220 ","End":"01:39.860","Text":"Let\u0027s go to our table."},{"Start":"01:39.860 ","End":"01:48.205","Text":"Here\u0027s our table and we\u0027re looking for 0.95 within the table right here."},{"Start":"01:48.205 ","End":"01:55.655","Text":"These then are the 2 values within the table that are closest."},{"Start":"01:55.655 ","End":"01:57.875","Text":"We\u0027re at 1.6,"},{"Start":"01:57.875 ","End":"02:04.915","Text":"z equals to 1.6 between for 1.64 and 1.65."},{"Start":"02:04.915 ","End":"02:09.710","Text":"Now let\u0027s just scroll down and see if we can get"},{"Start":"02:09.710 ","End":"02:16.215","Text":"a more exact value for z at Phi of z being equal to 95,"},{"Start":"02:16.215 ","End":"02:21.045","Text":"then z would be equal to 1.645."},{"Start":"02:21.045 ","End":"02:24.910","Text":"We\u0027ll be using this value right here."},{"Start":"02:25.370 ","End":"02:31.185","Text":"We know then that z_0.95,"},{"Start":"02:31.185 ","End":"02:36.660","Text":"that equals to 1.645."},{"Start":"02:36.660 ","End":"02:40.055","Text":"We found z. Now we have to calculate x."},{"Start":"02:40.055 ","End":"02:42.650","Text":"Now, how did we get from x to z?"},{"Start":"02:42.650 ","End":"02:51.160","Text":"Well, we said that z was equal to x minus Mu divided by Sigma."},{"Start":"02:51.160 ","End":"02:57.305","Text":"Let\u0027s just plug in the numbers and work backwards to calculate what x is,"},{"Start":"02:57.305 ","End":"03:05.585","Text":"so we know that z equals to 1.645 and then it has to be equal to"},{"Start":"03:05.585 ","End":"03:16.065","Text":"x_0.95 minus now Mu is 3,300 divided by 400."},{"Start":"03:16.065 ","End":"03:20.420","Text":"Again, we have an equation with only 1 variable."},{"Start":"03:20.420 ","End":"03:22.520","Text":"Let\u0027s just solve that."},{"Start":"03:22.520 ","End":"03:28.520","Text":"Now, we\u0027ll just multiply both sides by 400 and that will be"},{"Start":"03:28.520 ","End":"03:38.450","Text":"658 and that will be equal to x_0.95 minus 3,300,"},{"Start":"03:38.450 ","End":"03:43.335","Text":"and x_0.95 well that"},{"Start":"03:43.335 ","End":"03:50.325","Text":"equals to 3,958 grams."},{"Start":"03:50.325 ","End":"03:58.480","Text":"That then is our 95th percentile."},{"Start":"03:59.570 ","End":"04:05.930","Text":"That means that only 5 percent of the babies weigh more"},{"Start":"04:05.930 ","End":"04:12.500","Text":"than this value and 95 percent of the babies weigh less than this value."},{"Start":"04:12.500 ","End":"04:17.230","Text":"In this section, we\u0027re asked to find the bottom 10th percentile."},{"Start":"04:17.230 ","End":"04:20.805","Text":"Let\u0027s first draw our density function."},{"Start":"04:20.805 ","End":"04:23.130","Text":"Here\u0027s our density function,"},{"Start":"04:23.130 ","End":"04:26.660","Text":"and we\u0027re looking for a specific x right here."},{"Start":"04:26.660 ","End":"04:31.339","Text":"We\u0027ll call this x_0.1 for the volume 10th percentile,"},{"Start":"04:31.339 ","End":"04:36.590","Text":"where 10 percent of the area under the curve is"},{"Start":"04:36.590 ","End":"04:43.865","Text":"below this value right here and 90 percent is above this value right here."},{"Start":"04:43.865 ","End":"04:49.379","Text":"Now, because we know that this value is below the average,"},{"Start":"04:49.379 ","End":"04:54.035","Text":"then we can expect a negative z value."},{"Start":"04:54.035 ","End":"04:59.025","Text":"The comparable z value for x 0.1 will be negative."