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-d
- Exercise 2
- Exercise 3 - Parts a-b
- Exercise 3 - Parts c-d
- Exercise 4
- Exercise 5
- Exercise 6 - Parts a-b
- Exercise 6 - Parts c-e
- Exercise 7 - Part c
- Exercise 7 - Parts a-b
- Exercise 8 - Parts a-b
- Exercise 8 - Parts c-d
- Exercise 9 - Parts a-b
- Exercise 9 - Parts c-e
- Exercise 10
- Exercise 11
- Exercise 12 - Part c
- Exercise 12 - Parts a-b
- Exercise 13 - Part a
- Exercise 13 - Parts b-c
- Exercise 14

Exponential Probability
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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 - Part c
- Exercise 6 - Parts a-b
- 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 - Parts c-d
- Exercise 11 - Part b

Transformation of a Continuous Random Variable
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{"Free":0,"Sample":1,"Paid":2}

[{"Name":"General Probabilities without Integrals","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"3m 22s","ChapterTopicVideoID":12601,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12601.jpeg","UploadDate":"2018-07-23T15:53:32.2730000","DurationForVideoObject":"PT3M22S","Description":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":"00:17.190","Text":"we\u0027ll deal with the probability distribution of continuous random variables for example,"},{"Start":"00:17.190 ","End":"00:18.480","Text":"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":"00:23.400","Text":"As opposed to discrete variables,"},{"Start":"00:23.400 ","End":"00:28.230","Text":"continuous variables have an infinite number of values in a given range for example,"},{"Start":"00:28.230 ","End":"00:29.700","Text":"free roll the dice,"},{"Start":"00:29.700 ","End":"00:38.435","Text":"the random variable X which will be the result could have basically 6 values from 1-6."},{"Start":"00:38.435 ","End":"00:42.830","Text":"But if we take a random variable for example,"},{"Start":"00:42.830 ","End":"00:44.090","Text":"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":"00:51.020","Text":"For example, if we take a range of values from let\u0027s"},{"Start":"00:51.020 ","End":"00:55.750","Text":"say 150 centimeters to 200 centimeters well,"},{"Start":"00:55.750 ","End":"00:57.980","Text":"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\u0027s still an infinite number of"},{"Start":"01:11.720 ","End":"01:16.495","Text":"values that the random variable x can have."},{"Start":"01:16.495 ","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":"01:52.035","Text":"A density function must be non-negative"},{"Start":"01:52.035 ","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":"02:07.880","Text":"can be either negative or positive but the function f of x must be positive,"},{"Start":"02:07.880 ","End":"02:09.710","Text":"must be non-negative,"},{"Start":"02:09.710 ","End":"02:14.040","Text":"it has to be greater or equal to 0 and"},{"Start":"02:14.040 ","End":"02:19.355","Text":"the total area under this function is always equals to 1."},{"Start":"02:19.355 ","End":"02:24.290","Text":"Now, the current course doesn\u0027t use integration to calculate the areas so"},{"Start":"02:24.290 ","End":"02:29.810","Text":"we\u0027ll be using common geometric forms to use"},{"Start":"02:29.810 ","End":"02:34.580","Text":"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":"02:52.034","Text":"Here I\u0027ve given basically some equations of some general geometric forms for example,"},{"Start":"02:52.034 ","End":"02:54.015","Text":"the area of a triangle,"},{"Start":"02:54.015 ","End":"02:56.025","Text":"that\u0027s this guy right here,"},{"Start":"02:56.025 ","End":"02:57.830","Text":"the area of a rectangle,"},{"Start":"02:57.830 ","End":"03:03.590","Text":"that\u0027s this guy and I\u0027ve also given an equation for a straight line,"},{"Start":"03:03.590 ","End":"03:06.455","Text":"that\u0027s all of these guys right here."},{"Start":"03:06.455 ","End":"03:10.666","Text":"What\u0027s the general expression for a straight line,"},{"Start":"03:10.666 ","End":"03:16.146","Text":"and how to calculate the slope and the y-intercept?"},{"Start":"03:16.146 ","End":"03:22.200","Text":"Let\u0027s get to an example and see how we can work with these density functions."}],"ID":13080},{"Watched":false,"Name":"Example - Parts a-b","Duration":"5m 18s","ChapterTopicVideoID":12605,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12605.jpeg","UploadDate":"2018-07-23T15:54:46.7070000","DurationForVideoObject":"PT5M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"In our example, the diagram below shows the density function of the variable X"},{"Start":"00:04.830 ","End":"00:10.424","Text":"where X is the waiting time in minutes for a telephone response from customer service."},{"Start":"00:10.424 ","End":"00:14.530","Text":"In section a, we\u0027re asked to find the value of a."},{"Start":"00:14.660 ","End":"00:19.290","Text":"First of all, let\u0027s take a look at this straight line."},{"Start":"00:19.290 ","End":"00:20.640","Text":"This straight line right here,"},{"Start":"00:20.640 ","End":"00:22.815","Text":"that\u0027s f of x."},{"Start":"00:22.815 ","End":"00:26.970","Text":"Now, if we take a look at this right here,"},{"Start":"00:26.970 ","End":"00:34.455","Text":"this form, f of x is basically the hypotenuse of this triangle right here."},{"Start":"00:34.455 ","End":"00:36.780","Text":"We can see that right here."},{"Start":"00:36.780 ","End":"00:41.975","Text":"Now, what do we know about the area of a triangle?"},{"Start":"00:41.975 ","End":"00:44.375","Text":"How do we calculate the area of a triangle?"},{"Start":"00:44.375 ","End":"00:52.905","Text":"Well, the area of a triangle that will equal to base times the height divided by 2,"},{"Start":"00:52.905 ","End":"00:56.765","Text":"and in our case, the base equals 4."},{"Start":"00:56.765 ","End":"00:58.940","Text":"The height of the triangle."},{"Start":"00:58.940 ","End":"01:00.050","Text":"That\u0027s this guy right here."},{"Start":"01:00.050 ","End":"01:02.450","Text":"Well, that\u0027s a, that\u0027s what we\u0027re trying to find out."},{"Start":"01:02.450 ","End":"01:08.335","Text":"Divided by 2, that equals to 2a."},{"Start":"01:08.335 ","End":"01:13.040","Text":"That would be the area of the triangle right here."},{"Start":"01:13.040 ","End":"01:20.135","Text":"Now, what do we know about the area under the density function?"},{"Start":"01:20.135 ","End":"01:26.120","Text":"But we know the area under the density function has to be equal to 1."},{"Start":"01:26.120 ","End":"01:29.435","Text":"This thing has to be equal to 1."},{"Start":"01:29.435 ","End":"01:34.590","Text":"This is the area under the density function and that must equal to 1."},{"Start":"01:34.590 ","End":"01:38.590","Text":"A here has to equal to 1/2."},{"Start":"01:38.590 ","End":"01:43.665","Text":"This is the value of a,"},{"Start":"01:43.665 ","End":"01:46.440","Text":"a right here, equals 1/2."},{"Start":"01:46.440 ","End":"01:51.814","Text":"In section b, we\u0027re asked to write the formula for the density function."},{"Start":"01:51.814 ","End":"01:54.500","Text":"This is our density functions,"},{"Start":"01:54.500 ","End":"01:56.705","Text":"this is a diagram of a density function."},{"Start":"01:56.705 ","End":"02:04.430","Text":"But it shows us the density function only from x equals 0 to x equals 4."},{"Start":"02:04.430 ","End":"02:07.145","Text":"What about all other values of x?"},{"Start":"02:07.145 ","End":"02:12.350","Text":"Well, if we drop the density function for all values of x,"},{"Start":"02:12.350 ","End":"02:17.190","Text":"we see that f of x equals 0 for all x that\u0027s negative,"},{"Start":"02:17.190 ","End":"02:22.745","Text":"and it\u0027s also 0 for all events and x which are greater than 4,"},{"Start":"02:22.745 ","End":"02:26.540","Text":"and between 0 and 4 in that range."},{"Start":"02:26.540 ","End":"02:31.985","Text":"The function of the density function of X is this straight line right here."},{"Start":"02:31.985 ","End":"02:42.590","Text":"Let\u0027s just write this out saying that f of x equals some function for a straight line,"},{"Start":"02:42.590 ","End":"02:46.445","Text":"and we\u0027ll calculate that in a bed for all values of x"},{"Start":"02:46.445 ","End":"02:52.545","Text":"between 0 and 4 and it\u0027ll be 0 otherwise."},{"Start":"02:52.545 ","End":"02:59.015","Text":"All we have to do now is to calculate the equation for this line."},{"Start":"02:59.015 ","End":"03:02.720","Text":"Now, the general form of a line."},{"Start":"03:02.720 ","End":"03:09.965","Text":"The general equation of a line is y equals mx plus n,"},{"Start":"03:09.965 ","End":"03:16.355","Text":"where n is the y-intercept and m is the slope of the slide."},{"Start":"03:16.355 ","End":"03:19.355","Text":"First of all, let\u0027s take a look at n,"},{"Start":"03:19.355 ","End":"03:21.420","Text":"n is the y-intercept."},{"Start":"03:21.420 ","End":"03:25.850","Text":"Where does this density function intersect the y-axis?"},{"Start":"03:25.850 ","End":"03:29.609","Text":"Well, at 0, where x equals 0,"},{"Start":"03:29.609 ","End":"03:34.865","Text":"that\u0027s where f of x intersects the y-axis,"},{"Start":"03:34.865 ","End":"03:37.640","Text":"and what about the slope m?"},{"Start":"03:37.640 ","End":"03:47.970","Text":"Well, m is calculated as y2 minus y1 divided by x2 minus x1,"},{"Start":"03:47.970 ","End":"03:50.010","Text":"where x1, y1,"},{"Start":"03:50.010 ","End":"03:54.869","Text":"and x2, y2 are 2 points."},{"Start":"03:54.869 ","End":"03:58.250","Text":"Let\u0027s just write them out. This part right here."},{"Start":"03:58.250 ","End":"04:02.420","Text":"The point of origin would be x1, y1."},{"Start":"04:02.420 ","End":"04:04.340","Text":"That\u0027s 0,0."},{"Start":"04:04.340 ","End":"04:06.710","Text":"What about the second point?"},{"Start":"04:06.710 ","End":"04:09.080","Text":"Well, let\u0027s take a look at this point right here."},{"Start":"04:09.080 ","End":"04:12.590","Text":"That\u0027s 4, 1/2 X equals 4,"},{"Start":"04:12.590 ","End":"04:19.270","Text":"y equals 1/2 and why half because we\u0027ve calculated that in section a."},{"Start":"04:19.270 ","End":"04:25.605","Text":"Let\u0027s just plug in the numbers here y2 is 1/2 minus 0."},{"Start":"04:25.605 ","End":"04:33.335","Text":"That\u0027s 1/2 minus 0 divided by x2 is 4 minus x 1 that\u0027s 0."},{"Start":"04:33.335 ","End":"04:35.540","Text":"Let\u0027s do a quick calculation."},{"Start":"04:35.540 ","End":"04:41.180","Text":"That\u0027s 1/2 divided by 4 and that equals to an 1/8."},{"Start":"04:41.180 ","End":"04:45.980","Text":"We can say that the equation for this line right"},{"Start":"04:45.980 ","End":"04:53.510","Text":"here is m equals 1/8 times x plus 0."},{"Start":"04:53.510 ","End":"04:55.955","Text":"Now, what is y,"},{"Start":"04:55.955 ","End":"04:59.580","Text":"if not the density function of X?"},{"Start":"04:59.580 ","End":"05:03.650","Text":"Let\u0027s just simplify this and put that over here."},{"Start":"05:03.650 ","End":"05:06.280","Text":"That equals to 1/8 times x."},{"Start":"05:06.280 ","End":"05:11.645","Text":"We can say that the density function equals 1/8 times x,"},{"Start":"05:11.645 ","End":"05:18.720","Text":"where x is between 0 and 4 for any other value of x."}],"ID":13081},{"Watched":false,"Name":"Example - Part c","Duration":"3m 3s","ChapterTopicVideoID":12602,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12602.jpeg","UploadDate":"2018-07-23T15:53:45.6770000","DurationForVideoObject":"PT3M3S","Description":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":"00:14.850","Text":"Why is that, because the probability of"},{"Start":"00:14.850 ","End":"00:19.020","Text":"the chances x is the waiting time 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":"00:27.960","Text":"we see that what we\u0027re looking for is the area under"},{"Start":"00:27.960 ","End":"00:32.370","Text":"the density function between 0 and 2 and why is that?"},{"Start":"00:32.370 ","End":"00:37.005","Text":"Because the area under the density function is basically the probability,"},{"Start":"00:37.005 ","End":"00:40.300","Text":"so what we\u0027re looking for,"},{"Start":"00:40.300 ","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":"00:49.100","Text":"then we\u0027re looking for this area right here."},{"Start":"00:49.100 ","End":"00:58.095","Text":"All the area underneath the density function between 0 and 2."},{"Start":"00:58.095 ","End":"01:00.775","Text":"Let\u0027s get started, well,"},{"Start":"01:00.775 ","End":"01:04.234","Text":"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":"01:35.810 ","End":"01:39.675","Text":"Where x equals 2,"},{"Start":"01:39.675 ","End":"01:41.280","Text":"let\u0027s just plug that in."},{"Start":"01:41.280 ","End":"01:49.140","Text":"That\u0027s 1/8 times 2 and that equals to 2/8 and that equals to 1/4,"},{"Start":"01:49.140 ","End":"01:54.150","Text":"so 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":"02:08.525","Text":"and that equals to 1/4."},{"Start":"02:08.525 ","End":"02:14.750","Text":"This is the probability of the waiting time being less than 2 minutes."},{"Start":"02:14.750 ","End":"02:21.920","Text":"Now another thing that we should know is that the probability of x being less than 2,"},{"Start":"02:21.920 ","End":"02:29.780","Text":"that equals to the probability of x being less than or equal to 2 and 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":"02:44.270","Text":"Whether we include this,"},{"Start":"02:44.270 ","End":"02:48.380","Text":"the point to 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":13082},{"Watched":false,"Name":"Example - Part d","Duration":"6m 43s","ChapterTopicVideoID":12603,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12603.jpeg","UploadDate":"2018-07-23T15:54:14.1270000","DurationForVideoObject":"PT6M43S","Description":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":"00:09.790","Text":"or in short CDF."},{"Start":"00:09.790 ","End":"00:14.730","Text":"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":"00:26.295","Text":"the value t. We write it out like this,"},{"Start":"00:26.295 ","End":"00:30.460","Text":"the CDF for t,"},{"Start":"00:30.460 ","End":"00:36.370","Text":"well that equals to the probability of x being less than or equal to t. Similarly,"},{"Start":"00:36.370 ","End":"00:40.160","Text":"if we\u0027re looking for the probability of x being greater than t, well,"},{"Start":"00:40.160 ","End":"00:45.005","Text":"that\u0027s 1 minus this guy or 1 minus big F of t,"},{"Start":"00:45.005 ","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":"01:02.815","Text":"Well, that\u0027s just the value of the CDF at point b minus the value of the CDF at point a."},{"Start":"01:02.815 ","End":"01:10.220","Text":"Let\u0027s now go back to our example and see how we can work with this function."},{"Start":"01:10.220 ","End":"01:12.320","Text":"In section d, we\u0027re asked to build"},{"Start":"01:12.320 ","End":"01:17.715","Text":"a cumulative distribution function of a continuous random variable x."},{"Start":"01:17.715 ","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":"01:52.955","Text":"we basically see 3 areas of x that we have to deal with 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":"02:10.000","Text":"This equals to,"},{"Start":"02:10.250 ","End":"02:14.840","Text":"where x is negative,"},{"Start":"02:14.840 ","End":"02:22.575","Text":"where x is between 0 and 4 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":"02:31.279","Text":"this guy right here."},{"Start":"02:31.279 ","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":"02:48.395","Text":"Well, that\u0027s 0, because there\u0027s no area underneath the density function."},{"Start":"02:48.395 ","End":"02:54.720","Text":"Now, that would be for all x\u0027s that are negative until 0,"},{"Start":"02:54.720 ","End":"03:03.095","Text":"there would be no area underneath the density function that we can accumulate."},{"Start":"03:03.095 ","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":"03:29.720","Text":"assume t is 5, it\u0027s right here."},{"Start":"03:29.720 ","End":"03:36.125","Text":"Then this cumulative distribution function,"},{"Start":"03:36.125 ","End":"03:40.460","Text":"would be the probability of x being less than or equal to 5."},{"Start":"03:40.460 ","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 5 here."},{"Start":"03:57.230 ","End":"04:03.140","Text":"Now, what\u0027s the total area underneath the density function that we\u0027ve"},{"Start":"04:03.140 ","End":"04:06.110","Text":"accumulated while we\u0027ve accumulated all of"},{"Start":"04:06.110 ","End":"04:10.370","Text":"the area underneath the function or all the probability."},{"Start":"04:10.370 ","End":"04:13.765","Text":"That means that would be equal to 1."},{"Start":"04:13.765 ","End":"04:19.145","Text":"That means that for all values above 4,"},{"Start":"04:19.145 ","End":"04:23.510","Text":"the CDF would be equal to 1."},{"Start":"04:23.510 ","End":"04:25.345","Text":"Now what about here?"},{"Start":"04:25.345 ","End":"04:28.745","Text":"In this area between the area of 0 and 4?"},{"Start":"04:28.745 ","End":"04:35.720","Text":"Well, let\u0027s take the point here of value of x and call it t."},{"Start":"04:35.720 ","End":"04:39.125","Text":"This is the same t here and we want to know"},{"Start":"04:39.125 ","End":"04:43.460","Text":"what\u0027s the probability of x being less than or equal to t. Well,"},{"Start":"04:43.460 ","End":"04:50.120","Text":"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":"04:56.600","Text":"Well, that\u0027s base times height divided by 2."