},{"Start":"04:59.025 ","End":"05:00.560","Text":"Let\u0027s just write this out."},{"Start":"05:00.560 ","End":"05:03.130","Text":"We\u0027ll The comparable z value,"},{"Start":"05:03.130 ","End":"05:07.355","Text":"we\u0027ll call this minus z_0.1."},{"Start":"05:07.355 ","End":"05:11.435","Text":"Now, whenever we have a negative z value,"},{"Start":"05:11.435 ","End":"05:18.050","Text":"then we have to remember that this density function is symmetrical."},{"Start":"05:18.050 ","End":"05:23.510","Text":"As such, what we need to do is we need to find"},{"Start":"05:23.510 ","End":"05:30.255","Text":"the symmetrical value of z here in this area right here."},{"Start":"05:30.255 ","End":"05:36.075","Text":"This then on the x-axis would be x_0.9"},{"Start":"05:36.075 ","End":"05:42.140","Text":"where 10 percent of the area under"},{"Start":"05:42.140 ","End":"05:48.985","Text":"the curve is above this value and 90 percent is below this value right here,"},{"Start":"05:48.985 ","End":"05:56.290","Text":"and a comparable z value would be z_0.9."},{"Start":"05:56.990 ","End":"05:59.705","Text":"If that\u0027s the case,"},{"Start":"05:59.705 ","End":"06:03.410","Text":"then all we have to do is remember that we\u0027ve already"},{"Start":"06:03.410 ","End":"06:08.125","Text":"calculated in section A the value of z_0.9."},{"Start":"06:08.125 ","End":"06:14.020","Text":"Z_0.9, that equal 1.282."},{"Start":"06:15.650 ","End":"06:19.145","Text":"If that\u0027s the case for this value,"},{"Start":"06:19.145 ","End":"06:25.654","Text":"then z_0.1 well that sum minus value,"},{"Start":"06:25.654 ","End":"06:29.885","Text":"but it\u0027s the same value right here as this only with a minus sign."},{"Start":"06:29.885 ","End":"06:37.965","Text":"Z_0.1 is equal to minus 1.282. Why is that?"},{"Start":"06:37.965 ","End":"06:40.875","Text":"Again, because of the symmetry."},{"Start":"06:40.875 ","End":"06:49.145","Text":"This distance right here from 0 and the z-axis is equal to this distance right here."},{"Start":"06:49.145 ","End":"06:51.740","Text":"The same thing for the x-axis."},{"Start":"06:51.740 ","End":"06:57.060","Text":"This distance right here is equal to this distance right here."},{"Start":"06:57.610 ","End":"07:04.710","Text":"Now that we know the value of z_0.1,"},{"Start":"07:04.710 ","End":"07:07.935","Text":"then let\u0027s calculate x."},{"Start":"07:07.935 ","End":"07:09.990","Text":"We know that z,"},{"Start":"07:09.990 ","End":"07:13.740","Text":"well that equals to x minus Mu divided by Sigma,"},{"Start":"07:13.740 ","End":"07:15.920","Text":"so let\u0027s just plug in the numbers."},{"Start":"07:15.920 ","End":"07:23.660","Text":"That\u0027ll be minus 1.282 and that has to be equal to x_0.1,"},{"Start":"07:23.660 ","End":"07:25.010","Text":"that\u0027s what we\u0027re looking for,"},{"Start":"07:25.010 ","End":"07:30.715","Text":"minus 3,300 divided by 400."},{"Start":"07:30.715 ","End":"07:35.345","Text":"Let\u0027s just do a little bit of basic math here,"},{"Start":"07:35.345 ","End":"07:38.140","Text":"we\u0027ll multiply both sides by 400,"},{"Start":"07:38.140 ","End":"07:42.145","Text":"so we have minus 512.8,"},{"Start":"07:42.145 ","End":"07:48.675","Text":"that equals to x_0.1 minus 3,300."},{"Start":"07:48.675 ","End":"07:51.525","Text":"That means that x_0.1,"},{"Start":"07:51.525 ","End":"07:59.370","Text":"that equals to 2,787.2 grams."