},{"Start":"04:56.600 ","End":"05:00.090","Text":"Well, the base 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":"05:18.409","Text":"where x was between 0 and 4, and 0 otherwise."},{"Start":"05:18.409 ","End":"05:22.500","Text":"If we put t at a specific point,"},{"Start":"05:22.500 ","End":"05:27.620","Text":"that will be 1/8 times t. They\u0027ll be 1/8 times t,"},{"Start":"05:27.620 ","End":"05:29.945","Text":"that\u0027s the base, that\u0027s the height."},{"Start":"05:29.945 ","End":"05:32.765","Text":"All this would be divided by 2."},{"Start":"05:32.765 ","End":"05:38.340","Text":"That equals to t squared over 16."},{"Start":"05:38.560 ","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":"05:57.950","Text":"we asked what was the probability of x being less than 2?"},{"Start":"05:57.950 ","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":"06:14.985","Text":"look at what range it was in,"},{"Start":"06:14.985 ","End":"06:17.540","Text":"and right now, this is the range between 0 and 4."},{"Start":"06:17.540 ","End":"06:23.000","Text":"2 is between 0 and 4 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 over 16,"},{"Start":"06:29.300 ","End":"06:31.475","Text":"and that equals to 1 over 4."},{"Start":"06:31.475 ","End":"06:36.110","Text":"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":13083},{"Watched":false,"Name":"Example - Part e","Duration":"3m 53s","ChapterTopicVideoID":12604,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12604.jpeg","UploadDate":"2018-07-23T15:54:26.2570000","DurationForVideoObject":"PT3M53S","Description":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":"00:07.425","Text":"Now, the P percentile is a value,"},{"Start":"00:07.425 ","End":"00:09.930","Text":"we denoted as small x_p,"},{"Start":"00:09.930 ","End":"00:15.480","Text":"for which the chances of the variable being under it or p. In another words,"},{"Start":"00:15.480 ","End":"00:16.860","Text":"we write it like this."},{"Start":"00:16.860 ","End":"00:22.635","Text":"We write as the probability of x being less than or equal to small x_p."},{"Start":"00:22.635 ","End":"00:29.340","Text":"But that would equal to p. We\u0027re looking for this specific number right here,"},{"Start":"00:29.340 ","End":"00:35.400","Text":"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":"00:50.930 ","End":"00:53.480","Text":"Section If, for example,"},{"Start":"00:53.480 ","End":"00:57.785","Text":"we\u0027re asked 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":"01:08.255","Text":"we\u0027re looking for the probability of x being less than or equal to small x_p."},{"Start":"01:08.255 ","End":"01:12.080","Text":"Let\u0027s say 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":"01:20.230","Text":"like the cumulative distribution function of x."},{"Start":"01:20.230 ","End":"01:22.560","Text":"Let\u0027s just write this f,"},{"Start":"01:22.560 ","End":"01:25.550","Text":"the f of x small x,"},{"Start":"01:25.550 ","End":"01:31.490","Text":"which is equal to the probability of x being less than or equal to x."},{"Start":"01:31.490 ","End":"01:35.720","Text":"That equals to now if we remember,"},{"Start":"01:35.720 ","End":"01:37.580","Text":"we did it like this."},{"Start":"01:37.580 ","End":"01:39.950","Text":"X being less than 0."},{"Start":"01:39.950 ","End":"01:45.770","Text":"That was 0, t squared over 16 that was when x was"},{"Start":"01:45.770 ","End":"01:54.940","Text":"between 0 and 4 and 1 when x was greater than 4."},{"Start":"01:55.640 ","End":"01:58.130","Text":"What are we looking for?"},{"Start":"01:58.130 ","End":"02:01.955","Text":"Well, basically we are looking at this expression right here."},{"Start":"02:01.955 ","End":"02:06.250","Text":"We\u0027re looking for p."},{"Start":"02:06.250 ","End":"02:12.885","Text":"The percentile p is the 80th percentile so p here has to equal to 0.8."},{"Start":"02:12.885 ","End":"02:20.140","Text":"We\u0027re looking at the probability of x being less than or equal to some x,"},{"Start":"02:20.140 ","End":"02:23.685","Text":"sub 0.8 sub p,"},{"Start":"02:23.685 ","End":"02:26.580","Text":"that equals to 0.8."},{"Start":"02:26.580 ","End":"02:30.155","Text":"That means to only have to do is to take this expression"},{"Start":"02:30.155 ","End":"02:33.995","Text":"and equated to this percentile right here,"},{"Start":"02:33.995 ","End":"02:38.345","Text":"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":"02:43.130","Text":"We\u0027re looking at,"},{"Start":"02:43.130 ","End":"02:47.825","Text":"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 that equals 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":"03:17.000","Text":"x is between 04,"},{"Start":"03:17.000 ","End":"03:24.560","Text":"so that would come out to 3.58 and what does this mean?"},{"Start":"03:24.560 ","End":"03:31.085","Text":"This means that when x is equal to 3.58 within this range,"},{"Start":"03:31.085 ","End":"03:32.930","Text":"it\u0027s between 0 and 4."},{"Start":"03:32.930 ","End":"03:35.765","Text":"When x equals 3.58,"},{"Start":"03:35.765 ","End":"03:40.880","Text":"the probability of x being less than this number,"},{"Start":"03:40.880 ","End":"03:42.470","Text":"less than or equal to this number."},{"Start":"03:42.470 ","End":"03:46.445","Text":"Well, that would be 0.8 or 80 percent."},{"Start":"03:46.445 ","End":"03:53.430","Text":"This number right here is the 80th percentile of the distribution."}],"ID":13084},{"Watched":false,"Name":"Exercise 1 - Parts a-b","Duration":"7m 5s","ChapterTopicVideoID":12608,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12608.jpeg","UploadDate":"2018-07-23T15:55:34.7600000","DurationForVideoObject":"PT7M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.830","Text":"In our example, the diagram below shows the density function of the variable X"},{"Start":"00:04.830 ","End":"00:10.424","Text":"where X is the waiting time in minutes for a telephone response from customer service."},{"Start":"00:10.424 ","End":"00:14.530","Text":"In section a, we\u0027re asked to find the value of a."},{"Start":"00:14.660 ","End":"00:19.290","Text":"First of all, let\u0027s take a look at this straight line."},{"Start":"00:19.290 ","End":"00:20.640","Text":"This straight line right here,"},{"Start":"00:20.640 ","End":"00:22.815","Text":"that\u0027s f of x."},{"Start":"00:22.815 ","End":"00:26.970","Text":"Now, if we take a look at this right here,"},{"Start":"00:26.970 ","End":"00:34.455","Text":"this form, f of x is basically the hypotenuse of this triangle right here."},{"Start":"00:34.455 ","End":"00:36.780","Text":"We can see that right here."},{"Start":"00:36.780 ","End":"00:41.975","Text":"Now, what do we know about the area of a triangle?"},{"Start":"00:41.975 ","End":"00:44.375","Text":"How do we calculate the area of a triangle?"},{"Start":"00:44.375 ","End":"00:52.905","Text":"Well, the area of a triangle that will equal to base times the height divided by 2,"},{"Start":"00:52.905 ","End":"00:56.765","Text":"and in our case, the base equals 4."},{"Start":"00:56.765 ","End":"00:58.940","Text":"The height of the triangle."},{"Start":"00:58.940 ","End":"01:00.050","Text":"That\u0027s this guy right here."},{"Start":"01:00.050 ","End":"01:02.450","Text":"Well, that\u0027s a, that\u0027s what we\u0027re trying to find out."},{"Start":"01:02.450 ","End":"01:08.335","Text":"Divided by 2, that equals to 2a."},{"Start":"01:08.335 ","End":"01:13.040","Text":"That would be the area of the triangle right here."},{"Start":"01:13.040 ","End":"01:20.135","Text":"Now, what do we know about the area under the density function?"},{"Start":"01:20.135 ","End":"01:26.120","Text":"But we know the area under the density function has to be equal to 1."},{"Start":"01:26.120 ","End":"01:29.435","Text":"This thing has to be equal to 1."},{"Start":"01:29.435 ","End":"01:34.590","Text":"This is the area under the density function and that must equal to 1."},{"Start":"01:34.590 ","End":"01:38.590","Text":"A here has to equal to 1/2."},{"Start":"01:38.590 ","End":"01:43.665","Text":"This is the value of a,"},{"Start":"01:43.665 ","End":"01:46.440","Text":"a right here, equals 1/2."},{"Start":"01:46.440 ","End":"01:51.814","Text":"In section b, we\u0027re asked to write the formula for the density function."},{"Start":"01:51.814 ","End":"01:54.500","Text":"This is our density functions,"},{"Start":"01:54.500 ","End":"01:56.705","Text":"this is a diagram of a density function."},{"Start":"01:56.705 ","End":"02:04.430","Text":"But it shows us the density function only from x equals 0 to x equals 4."},{"Start":"02:04.430 ","End":"02:07.145","Text":"What about all other values of x?"},{"Start":"02:07.145 ","End":"02:12.350","Text":"Well, if we drop the density function for all values of x,"},{"Start":"02:12.350 ","End":"02:17.190","Text":"we see that f of x equals 0 for all x that\u0027s negative,"},{"Start":"02:17.190 ","End":"02:22.745","Text":"and it\u0027s also 0 for all events and x which are greater than 4,"},{"Start":"02:22.745 ","End":"02:26.540","Text":"and between 0 and 4 in that range."},{"Start":"02:26.540 ","End":"02:31.985","Text":"The function of the density function of X is this straight line right here."},{"Start":"02:31.985 ","End":"02:42.590","Text":"Let\u0027s just write this out saying that f of x equals some function for a straight line,"},{"Start":"02:42.590 ","End":"02:46.445","Text":"and we\u0027ll calculate that in a bed for all values of x"},{"Start":"02:46.445 ","End":"02:52.545","Text":"between 0 and 4 and it\u0027ll be 0 otherwise."},{"Start":"02:52.545 ","End":"02:59.015","Text":"All we have to do now is to calculate the equation for this line."},{"Start":"02:59.015 ","End":"03:02.720","Text":"Now, the general form of a line."},{"Start":"03:02.720 ","End":"03:09.965","Text":"The general equation of a line is y equals mx plus n,"},{"Start":"03:09.965 ","End":"03:16.355","Text":"where n is the y-intercept and m is the slope of the slide."},{"Start":"03:16.355 ","End":"03:19.355","Text":"First of all, let\u0027s take a look at n,"},{"Start":"03:19.355 ","End":"03:21.420","Text":"n is the y-intercept."},{"Start":"03:21.420 ","End":"03:25.850","Text":"Where does this density function intersect the y-axis?"},{"Start":"03:25.850 ","End":"03:29.609","Text":"Well, at 0, where x equals 0,"},{"Start":"03:29.609 ","End":"03:34.865","Text":"that\u0027s where f of x intersects the y-axis,"},{"Start":"03:34.865 ","End":"03:37.640","Text":"and what about the slope m?"},{"Start":"03:37.640 ","End":"03:47.970","Text":"Well, m is calculated as y2 minus y1 divided by x2 minus x1,"},{"Start":"03:47.970 ","End":"03:50.010","Text":"where x1, y1,"},{"Start":"03:50.010 ","End":"03:54.869","Text":"and x2, y2 are 2 points."},{"Start":"03:54.869 ","End":"03:58.250","Text":"Let\u0027s just write them out. This part right here."},{"Start":"03:58.250 ","End":"04:02.420","Text":"The point of origin would be x1, y1."},{"Start":"04:02.420 ","End":"04:04.340","Text":"That\u0027s 0,0."},{"Start":"04:04.340 ","End":"04:06.710","Text":"What about the second point?"},{"Start":"04:06.710 ","End":"04:09.080","Text":"Well, let\u0027s take a look at this point right here."},{"Start":"04:09.080 ","End":"04:12.590","Text":"That\u0027s 4, 1/2 X equals 4,"},{"Start":"04:12.590 ","End":"04:19.270","Text":"y equals 1/2 and why half because we\u0027ve calculated that in section a."},{"Start":"04:19.270 ","End":"04:25.605","Text":"Let\u0027s just plug in the numbers here y2 is 1/2 minus 0."},{"Start":"04:25.605 ","End":"04:33.335","Text":"That\u0027s 1/2 minus 0 divided by x2 is 4 minus x 1 that\u0027s 0."},{"Start":"04:33.335 ","End":"04:35.540","Text":"Let\u0027s do a quick calculation."},{"Start":"04:35.540 ","End":"04:41.180","Text":"That\u0027s 1/2 divided by 4 and that equals to an 1/8."},{"Start":"04:41.180 ","End":"04:45.980","Text":"We can say that the equation for this line right"},{"Start":"04:45.980 ","End":"04:53.510","Text":"here is m equals 1/8 times x plus 0."},{"Start":"04:53.510 ","End":"04:55.955","Text":"Now, what is y,"},{"Start":"04:55.955 ","End":"04:59.580","Text":"if not the density function of X?"},{"Start":"04:59.580 ","End":"05:03.650","Text":"Let\u0027s just simplify this and put that over here."},{"Start":"05:03.650 ","End":"05:06.280","Text":"That equals to 1/8 times x."},{"Start":"05:06.280 ","End":"05:11.645","Text":"We can say that the density function equals 1/8 times x,"},{"Start":"05:11.645 ","End":"05:18.720","Text":"where x is between 0 and 4 for any other value of x."}],"ID":13085},{"Watched":false,"Name":"Exercise 1 - Part c","Duration":"3m 58s","ChapterTopicVideoID":12606,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12606.jpeg","UploadDate":"2018-07-23T15:54:59.5170000","DurationForVideoObject":"PT3M58S","Description":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":"00:07.590","Text":"Well, let\u0027s use our CDF to do that."},{"Start":"00:07.590 ","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":"That equals to following."},{"Start":"00:21.450 ","End":"00:23.715","Text":"Now it\u0027s 0,"},{"Start":"00:23.715 ","End":"00:26.610","Text":"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":"01:11.909","Text":"that\u0027s F at 4."},{"Start":"01:11.909 ","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":"01:20.099","Text":"we\u0027ll be using this expression."},{"Start":"01:20.099 ","End":"01:28.935","Text":"That\u0027ll be 0.25 plus 4 minus 2 times 3/16,"},{"Start":"01:28.935 ","End":"01:31.720","Text":"that equals to 5/8."},{"Start":"01:31.720 ","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":"01:45.530","Text":"That\u0027s 1 minus, now,"},{"Start":"01:45.530 ","End":"01:51.245","Text":"1.5 is in this range between 1 and 2 so we\u0027ll be using this expression."},{"Start":"01:51.245 ","End":"01:58.320","Text":"That will be 1.5 minus 1 times 0.25,"},{"Start":"01:58.480 ","End":"02:06.170","Text":"that will equal to 7/8."},{"Start":"02:06.170 ","End":"02:08.060","Text":"Let\u0027s take a look at this probability,"},{"Start":"02:08.060 ","End":"02:15.475","Text":"probability of x being between 1.5 and 5."},{"Start":"02:15.475 ","End":"02:23.655","Text":"That equals to F at 5 minus F at 1.5."},{"Start":"02:23.655 ","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":"02:37.770","Text":"That will be 0.25 plus 5 minus 2 times 3/16."},{"Start":"02:37.770 ","End":"02:41.330","Text":"Minus F at 1.5."},{"Start":"02:41.330 ","End":"02:44.675","Text":"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 at 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":"that\u0027s right here. That equals to 1."},{"Start":"03:24.515 ","End":"03:33.225","Text":"The probability of x being less than 10 is 1 minus F at 5."},{"Start":"03:33.225 ","End":"03:36.780","Text":"Now, F at 5 is right here."},{"Start":"03:36.780 ","End":"03:38.735","Text":"We will be using this expression right here."},{"Start":"03:38.735 ","End":"03:44.789","Text":"That\u0027s 0.25 plus 5"},{"Start":"03:44.789 ","End":"03:50.150","Text":"minus 2 times 3/16."},{"Start":"03:50.150 ","End":"03:59.560","Text":"That equals to 3/16."}],"ID":13086},{"Watched":false,"Name":"Exercise 1 - Part d","Duration":"4m 22s","ChapterTopicVideoID":12607,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12607.jpeg","UploadDate":"2018-07-23T15:55:11.5900000","DurationForVideoObject":"PT4M22S","Description":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":"00:12.240","Text":"The median is a specific value of X,"},{"Start":"00:12.240 ","End":"00:18.795","Text":"where half 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":"00:30.870","Text":"This is a specific percentile, very special percentile,"},{"Start":"00:30.870 ","End":"00:34.980","Text":"the median, and we write it like this."},{"Start":"00:34.980 ","End":"00:41.765","Text":"That\u0027s the probability of X being less than or equal to a specific value."},{"Start":"00:41.765 ","End":"00:45.345","Text":"That would be 0.5."},{"Start":"00:45.345 ","End":"00:48.450","Text":"This is a definition of a percentile,"},{"Start":"00:48.450 ","End":"00:55.800","Text":"where the probability that X would be less than that,"},{"Start":"00:55.800 ","End":"01:00.550","Text":"this specific value would be a half."},{"Start":"01:01.520 ","End":"01:05.370","Text":"This also can be solved using the CDF."},{"Start":"01:05.370 ","End":"01:07.545","Text":"Let\u0027s write out the CDF."},{"Start":"01:07.545 ","End":"01:10.680","Text":"The CDF, F of t,"},{"Start":"01:10.680 ","End":"01:17.210","Text":"that\u0027s the probability of x being less than or equal to t. In our case,"},{"Start":"01:17.210 ","End":"01:22.965","Text":"that was 0 when x was less than 1."},{"Start":"01:22.965 ","End":"01:31.140","Text":"It was t minus 1 times a quarter when x was between 1 and 2."},{"Start":"01:31.140 ","End":"01:38.565","Text":"It was a quarter 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":"01:52.300","Text":"How can we use this CDF to solve"},{"Start":"01:52.300 ","End":"01:57.650","Text":"this probability or to find the median of x or this distribution?"},{"Start":"01:57.650 ","End":"02:03.025","Text":"Let\u0027s use also the density function."},{"Start":"02:03.025 ","End":"02:05.530","Text":"Here\u0027s a density function,"},{"Start":"02:05.530 ","End":"02:09.400","Text":"and we can see the following."