},{"Start":"07:59.370 ","End":"08:01.265","Text":"This is the answer."},{"Start":"08:01.265 ","End":"08:08.225","Text":"This means that only 10 percent of the babies weigh less than this value,"},{"Start":"08:08.225 ","End":"08:12.945","Text":"and 90 percent of the babies weigh more than this value."},{"Start":"08:12.945 ","End":"08:17.589","Text":"That\u0027s the meaning of the bottom 10th percentile."}],"ID":13768},{"Watched":false,"Name":"Exercise 5 - Parts a-b","Duration":"10m 33s","ChapterTopicVideoID":13281,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this question we\u0027re given that the marks in an intelligence tests have"},{"Start":"00:03.450 ","End":"00:08.895","Text":"a normal probability distribution with an average of 100 and a variance of 225."},{"Start":"00:08.895 ","End":"00:14.610","Text":"We\u0027re asked, what\u0027s the upper 10th percentile of the marks on the intelligence test?"},{"Start":"00:14.610 ","End":"00:17.700","Text":"The first thing that we want to know is that we\u0027re dealing with"},{"Start":"00:17.700 ","End":"00:20.850","Text":"a normal probability distribution and we are,"},{"Start":"00:20.850 ","End":"00:22.980","Text":"that\u0027s what we\u0027re given right here."},{"Start":"00:22.980 ","End":"00:28.200","Text":"The second thing that we need to do is we need to define our random variable."},{"Start":"00:28.200 ","End":"00:29.700","Text":"Now let\u0027s call that x,"},{"Start":"00:29.700 ","End":"00:35.060","Text":"and that\u0027ll be the marks that\u0027s given on the intelligence test."},{"Start":"00:35.060 ","End":"00:41.300","Text":"Now we know that that has a normal probability distribution where our average Mu,"},{"Start":"00:41.300 ","End":"00:43.255","Text":"well that equals to 100,"},{"Start":"00:43.255 ","End":"00:44.700","Text":"that\u0027s given right here,"},{"Start":"00:44.700 ","End":"00:51.434","Text":"and the variance sigma squared,"},{"Start":"00:51.434 ","End":"00:55.170","Text":"that equals to 225."},{"Start":"00:55.170 ","End":"01:02.120","Text":"Now here, I\u0027d preferred if you write this out like this."},{"Start":"01:02.120 ","End":"01:05.525","Text":"We know that if sigma squared is 225,"},{"Start":"01:05.525 ","End":"01:08.080","Text":"then sigma then equals 15."},{"Start":"01:08.080 ","End":"01:13.305","Text":"Here we\u0027d write this as 15 squared."},{"Start":"01:13.305 ","End":"01:15.830","Text":"Now that we have the information,"},{"Start":"01:15.830 ","End":"01:18.320","Text":"let\u0027s just draw our density function."},{"Start":"01:18.320 ","End":"01:21.319","Text":"Here\u0027s our density function,"},{"Start":"01:21.319 ","End":"01:23.060","Text":"and on the x-axis, well,"},{"Start":"01:23.060 ","End":"01:24.650","Text":"we know that Mu equals 100,"},{"Start":"01:24.650 ","End":"01:26.195","Text":"that\u0027s our average right here,"},{"Start":"01:26.195 ","End":"01:29.720","Text":"and the comparable value on the z axis is 0."},{"Start":"01:29.720 ","End":"01:33.920","Text":"Now, we\u0027re looking for the upper 10th percentile of the marks."},{"Start":"01:33.920 ","End":"01:38.525","Text":"That means that we\u0027re looking for x_0.9,"},{"Start":"01:38.525 ","End":"01:43.070","Text":"that\u0027s the value where 10 percent of"},{"Start":"01:43.070 ","End":"01:50.890","Text":"the area under the density function falls above this value,"},{"Start":"01:50.890 ","End":"01:57.440","Text":"and 90 percent of the area under the density function falls below this value."