},{"Start":"02:09.400 ","End":"02:13.310","Text":"First of all, when we calculated"},{"Start":"02:13.310 ","End":"02:18.115","Text":"the probabilities or the area underneath the density function,"},{"Start":"02:18.115 ","End":"02:26.575","Text":"we found out 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":"02:48.785","Text":"all the probabilities below t have to equal to a half."},{"Start":"02:48.785 ","End":"02:51.210","Text":"How do we do that?"},{"Start":"02:51.210 ","End":"02:53.805","Text":"Since we know that t is in this range,"},{"Start":"02:53.805 ","End":"02:59.015","Text":"from 2-6, then we\u0027ll be using this expression right here."},{"Start":"02:59.015 ","End":"03:00.755","Text":"Let\u0027s just write this out."},{"Start":"03:00.755 ","End":"03:07.930","Text":"That\u0027ll be a quarter plus t minus 2 times 3/16."},{"Start":"03:07.930 ","End":"03:10.880","Text":"We know in order to find the median,"},{"Start":"03:10.880 ","End":"03:15.480","Text":"that has to be equal to a half."},{"Start":"03:16.310 ","End":"03:24.755","Text":"Now let\u0027s solve for t. That means that t minus 2 times 3/16,"},{"Start":"03:24.755 ","End":"03:28.285","Text":"that has to be equal to a quarter,"},{"Start":"03:28.285 ","End":"03:32.740","Text":"and that means that t minus 2,"},{"Start":"03:32.740 ","End":"03:37.355","Text":"that equals to 16/12,"},{"Start":"03:37.355 ","End":"03:40.405","Text":"which equals to 4/3."},{"Start":"03:40.405 ","End":"03:47.010","Text":"That means that t now equals to 2 plus 4/3."},{"Start":"03:47.010 ","End":"03:50.520","Text":"That equals to 3 and 1/3."},{"Start":"03:50.520 ","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":"04:00.020","Text":"this point right here,"},{"Start":"04:00.020 ","End":"04:08.930","Text":"half of the probability lies below this value 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":13087},{"Watched":false,"Name":"Exercise 2","Duration":"8m 51s","ChapterTopicVideoID":12609,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12609.jpeg","UploadDate":"2018-07-23T15:56:07.8570000","DurationForVideoObject":"PT8M51S","Description":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":"00:06.720","Text":"that has the following density function: f of x,"},{"Start":"00:06.720 ","End":"00:07.895","Text":"that\u0027s a density function,"},{"Start":"00:07.895 ","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":"00:19.980","Text":"Now, it\u0027s also known that the probability of x being between 0 and 1,"},{"Start":"00:19.980 ","End":"00:24.525","Text":"well that\u0027s 1/4, and we\u0027re asked to write the density function of x."},{"Start":"00:24.525 ","End":"00:28.510","Text":"Let\u0027s first draw this function."},{"Start":"00:28.790 ","End":"00:33.270","Text":"We have the density function right here."},{"Start":"00:33.270 ","End":"00:35.960","Text":"Where on the x-axis,"},{"Start":"00:35.960 ","End":"00:41.780","Text":"we see the points 0 and we see b. X is between 0 and b."},{"Start":"00:41.780 ","End":"00:43.450","Text":"That\u0027s what we have here."},{"Start":"00:43.450 ","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":"01:15.320","Text":"We want to write the density function of x. That\u0027s fine."},{"Start":"01:15.320 ","End":"01:21.575","Text":"What do we know here about density functions and how can we calculate c?"},{"Start":"01:21.575 ","End":"01:26.060","Text":"Well, we know that the area under the density function,"},{"Start":"01:26.060 ","End":"01:30.870","Text":"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":"01:41.715","Text":"the probability is 1/4."},{"Start":"01:41.715 ","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":"01:53.735","Text":"the area under the density function equals 1/4."},{"Start":"01:53.735 ","End":"01:59.850","Text":"Because the area under the density function is the probability."},{"Start":"01:59.850 ","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":"02:13.550","Text":"The base is b times the height."},{"Start":"02:13.550 ","End":"02:19.085","Text":"Well, the height is the value of the density function at point b."},{"Start":"02:19.085 ","End":"02:27.365","Text":"That would be c times b divided by 2 and that equals to 1."},{"Start":"02:27.365 ","End":"02:30.335","Text":"Now, what does that mean?"},{"Start":"02:30.335 ","End":"02:33.940","Text":"That means that b squared times c,"},{"Start":"02:33.940 ","End":"02:36.390","Text":"that equals to 2."},{"Start":"02:36.390 ","End":"02:40.490","Text":"Now, what else are we given?"},{"Start":"02:40.490 ","End":"02:43.857","Text":"Let\u0027s call this the first equation right here."},{"Start":"02:43.857 ","End":"02:50.150","Text":"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, that equals to c,"},{"Start":"03:03.680 ","End":"03:06.125","Text":"c times 1 equals c,"},{"Start":"03:06.125 ","End":"03:10.740","Text":"divided by 2 and 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":"03:18.270","Text":"That means that c equals 1/2."},{"Start":"03:18.270 ","End":"03:21.390","Text":"We automatically got c here."},{"Start":"03:21.390 ","End":"03:25.780","Text":"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":"03:35.430","Text":"now c is 1/2, equals 2."},{"Start":"03:35.430 ","End":"03:39.385","Text":"So b squared equals 4."},{"Start":"03:39.385 ","End":"03:43.895","Text":"When we take the square root of both sides,"},{"Start":"03:43.895 ","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":"03:51.835","Text":"That\u0027s the way it\u0027s being defined."},{"Start":"03:51.835 ","End":"03:58.790","Text":"That means that b equals 2."},{"Start":"03:58.970 ","End":"04:02.070","Text":"Now let\u0027s take a look."},{"Start":"04:02.070 ","End":"04:06.500","Text":"Let\u0027s write the density function of x."},{"Start":"04:06.500 ","End":"04:08.825","Text":"Well, that\u0027s f of x."},{"Start":"04:08.825 ","End":"04:13.970","Text":"Now that will equal c. What\u0027s c?"},{"Start":"04:13.970 ","End":"04:17.930","Text":"C is 1/2, so it\u0027s 1/2 times x,"},{"Start":"04:17.930 ","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":"04:39.005","Text":"Let\u0027s call this t for now."},{"Start":"04:39.005 ","End":"04:47.780","Text":"Where half of the probability lies below this point and half 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":"05:02.740","Text":"We\u0027re looking for this area right here to be equal to 1/2."},{"Start":"05:02.740 ","End":"05:07.430","Text":"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":"05:32.885","Text":"Well, that\u0027s t,"},{"Start":"05:32.885 ","End":"05:34.085","Text":"and 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":"05:48.120","Text":"to 1/2 times t. We substitute x for t divided by 2."},{"Start":"05:48.120 ","End":"05:52.160","Text":"The base times the height divided by 2,"},{"Start":"05:52.160 ","End":"05:59.330","Text":"that\u0027s the area of this triangle and that has to equal to 1/2."},{"Start":"05:59.330 ","End":"06:00.935","Text":"That was 0.5."},{"Start":"06:00.935 ","End":"06:02.860","Text":"In essence,"},{"Start":"06:02.860 ","End":"06:05.310","Text":"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":"06:11.430","Text":"that equals to 1, or t squared,"},{"Start":"06:11.430 ","End":"06:14.190","Text":"well, that equals to 2."},{"Start":"06:14.190 ","End":"06:24.290","Text":"That means that t equals to the square root of 2 or 1.41 in approximation."},{"Start":"06:24.290 ","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 half of"},{"Start":"06:35.540 ","End":"06:41.585","Text":"the probability below this point and half of the probability we find above this point."},{"Start":"06:41.585 ","End":"06:46.130","Text":"This is square root of 2 here we found 0.5 of the probability,"},{"Start":"06:46.130 ","End":"06:49.120","Text":"that\u0027s this area right here,"},{"Start":"06:49.120 ","End":"06:52.880","Text":"and here that\u0027s the other half right here."},{"Start":"06:52.880 ","End":"06:55.500","Text":"That\u0027s 0.5 right here."},{"Start":"06:56.150 ","End":"07:02.180","Text":"Section c, we\u0027re asked what is the chances of x being less than 0.5 or 1/2?"},{"Start":"07:02.180 ","End":"07:08.240","Text":"Well, that\u0027s the probability of x being less than 1/2."},{"Start":"07:08.240 ","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 can just plug in the number,"},{"Start":"07:18.230 ","End":"07:19.655","Text":"but since I don\u0027t, well,"},{"Start":"07:19.655 ","End":"07:29.150","Text":"let\u0027s use the density function and find the area under the density function from 0-1/2."},{"Start":"07:29.150 ","End":"07:32.910","Text":"Now, this is our point 1/2."},{"Start":"07:32.910 ","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":"It\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":"07:59.760","Text":"where x was between 0 and b, and 0 otherwise."},{"Start":"07:59.760 ","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":"08:11.318","Text":"well, the base is 1/2."},{"Start":"08:11.318 ","End":"08:13.726","Text":"What\u0027s the height? Well,"},{"Start":"08:13.726 ","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":"08:36.270","Text":"This is the height and we divide that by 2."},{"Start":"08:36.270 ","End":"08:38.625","Text":"Base times height divided by 2."},{"Start":"08:38.625 ","End":"08:43.340","Text":"That equals to 1/16,"},{"Start":"08:43.340 ","End":"08:45.230","Text":"so 1 divided by 16."},{"Start":"08:45.230 ","End":"08:51.330","Text":"This is the probability of x being less than 1/2."}],"ID":13088},{"Watched":false,"Name":"Exercise 3 - Parts a-b","Duration":"12m 12s","ChapterTopicVideoID":12610,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12610.jpeg","UploadDate":"2018-07-23T15:56:48.2470000","DurationForVideoObject":"PT12M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.385","Text":"The diagram below shows the density function of the random variable Y."},{"Start":"00:05.385 ","End":"00:08.190","Text":"This axis right here is y,"},{"Start":"00:08.190 ","End":"00:11.730","Text":"and this axis right here is f of y."},{"Start":"00:11.730 ","End":"00:15.150","Text":"We\u0027re asked to calculate, in section a,"},{"Start":"00:15.150 ","End":"00:22.664","Text":"the value of c. This is the value that we want to calculate."},{"Start":"00:22.664 ","End":"00:25.650","Text":"What\u0027s the area of a triangle?"},{"Start":"00:25.650 ","End":"00:32.595","Text":"The area of a triangle equals the base times the height divided by 2."},{"Start":"00:32.595 ","End":"00:39.540","Text":"In our case, the base is 10. What\u0027s the height?"},{"Start":"00:39.540 ","End":"00:45.440","Text":"The height is c. The base times the height divided by 2."},{"Start":"00:45.440 ","End":"00:49.790","Text":"We know that because this is a density function,"},{"Start":"00:49.790 ","End":"00:55.295","Text":"then this area, the total area under the density function has to equal to 1."},{"Start":"00:55.295 ","End":"00:57.640","Text":"All that equals to 1."},{"Start":"00:57.640 ","End":"01:02.675","Text":"That means that 10c equals to 2."},{"Start":"01:02.675 ","End":"01:06.650","Text":"Therefore c equals to 2 over 10,"},{"Start":"01:06.650 ","End":"01:11.670","Text":"or 1 over 5, or 0.2."},{"Start":"01:11.670 ","End":"01:16.910","Text":"That is the value of c. Section b,"},{"Start":"01:16.910 ","End":"01:20.750","Text":"we\u0027re asked to write the cumulative distribution function of Y."},{"Start":"01:20.750 ","End":"01:24.995","Text":"That means that we\u0027re looking at F of t,"},{"Start":"01:24.995 ","End":"01:32.855","Text":"or the probability that y is less than or equal to t. In order to calculate this,"},{"Start":"01:32.855 ","End":"01:36.830","Text":"we\u0027re going to have to look at the various ranges of"},{"Start":"01:36.830 ","End":"01:42.290","Text":"y and calculate the CDF for each 1 of these ranges."},{"Start":"01:42.290 ","End":"01:44.165","Text":"What are the ranges of y?"},{"Start":"01:44.165 ","End":"01:45.890","Text":"Well, first of all,"},{"Start":"01:45.890 ","End":"01:48.920","Text":"y is less than 0, y is negative."},{"Start":"01:48.920 ","End":"01:50.435","Text":"That\u0027s the 1st range."},{"Start":"01:50.435 ","End":"01:54.445","Text":"2nd range is where y is between 0 and 5."},{"Start":"01:54.445 ","End":"01:59.010","Text":"3rd range, where y is between 5 and 10."},{"Start":"01:59.010 ","End":"02:02.450","Text":"The 4th range where y is greater than 10."},{"Start":"02:02.450 ","End":"02:05.680","Text":"Well, let\u0027s just write these guys down."},{"Start":"02:05.680 ","End":"02:10.625","Text":"This thing, the probability of y being less than or equal to t,"},{"Start":"02:10.625 ","End":"02:13.080","Text":"that would equal to"},{"Start":"02:14.080 ","End":"02:20.240","Text":"a whole bunch of calculations here and we\u0027ll get to that right under the ranges where y,"},{"Start":"02:20.240 ","End":"02:22.535","Text":"first of all, is less than 0,"},{"Start":"02:22.535 ","End":"02:28.365","Text":"y is between 0 and 5,"},{"Start":"02:28.365 ","End":"02:32.370","Text":"y is between 5 and 10,"},{"Start":"02:32.370 ","End":"02:36.820","Text":"and then y is being greater than 10."},{"Start":"02:37.550 ","End":"02:40.445","Text":"Let\u0027s first take a look at this range."},{"Start":"02:40.445 ","End":"02:43.250","Text":"Y is negative, y is less than 0."},{"Start":"02:43.250 ","End":"02:46.190","Text":"Well, when y is less than 0,"},{"Start":"02:46.190 ","End":"02:50.960","Text":"we have no area under the density function."},{"Start":"02:50.960 ","End":"02:55.130","Text":"That means that we haven\u0027t accumulated any probability."},{"Start":"02:55.130 ","End":"02:58.280","Text":"The probability of y being less than or equal to t,"},{"Start":"02:58.280 ","End":"03:00.520","Text":"that would be equal to 0."},{"Start":"03:00.520 ","End":"03:04.135","Text":"Let\u0027s take a look at the other side when y is greater than 10."},{"Start":"03:04.135 ","End":"03:06.715","Text":"Well, when y is greater than 10,"},{"Start":"03:06.715 ","End":"03:08.515","Text":"let\u0027s say somewhere here,"},{"Start":"03:08.515 ","End":"03:11.710","Text":"then we\u0027ve accumulated all the probability,"},{"Start":"03:11.710 ","End":"03:17.259","Text":"or we\u0027ve basically summed up all the area under the density function."},{"Start":"03:17.259 ","End":"03:19.850","Text":"That equals 1."},{"Start":"03:20.090 ","End":"03:28.310","Text":"Let\u0027s go into each 1 of these ranges and calculate the equation for the CDF."},{"Start":"03:28.880 ","End":"03:35.125","Text":"Let\u0027s take a look now at this range where y is between 0 and 5."},{"Start":"03:35.125 ","End":"03:38.830","Text":"What we want to do now is to calculate the probability of"},{"Start":"03:38.830 ","End":"03:43.085","Text":"y being less than or equal to t in this range."},{"Start":"03:43.085 ","End":"03:45.085","Text":"Let\u0027s define a point here."},{"Start":"03:45.085 ","End":"03:48.265","Text":"We\u0027ll call it t within this range."},{"Start":"03:48.265 ","End":"03:51.550","Text":"When we want to find the probability, in essence,"},{"Start":"03:51.550 ","End":"03:54.310","Text":"we want to find the area under"},{"Start":"03:54.310 ","End":"03:58.240","Text":"the density function between the points"},{"Start":"03:58.240 ","End":"04:03.175","Text":"0 and t. That would be defined as our probability."},{"Start":"04:03.175 ","End":"04:05.574","Text":"If that\u0027s the case,"},{"Start":"04:05.574 ","End":"04:11.980","Text":"then what we\u0027re looking at is the area under this graph right here,"},{"Start":"04:11.980 ","End":"04:19.630","Text":"under this line between 0 and t. What\u0027s this area?"},{"Start":"04:19.630 ","End":"04:22.390","Text":"Well, that\u0027s the area of a triangle and that equals to"},{"Start":"04:22.390 ","End":"04:27.030","Text":"the base times height divided by 2."},{"Start":"04:27.030 ","End":"04:28.800","Text":"The base is t,"},{"Start":"04:28.800 ","End":"04:30.875","Text":"that\u0027s easy, but what about the height?"},{"Start":"04:30.875 ","End":"04:36.620","Text":"Well, the height of this function right here, of this triangle,"},{"Start":"04:36.620 ","End":"04:44.780","Text":"is equal to the value of the density function at point t. What\u0027s our density function?"},{"Start":"04:44.780 ","End":"04:47.400","Text":"Well, that\u0027s this straight line right here."},{"Start":"04:47.400 ","End":"04:51.800","Text":"Let\u0027s calculate the equation for this straight line."},{"Start":"04:51.800 ","End":"05:00.800","Text":"Well, the general form of a straight line is y equals mx plus n. N is the y-intercept,"},{"Start":"05:00.800 ","End":"05:05.690","Text":"and we see that this line intercepts the y-axis at point 0,"},{"Start":"05:05.690 ","End":"05:08.120","Text":"so n equals 0."},{"Start":"05:08.120 ","End":"05:11.640","Text":"What about m? M is the slope."},{"Start":"05:11.640 ","End":"05:13.220","Text":"How do we define the slope?"},{"Start":"05:13.220 ","End":"05:19.655","Text":"That\u0027s y_2 minus y_1 divided by x_2 minus x_1."},{"Start":"05:19.655 ","End":"05:22.725","Text":"Well, what are 2 points here?"},{"Start":"05:22.725 ","End":"05:24.750","Text":"X_1, y_1, and x_2, y_2."},{"Start":"05:24.750 ","End":"05:26.475","Text":"Well, the first point x_1,"},{"Start":"05:26.475 ","End":"05:28.350","Text":"y_1, well that\u0027s 0,0."},{"Start":"05:28.350 ","End":"05:30.600","Text":"That\u0027s the point of origin."},{"Start":"05:30.600 ","End":"05:32.295","Text":"What about x_2,"},{"Start":"05:32.295 ","End":"05:34.610","Text":"y_2, that\u0027s this point right here."},{"Start":"05:34.610 ","End":"05:40.115","Text":"We\u0027ll define that as x equals 5 and y being equal to 0.2."