},{"Start":"01:57.540 ","End":"02:05.005","Text":"What we have to do right now is to see how we can calculate this."},{"Start":"02:05.005 ","End":"02:08.950","Text":"Now usually we\u0027re given our x and then we have to go and"},{"Start":"02:08.950 ","End":"02:13.705","Text":"standardize it in order to find a specific probability."},{"Start":"02:13.705 ","End":"02:15.940","Text":"Now, here we have to work backwards,"},{"Start":"02:15.940 ","End":"02:18.970","Text":"we\u0027re given the probability that\u0027s 0.9,"},{"Start":"02:18.970 ","End":"02:21.040","Text":"we have to go to"},{"Start":"02:21.040 ","End":"02:28.135","Text":"our standard table and see what our z is that\u0027s comparable to our probability,"},{"Start":"02:28.135 ","End":"02:33.350","Text":"and from z, we have to calculate our x."},{"Start":"02:33.860 ","End":"02:42.310","Text":"First we need to know what is the comparable z value for our x."},{"Start":"02:42.310 ","End":"02:49.530","Text":"That means that we\u0027re looking now for z_0.9."},{"Start":"02:49.530 ","End":"02:56.865","Text":"We\u0027re looking then for Phi of z_0.9,"},{"Start":"02:56.865 ","End":"03:00.945","Text":"and we know that this is equal to 0.9."},{"Start":"03:00.945 ","End":"03:06.780","Text":"Let\u0027s go to our standard table and see what our z value is."},{"Start":"03:07.700 ","End":"03:13.910","Text":"Again, we\u0027re looking for Phi of a specific z."},{"Start":"03:13.910 ","End":"03:16.190","Text":"We call that z_0.9,"},{"Start":"03:16.190 ","End":"03:19.520","Text":"and we want that to be equal to 0.9."},{"Start":"03:19.520 ","End":"03:23.390","Text":"We\u0027re looking for this value, 0.9,"},{"Start":"03:23.390 ","End":"03:29.445","Text":"within the table and then we can go and see what our z value is."},{"Start":"03:29.445 ","End":"03:37.520","Text":"0.9, what\u0027s the closest value that we have to 0.9 within the table right here?"},{"Start":"03:37.520 ","End":"03:42.790","Text":"Well, that looks like this value right here."},{"Start":"03:42.790 ","End":"03:51.330","Text":"Now this Phi value occurs in the intersection of 1.28."},{"Start":"03:51.330 ","End":"03:58.440","Text":"Z_0.9, well,"},{"Start":"03:58.440 ","End":"04:01.390","Text":"that equals to 1.28."},{"Start":"04:01.390 ","End":"04:05.690","Text":"Now let\u0027s see if we can get a more exact figure for z."},{"Start":"04:05.690 ","End":"04:07.870","Text":"Let\u0027s scroll down,"},{"Start":"04:07.870 ","End":"04:12.575","Text":"and here for 5z being equal to 0.9,"},{"Start":"04:12.575 ","End":"04:17.275","Text":"we get a z value of 1.282."},{"Start":"04:17.275 ","End":"04:22.340","Text":"Let\u0027s just write that down here, that\u0027s 1.282."},{"Start":"04:22.340 ","End":"04:27.670","Text":"Now really doesn\u0027t matter if you write 1.28 or 1.282,"},{"Start":"04:27.670 ","End":"04:33.145","Text":"the minute you understand this principle then all is well."},{"Start":"04:33.145 ","End":"04:39.855","Text":"Let\u0027s go back to our question and continue with our calculations."},{"Start":"04:39.855 ","End":"04:45.900","Text":"Now we know then that z_0.9,"},{"Start":"04:45.900 ","End":"04:48.580","Text":"well, that equals to 1.282."},{"Start":"04:49.670 ","End":"04:54.340","Text":"Let\u0027s write that down, that\u0027s 1.282."},{"Start":"04:56.240 ","End":"04:59.440","Text":"How do we get from z to our x?"},{"Start":"04:59.440 ","End":"05:04.280","Text":"Well, we know that the standardization transformation is this,"},{"Start":"05:04.280 ","End":"05:10.440","Text":"that\u0027s z equals x minus Mu divided by Sigma."},{"Start":"05:10.440 ","End":"05:11.955","Text":"Well, in our case,"},{"Start":"05:11.955 ","End":"05:14.440","Text":"that\u0027ll be 1.282,"},{"Start":"05:14.440 ","End":"05:19.520","Text":"and that has to be equal to x_0.9 minus Mu,"},{"Start":"05:19.520 ","End":"05:23.390","Text":"now Mu is 100, divided by sigma."},{"Start":"05:23.390 ","End":"05:25.690","Text":"Well, sigma is 15."},{"Start":"05:25.690 ","End":"05:30.365","Text":"The minute we have this then it\u0027s just basic math."},{"Start":"05:30.365 ","End":"05:32.375","Text":"Let\u0027s just continue here,"},{"Start":"05:32.375 ","End":"05:33.920","Text":"let\u0027s just scroll this up."},{"Start":"05:33.920 ","End":"05:42.210","Text":"Right now we\u0027ll multiply both sides by 15 so we\u0027ll get 19.23."},{"Start":"05:42.210 ","End":"05:46.770","Text":"Well, that equals to x_0.9 minus 100."},{"Start":"05:46.770 ","End":"05:52.870","Text":"X_0.9 that equals to 119.23."},{"Start":"05:53.900 ","End":"05:59.909","Text":"This then is the upper 10th percentile,"},{"Start":"05:59.909 ","End":"06:01.775","Text":"let\u0027s just scroll this down here a bit."},{"Start":"06:01.775 ","End":"06:06.846","Text":"This is the upper 10th percentile of the marks on the intelligence test,"},{"Start":"06:06.846 ","End":"06:10.685","Text":"where 10 percent of the grades are"},{"Start":"06:10.685 ","End":"06:16.370","Text":"above this value and 90 percent of the grades are below this value."},{"Start":"06:16.370 ","End":"06:18.590","Text":"In this section, we\u0027re asked what\u0027s"},{"Start":"06:18.590 ","End":"06:22.175","Text":"the bottom 10th percentile of the probability distribution?"},{"Start":"06:22.175 ","End":"06:24.935","Text":"Let\u0027s draw our density function."},{"Start":"06:24.935 ","End":"06:27.200","Text":"Here\u0027s our density function,"},{"Start":"06:27.200 ","End":"06:29.900","Text":"what we\u0027re looking for is some value of x,"},{"Start":"06:29.900 ","End":"06:33.935","Text":"that\u0027s right here, call this x_0.1,"},{"Start":"06:33.935 ","End":"06:36.770","Text":"where 10 percent of the area under"},{"Start":"06:36.770 ","End":"06:43.970","Text":"the density function falls below this value right here on the x-axis,"},{"Start":"06:43.970 ","End":"06:49.985","Text":"that means that 90 percent of the area under the density function falls above this value."},{"Start":"06:49.985 ","End":"06:53.030","Text":"Now, this is the value that we\u0027re looking for."},{"Start":"06:53.030 ","End":"06:59.750","Text":"Now, usually we have our x and we have to standardize it,"},{"Start":"06:59.750 ","End":"07:06.065","Text":"then we have to go to the standard table to calculate the probability."},{"Start":"07:06.065 ","End":"07:08.810","Text":"Now here, we have to work backwards."},{"Start":"07:08.810 ","End":"07:11.225","Text":"We\u0027re given the probability,"},{"Start":"07:11.225 ","End":"07:15.575","Text":"we have to calculate our z or standards score,"},{"Start":"07:15.575 ","End":"07:19.280","Text":"from there we have to calculate our x."},{"Start":"07:19.280 ","End":"07:26.375","Text":"Now, this is very similar to what we did in section a."},{"Start":"07:26.375 ","End":"07:28.475","Text":"Where in section a,"},{"Start":"07:28.475 ","End":"07:33.515","Text":"we had to calculate the top 10 percentile."