},{"Start":"05:40.115 ","End":"05:44.205","Text":"That\u0027s c. We\u0027ve defined that in section a."},{"Start":"05:44.205 ","End":"05:46.559","Text":"Let\u0027s just plug in the numbers."},{"Start":"05:46.559 ","End":"05:53.610","Text":"That\u0027s 0.2 minus 0 divided by 5 minus 0."},{"Start":"05:53.610 ","End":"05:57.195","Text":"That equals to 0.2 divided by 5,"},{"Start":"05:57.195 ","End":"06:00.450","Text":"that equals to 0.04."},{"Start":"06:00.450 ","End":"06:05.569","Text":"When we plug this back into the equation for the straight line,"},{"Start":"06:05.569 ","End":"06:12.299","Text":"we\u0027ll get y equals 0.04 times x,"},{"Start":"06:12.299 ","End":"06:15.480","Text":"plus n, now n is 0."},{"Start":"06:15.480 ","End":"06:21.350","Text":"This is the equation of this straight line right here."},{"Start":"06:21.350 ","End":"06:26.675","Text":"If we plug in the value t for this equation,"},{"Start":"06:26.675 ","End":"06:32.100","Text":"we\u0027ll get the height of the triangle."},{"Start":"06:32.100 ","End":"06:37.160","Text":"That means that the height equals to 0.04 times"},{"Start":"06:37.160 ","End":"06:42.650","Text":"t. Let\u0027s calculate the area of this triangle."},{"Start":"06:42.650 ","End":"06:47.795","Text":"Remember this area is the probability that we\u0027re looking for."},{"Start":"06:47.795 ","End":"06:52.190","Text":"The area, equals the base,"},{"Start":"06:52.190 ","End":"06:55.135","Text":"the base is t times the height,"},{"Start":"06:55.135 ","End":"07:00.020","Text":"that\u0027s 0.04 times t divided by 2."},{"Start":"07:00.020 ","End":"07:05.555","Text":"That equals to 0.02 times t squared."},{"Start":"07:05.555 ","End":"07:13.290","Text":"This would be the equation for the cumulative distribution function."},{"Start":"07:13.780 ","End":"07:18.260","Text":"At the range where y is between 0 and 5."},{"Start":"07:18.260 ","End":"07:20.900","Text":"Let\u0027s write that down here."},{"Start":"07:20.900 ","End":"07:25.860","Text":"That\u0027s 0.02 times t squared."},{"Start":"07:26.030 ","End":"07:29.570","Text":"We have only 1 more range to deal with,"},{"Start":"07:29.570 ","End":"07:32.660","Text":"and that\u0027s the range between 5 and 10."},{"Start":"07:32.660 ","End":"07:37.280","Text":"In order to calculate the CDF in this range,"},{"Start":"07:37.280 ","End":"07:39.260","Text":"we\u0027ll be using the same type of logic."},{"Start":"07:39.260 ","End":"07:41.765","Text":"We\u0027ll define a value of y."},{"Start":"07:41.765 ","End":"07:45.270","Text":"Let\u0027s call it t, in that range."},{"Start":"07:45.270 ","End":"07:54.915","Text":"What we want to do is we want to calculate the area under the density function from 0-t."},{"Start":"07:54.915 ","End":"07:58.475","Text":"This isn\u0027t a straight forward area to calculate."},{"Start":"07:58.475 ","End":"08:02.150","Text":"On the other hand, this is."},{"Start":"08:02.150 ","End":"08:09.470","Text":"Because the total area under the density function equals 1,"},{"Start":"08:09.470 ","End":"08:16.745","Text":"if we calculate the area of the yellow triangle and we subtract that from 1,"},{"Start":"08:16.745 ","End":"08:20.790","Text":"then we\u0027ll get the area that we\u0027re looking for."},{"Start":"08:21.170 ","End":"08:27.170","Text":"The area of a triangle as we said is the base times the height divided by 2."},{"Start":"08:27.170 ","End":"08:30.020","Text":"The base here is pretty easy to calculate."},{"Start":"08:30.020 ","End":"08:33.463","Text":"That\u0027s just 10 minus t. What about the height?"},{"Start":"08:33.463 ","End":"08:37.370","Text":"Well, the height of this triangle, in actual fact,"},{"Start":"08:37.370 ","End":"08:41.605","Text":"is the value of the density function at point t."},{"Start":"08:41.605 ","End":"08:47.845","Text":"That means we\u0027re going to have to calculate the equation for this line right here."},{"Start":"08:47.845 ","End":"08:51.319","Text":"Let\u0027s do that. Well, let\u0027s first start with the slope."},{"Start":"08:51.319 ","End":"08:59.020","Text":"Slope is defined as y_2 minus y_1 divided by x_2 minus x_1."},{"Start":"08:59.060 ","End":"09:02.700","Text":"What are the 2 points that we\u0027re going to use?"},{"Start":"09:02.700 ","End":"09:04.125","Text":"1 would be 10,0,"},{"Start":"09:04.125 ","End":"09:06.225","Text":"that\u0027s this point right here."},{"Start":"09:06.225 ","End":"09:10.000","Text":"The other point right here will be 5,0.2,"},{"Start":"09:10.240 ","End":"09:13.130","Text":"c equals 0.2."},{"Start":"09:13.130 ","End":"09:14.900","Text":"We\u0027ve calculated that."},{"Start":"09:14.900 ","End":"09:17.450","Text":"Let\u0027s just plug in the numbers here."},{"Start":"09:17.450 ","End":"09:18.995","Text":"Y_2 is 0,"},{"Start":"09:18.995 ","End":"09:23.810","Text":"that\u0027s 0 minus 0.2 divided by 10 minus 5."},{"Start":"09:23.810 ","End":"09:31.640","Text":"That equals to minus 0.2 divided by 5 and that equals to minus 0.04."},{"Start":"09:33.040 ","End":"09:38.090","Text":"The equation of a line when we know the slope and we have,"},{"Start":"09:38.090 ","End":"09:39.830","Text":"1 of the points is this."},{"Start":"09:39.830 ","End":"09:43.215","Text":"That\u0027s y minus y_1,"},{"Start":"09:43.215 ","End":"09:49.415","Text":"that would equal to m times x minus x_1."},{"Start":"09:49.415 ","End":"09:52.340","Text":"We\u0027ll be using this point right here, 10,0."},{"Start":"09:52.340 ","End":"09:54.230","Text":"When we plug in the numbers,"},{"Start":"09:54.230 ","End":"09:58.230","Text":"that\u0027ll be y minus 0, y_1 is 0."},{"Start":"09:58.230 ","End":"10:07.420","Text":"It will equal to minus 0.04 times x minus 10."},{"Start":"10:07.790 ","End":"10:13.875","Text":"At point t, that would be the height of this triangle."},{"Start":"10:13.875 ","End":"10:18.290","Text":"Let\u0027s just plug in t instead of x,"},{"Start":"10:18.290 ","End":"10:24.575","Text":"that\u0027ll be minus 0.04, t minus 10."},{"Start":"10:24.575 ","End":"10:26.860","Text":"In order to make things a little bit simpler,"},{"Start":"10:26.860 ","End":"10:30.270","Text":"and we\u0027ll see why in a bit that would equal to"},{"Start":"10:30.270 ","End":"10:38.365","Text":"0.04 times 10 minus t. All I did,"},{"Start":"10:38.365 ","End":"10:43.990","Text":"is I switched between the 10 and the t but then I took away the minus sign right here."},{"Start":"10:43.990 ","End":"10:50.080","Text":"This would be the height of the yellow triangle."},{"Start":"10:52.670 ","End":"11:00.440","Text":"First of all, let\u0027s calculate the area of the yellow triangle."},{"Start":"11:00.450 ","End":"11:06.630","Text":"That\u0027s the base is 10 minus t times the height."},{"Start":"11:06.630 ","End":"11:16.950","Text":"That\u0027s 0.04 times 10 minus t. That\u0027s divided by 2, excuse me."},{"Start":"11:16.950 ","End":"11:26.160","Text":"That would equal to 10 minus t squared times 0.02."},{"Start":"11:26.390 ","End":"11:29.660","Text":"That was the area,"},{"Start":"11:29.660 ","End":"11:33.020","Text":"this here was the area of the yellow triangle,"},{"Start":"11:33.020 ","End":"11:34.100","Text":"and we don\u0027t want this,"},{"Start":"11:34.100 ","End":"11:37.865","Text":"we want 1 minus that in order to capture this area."},{"Start":"11:37.865 ","End":"11:48.995","Text":"The CDF, would be 1 minus 10 minus t squared times 0.02."},{"Start":"11:48.995 ","End":"11:52.880","Text":"This is what we want to input here."},{"Start":"11:52.880 ","End":"12:02.840","Text":"The CDF for this range would be 1 minus 10 minus t squared times 0.02."},{"Start":"12:02.840 ","End":"12:12.150","Text":"This then is the CDF for the function of y for all of its ranges."}],"ID":13089},{"Watched":false,"Name":"Exercise 3 - Parts c-d","Duration":"8m 53s","ChapterTopicVideoID":12611,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12611.jpeg","UploadDate":"2018-07-23T15:57:15.0370000","DurationForVideoObject":"PT8M53S","Description":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":"00:07.695","Text":"The probability of Y being greater than 4 and so forth."},{"Start":"00:07.695 ","End":"00:15.710","Text":"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 1,"},{"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":"00:29.600","Text":"Well, we don\u0027t have to do a lot of thinking here that equals 0. Why is that?"},{"Start":"00:29.600 ","End":"00:34.565","Text":"Because the probability of Y 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":"00:49.340","Text":"Well, less than or equal to 3,"},{"Start":"00:49.340 ","End":"00:52.265","Text":"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":"00:59.660","Text":"Again, we\u0027re dealing with this range right here."