},{"Start":"07:33.515 ","End":"07:36.490","Text":"Let\u0027s just see what we did there."},{"Start":"07:36.490 ","End":"07:38.625","Text":"In section a,"},{"Start":"07:38.625 ","End":"07:44.330","Text":"we\u0027re asked to find the top 10th percentile. What did that mean?"},{"Start":"07:44.330 ","End":"07:49.280","Text":"That meant that 10 percent of the area below"},{"Start":"07:49.280 ","End":"07:57.335","Text":"the density function had to be above this value right here, x_0.9."},{"Start":"07:57.335 ","End":"08:00.110","Text":"Now here we\u0027re asked to find"},{"Start":"08:00.110 ","End":"08:04.760","Text":"the symmetrical point on the other side of the distribution function."},{"Start":"08:04.760 ","End":"08:08.840","Text":"Right here, where 10 percent of the area under"},{"Start":"08:08.840 ","End":"08:14.725","Text":"the distribution or the density function has to be below this specific value."},{"Start":"08:14.725 ","End":"08:19.130","Text":"Let\u0027s see what our corresponding z value is right here."},{"Start":"08:19.130 ","End":"08:24.970","Text":"Well, in section a, z_0.9 was 1.282."},{"Start":"08:24.970 ","End":"08:26.795","Text":"Now because of symmetry,"},{"Start":"08:26.795 ","End":"08:36.730","Text":"then the corresponding z value here for x_0.1 would be minus 1.282."},{"Start":"08:36.730 ","End":"08:39.870","Text":"We don\u0027t really need to go to the table."},{"Start":"08:39.870 ","End":"08:46.145","Text":"In any case, the table doesn\u0027t support negative z values if you recall."},{"Start":"08:46.145 ","End":"08:49.760","Text":"Now, once we have our z value,"},{"Start":"08:49.760 ","End":"08:54.155","Text":"then it\u0027s very easy to calculate our x because if you recall,"},{"Start":"08:54.155 ","End":"09:00.800","Text":"the standardization transformation is equal to x minus Mu divided by Sigma,"},{"Start":"09:00.800 ","End":"09:06.380","Text":"where in our case, that\u0027s z_0.1."},{"Start":"09:06.380 ","End":"09:16.325","Text":"Z_0.1 equals minus 1.282 and that equals to x_0.1 minus 100,"},{"Start":"09:16.325 ","End":"09:18.605","Text":"that\u0027s our Mu divided by Sigma."},{"Start":"09:18.605 ","End":"09:20.210","Text":"Well, Sigma is 15."},{"Start":"09:20.210 ","End":"09:25.115","Text":"Again, let\u0027s just multiply both sides by 15,"},{"Start":"09:25.115 ","End":"09:30.930","Text":"that would be minus 19.23 and that equals"},{"Start":"09:30.930 ","End":"09:37.775","Text":"to x_0.1 minus a 100 or x_0.1,"},{"Start":"09:37.775 ","End":"09:45.750","Text":"that equals to 80.77."},{"Start":"09:45.750 ","End":"09:51.110","Text":"This then is the bottom 10th percentile of the probability distribution."},{"Start":"09:51.110 ","End":"09:55.445","Text":"Now, we could have calculated this a little bit differently."},{"Start":"09:55.445 ","End":"10:04.875","Text":"We know from section a that the distance between the average and the 90th percentile,"},{"Start":"10:04.875 ","End":"10:09.040","Text":"well, that\u0027s 19.23."},{"Start":"10:10.190 ","End":"10:13.970","Text":"That\u0027s this distance right here."},{"Start":"10:13.970 ","End":"10:17.690","Text":"Well, the symmetrical distance right here,"},{"Start":"10:17.690 ","End":"10:21.675","Text":"well, that\u0027s also 19.23."},{"Start":"10:21.675 ","End":"10:26.240","Text":"Now, having used this characteristics,"},{"Start":"10:26.240 ","End":"10:33.840","Text":"then we could have just subtracted 19.23 from 100 to get 80.77."}],"ID":13769},{"Watched":false,"Name":"Exercise 5 - Parts c-e","Duration":"13m 19s","ChapterTopicVideoID":13282,"CourseChapterTopicPlaylistID":64527,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.610","Text":"In this section, we\u0027re asked 20 percent of those taking"},{"Start":"00:02.610 ","End":"00:05.810","Text":"the test receive marks higher what number?"},{"Start":"00:05.810 ","End":"00:08.645","Text":"Let\u0027s look at our density function."},{"Start":"00:08.645 ","End":"00:10.524","Text":"This is our density function,"},{"Start":"00:10.524 ","End":"00:13.298","Text":"we\u0027re looking for some value of x,"},{"Start":"00:13.298 ","End":"00:15.165","Text":"we\u0027ll call that x right here,"},{"Start":"00:15.165 ","End":"00:23.700","Text":"where 80 percent of the area underneath the density function"},{"Start":"00:23.700 ","End":"00:26.475","Text":"falls below this value and"},{"Start":"00:26.475 ","End":"00:32.565","Text":"20 percent of the area under the density function fall above this value."},{"Start":"00:32.565 ","End":"00:34.395","Text":"That\u0027ll be 0.2 here,"},{"Start":"00:34.395 ","End":"00:36.405","Text":"that\u0027ll be 0.8 here,"},{"Start":"00:36.405 ","End":"00:41.080","Text":"and we\u0027ll call this value x_0.8 because,"},{"Start":"00:41.080 ","End":"00:44.705","Text":"in essence, we\u0027re talking about the 80th percentile."},{"Start":"00:44.705 ","End":"00:48.770","Text":"Now, usually, we have our x,"},{"Start":"00:48.770 ","End":"00:51.995","Text":"we have to standardize it into a standard score,"},{"Start":"00:51.995 ","End":"00:55.280","Text":"and then go to a standard table to calculate the probability."},{"Start":"00:55.280 ","End":"00:56.750","Text":"Now, here we have to work backwards."},{"Start":"00:56.750 ","End":"00:58.961","Text":"We\u0027re given the probability,"},{"Start":"00:58.961 ","End":"01:05.130","Text":"we have to see what our comparable z value is,"},{"Start":"01:05.130 ","End":"01:08.150","Text":"and then we have to calculate our x."},{"Start":"01:08.150 ","End":"01:14.240","Text":"The first thing that we need to do is to see what our comparable z value is,"},{"Start":"01:14.240 ","End":"01:16.445","Text":"that\u0027ll be z_0.8,"},{"Start":"01:16.445 ","End":"01:19.470","Text":"that\u0027s what we will call it."},{"Start":"01:21.020 ","End":"01:29.890","Text":"What we\u0027re looking for then is Phi of z_0.8,"},{"Start":"01:31.160 ","End":"01:34.155","Text":"so we\u0027re looking for this z value,"},{"Start":"01:34.155 ","End":"01:39.150","Text":"which makes Phi of this value equals to 0.8."},{"Start":"01:39.150 ","End":"01:40.875","Text":"Let\u0027s go to the table."},{"Start":"01:40.875 ","End":"01:43.500","Text":"This is our table and as I said,"},{"Start":"01:43.500 ","End":"01:51.315","Text":"we\u0027re looking for our z value where Phi of the z value,"},{"Start":"01:51.315 ","End":"01:55.815","Text":"z_0.8, that equals to 0.8."},{"Start":"01:55.815 ","End":"02:00.635","Text":"What\u0027s the closest value to 0.8 that we have within the table?"},{"Start":"02:00.635 ","End":"02:04.870","Text":"That\u0027s right here,"},{"Start":"02:05.210 ","End":"02:09.600","Text":"the comparable z value or the z value that we\u0027re looking for,"},{"Start":"02:09.600 ","End":"02:12.600","Text":"z_0.8, well,"},{"Start":"02:12.600 ","End":"02:17.249","Text":"that equals to 0.84,"},{"Start":"02:17.249 ","End":"02:23.015","Text":"that\u0027s this line in this column right here, that\u0027s 0.84."},{"Start":"02:23.015 ","End":"02:26.600","Text":"Now, if we really want to get precise,"},{"Start":"02:26.600 ","End":"02:31.440","Text":"let\u0027s see if we have a comparable z value,"},{"Start":"02:31.440 ","End":"02:33.310","Text":"and we don\u0027t,"},{"Start":"02:33.310 ","End":"02:43.055","Text":"this smaller table doesn\u0027t give us the z value for Phi of z being equal to 0.8."},{"Start":"02:43.055 ","End":"02:51.360","Text":"But I can tell you that this value right here is 0.842."},{"Start":"02:52.940 ","End":"02:58.810","Text":"Let\u0027s now go back to our question and see how we can continue."},{"Start":"02:58.810 ","End":"03:03.620","Text":"Z_0.8, well,"},{"Start":"03:03.620 ","End":"03:09.965","Text":"that equals to 0.842."},{"Start":"03:09.965 ","End":"03:13.730","Text":"Now, our standardization transformation,"},{"Start":"03:13.730 ","End":"03:17.610","Text":"that equals to x minus Mu divided by Sigma."},{"Start":"03:17.610 ","End":"03:23.727","Text":"In our case, we have 0.842, well,"},{"Start":"03:23.727 ","End":"03:27.770","Text":"that equals to x_0.8 minus Mu,"},{"Start":"03:27.770 ","End":"03:32.130","Text":"well, that\u0027s 100 divided by Sigma, well, that\u0027s 15."},{"Start":"03:32.130 ","End":"03:39.020","Text":"Let\u0027s multiply both sides by 15 and we get 12.6,"},{"Start":"03:39.020 ","End":"03:43.025","Text":"that equals to x_0.8 minus 100."},{"Start":"03:43.025 ","End":"03:48.080","Text":"That means that x_0.8, well,"},{"Start":"03:48.080 ","End":"03:55.915","Text":"that equals to 112.6."},{"Start":"03:55.915 ","End":"04:01.460","Text":"This then is the value where 80 percent of the marks"},{"Start":"04:01.460 ","End":"04:08.460","Text":"fall below this value and 20 percent of the marks are higher than this value."},{"Start":"04:08.840 ","End":"04:11.070","Text":"In this section, we\u0027re asked,"},{"Start":"04:11.070 ","End":"04:13.170","Text":"what\u0027s the 20th percentile?"},{"Start":"04:13.170 ","End":"04:16.365","Text":"Let\u0027s draw our density function."},{"Start":"04:16.365 ","End":"04:21.155","Text":"Here\u0027s our density function and we\u0027re looking for some value of x,"},{"Start":"04:21.155 ","End":"04:24.050","Text":"we\u0027ll call that x right here,"},{"Start":"04:24.050 ","End":"04:28.430","Text":"and we\u0027re looking for a value where 20 percent of the area under"},{"Start":"04:28.430 ","End":"04:33.785","Text":"the density function has to be below this value right here."},{"Start":"04:33.785 ","End":"04:36.770","Text":"That means that 80 percent of the area"},{"Start":"04:36.770 ","End":"04:40.055","Text":"under the density function has to be above this value,"},{"Start":"04:40.055 ","End":"04:43.100","Text":"and we\u0027ll call this x_0.2 because,"},{"Start":"04:43.100 ","End":"04:47.490","Text":"in essence, we\u0027re looking for the 20th percentile."},{"Start":"04:47.540 ","End":"04:51.335","Text":"Usually, we have our x,"},{"Start":"04:51.335 ","End":"04:54.305","Text":"we have to standardize it into a standard score,"},{"Start":"04:54.305 ","End":"05:01.055","Text":"and then we have to go to our table to find the appropriate probability."},{"Start":"05:01.055 ","End":"05:03.680","Text":"Now, here we have to work backwards, and why is that?"},{"Start":"05:03.680 ","End":"05:06.320","Text":"Well, because we have our probability,"},{"Start":