},{"Start":"00:59.660 ","End":"01:05.360","Text":"Let\u0027s just put 3 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":"01:42.270","Text":"that\u0027s this range right here."},{"Start":"01:42.270 ","End":"01:43.980","Text":"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":"01:51.915","Text":"now what\u0027s F at 7.5 that\u0027s right here."},{"Start":"01:51.915 ","End":"01:55.725","Text":"All we need to do is plug in 7.5 here."},{"Start":"01:55.725 ","End":"02:05.710","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":"02:23.300","Text":"The probability of Y being greater than 4."},{"Start":"02:23.300 ","End":"02:32.390","Text":"Well, that equals to 1 minus F at 4."},{"Start":"02:32.390 ","End":"02:35.480","Text":"Now, that equals to 1 minus,"},{"Start":"02:35.480 ","End":"02:37.010","Text":"what\u0027s F at 4 and again,"},{"Start":"02:37.010 ","End":"02:38.345","Text":"that\u0027s here in this range."},{"Start":"02:38.345 ","End":"02:44.775","Text":"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":"03:18.360","Text":"Now, we can calculate this much easier once we see the density function and the CDF."},{"Start":"03:18.530 ","End":"03:23.065","Text":"Here they are, that\u0027s a density function and that\u0027s 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":"03:37.795","Text":"The definition of the median is that we have a value of y,"},{"Start":"03:37.795 ","End":"03:41.465","Text":"where 50 percent of the probability falls"},{"Start":"03:41.465 ","End":"03:46.055","Text":"above 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":"03:56.285","Text":"We\u0027ll see that this function is symmetrical."},{"Start":"03:56.285 ","End":"03:59.630","Text":"We have the range from 0-10,"},{"Start":"03:59.630 ","End":"04:05.435","Text":"and it\u0027s symmetrical around the value 5. What does that mean?"},{"Start":"04:05.435 ","End":"04:10.915","Text":"That means that the area here in the range between 0 and 5,"},{"Start":"04:10.915 ","End":"04:16.105","Text":"equals to the area in the range between 5 and 10."},{"Start":"04:16.105 ","End":"04:20.880","Text":"Having said that, then this equals 0.5."},{"Start":"04:20.880 ","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":"04:31.565","Text":"That\u0027s exactly the definition of a median,"},{"Start":"04:31.565 ","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":"04:40.790","Text":"You can say that the median of y is 5."},{"Start":"04:40.790 ","End":"04:47.340","Text":"We 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":"05:01.490","Text":"That\u0027s the probability of Y being less than or equal to y sub 0.1."},{"Start":"05:01.490 ","End":"05:04.930","Text":"That has to equal to 0.1."},{"Start":"05:04.930 ","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":"05:18.265","Text":"and this area has to be equal to 0.5."},{"Start":"05:18.265 ","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":"05:38.795","Text":"then we have to use this expression of the CDF. Let\u0027s do that."},{"Start":"05:38.795 ","End":"05:43.385","Text":"We\u0027ll be using 0.02 times t squared."},{"Start":"05:43.385 ","End":"05:52.920","Text":"We want that to be equal to 0.1 in order to calculate the bottom 10th percentile."},{"Start":"05:52.920 ","End":"05:58.325","Text":"That equals to t squared being equal to 5,"},{"Start":"05:58.325 ","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":"06:21.335","Text":"That\u0027s the probability of Y being less than or equal to small y sub 0.25,"},{"Start":"06:21.335 ","End":"06:24.200","Text":"and that has to equal to 0.25."},{"Start":"06:24.200 ","End":"06:34.129","Text":"Now again, the t that we\u0027re looking for is the area equaling 25 percent."},{"Start":"06:34.129 ","End":"06:37.865","Text":"Again, that\u0027s in this range between 0 and 5."},{"Start":"06:37.865 ","End":"06:44.655","Text":"We\u0027re looking now for this t with this area here equals 25 percent."},{"Start":"06:44.655 ","End":"06:47.810","Text":"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":"07:20.015","Text":"Well, instead of doing a whole bunch of calculations."},{"Start":"07:20.015 ","End":"07:28.035","Text":"We can infer what that percentile is and we can infer it from the bottom 10th percentile."},{"Start":"07:28.035 ","End":"07:31.940","Text":"If you recall, y sub 0.1,"},{"Start":"07:31.940 ","End":"07:34.895","Text":"that equal to 2.24."},{"Start":"07:34.895 ","End":"07:38.855","Text":"What that meant was that we found a point here,"},{"Start":"07:38.855 ","End":"07:47.435","Text":"where the area under the density function from 0 to that point was equal to 0.1."},{"Start":"07:47.435 ","End":"07:51.680","Text":"That\u0027s the bottom 10th percentile."},{"Start":"07:51.680 ","End":"07:56.240","Text":"Now, because the density function is symmetric,"},{"Start":"07:56.240 ","End":"08:02.780","Text":"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":"08:22.700","Text":"Again, because I can\u0027t stress this enough 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":"08:35.750","Text":"the upper 10th percentile,"},{"Start":"08:35.750 ","End":"08:40.165","Text":"as being equal to 10 minus 2.24,"},{"Start":"08:40.165 ","End":"08:43.405","Text":"that equals to 7.76."},{"Start":"08:43.405 ","End":"08:54.040","Text":"This number here would be the upper 10th percentile or y sub 0.9."}],"ID":13090},{"Watched":false,"Name":"Exercise 4 - Parts a-b","Duration":"9m 13s","ChapterTopicVideoID":12613,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12613.jpeg","UploadDate":"2018-07-23T15:57:54.8800000","DurationForVideoObject":"PT9M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.905","Text":"The diagram below shows the density function of the random variable X."},{"Start":"00:04.905 ","End":"00:08.868","Text":"We\u0027re asked in section a to calculate the value C,"},{"Start":"00:08.868 ","End":"00:14.430","Text":"it\u0027s this guy right here that defines diagram as a density function."},{"Start":"00:14.430 ","End":"00:18.390","Text":"In order for this to be a density function,"},{"Start":"00:18.390 ","End":"00:23.845","Text":"then we need for the area underneath this function to equal to 1."},{"Start":"00:23.845 ","End":"00:29.690","Text":"We can see that this function right here is made up of a series of rectangles."},{"Start":"00:29.690 ","End":"00:34.175","Text":"All we have to do is to calculate the area of"},{"Start":"00:34.175 ","End":"00:42.615","Text":"each rectangle and then equate that to 1 and solve for C. Let\u0027s do that."},{"Start":"00:42.615 ","End":"00:44.490","Text":"The first rectangle here,"},{"Start":"00:44.490 ","End":"00:48.090","Text":"the base is 3 and the height is 0.2."},{"Start":"00:48.090 ","End":"00:53.225","Text":"That\u0027s 3 times 0.2 and that equals to 0.6."},{"Start":"00:53.225 ","End":"00:55.280","Text":"What about the next rectangle?"},{"Start":"00:55.280 ","End":"01:01.720","Text":"That\u0027s the base is 2 times what\u0027s the height? The height is 0.05."},{"Start":"01:02.780 ","End":"01:06.440","Text":"That equals to 0.1."},{"Start":"01:06.440 ","End":"01:14.500","Text":"What about this rectangle right here where the base is 1 and the height is 0.15?"},{"Start":"01:15.370 ","End":"01:19.210","Text":"The area here is 0.15."},{"Start":"01:19.210 ","End":"01:21.090","Text":"Between 6 and 7,"},{"Start":"01:21.090 ","End":"01:24.375","Text":"we don\u0027t have anything and between 7 and C,"},{"Start":"01:24.375 ","End":"01:29.615","Text":"where the base is C minus 7 times what\u0027s the height here?"},{"Start":"01:29.615 ","End":"01:33.180","Text":"The height here is 0.05."},{"Start":"01:33.400 ","End":"01:37.130","Text":"What we have to do now is to add up"},{"Start":"01:37.130 ","End":"01:43.175","Text":"these areas and equate that to 1 and solve for C. Let\u0027s do that."},{"Start":"01:43.175 ","End":"01:49.940","Text":"That\u0027s 0.6 plus 0.1 plus"},{"Start":"01:49.940 ","End":"01:59.790","Text":"0.15 plus C minus 7 times 0.05 and that has to equal to 1."},{"Start":"01:59.790 ","End":"02:02.460","Text":"These guys,"},{"Start":"02:02.460 ","End":"02:08.580","Text":"that\u0027s 0.85 plus C minus"},{"Start":"02:08.580 ","End":"02:13.240","Text":"7 times 0.05 and that equals to 1."},{"Start":"02:13.240 ","End":"02:20.350","Text":"That means that C minus 7 times 0.05,"},{"Start":"02:20.350 ","End":"02:22.810","Text":"that equals to 0.15."},{"Start":"02:22.810 ","End":"02:27.400","Text":"C minus 7 equals to 3."},{"Start":"02:27.400 ","End":"02:32.390","Text":"That means that C equals to 10."},{"Start":"02:33.720 ","End":"02:39.530","Text":"In section b, we\u0027re asked to write the cumulative distribution function of x."},{"Start":"02:39.980 ","End":"02:45.090","Text":"In order to do that, we\u0027re going to have to divide x into many ranges and"},{"Start":"02:45.090 ","End":"02:49.805","Text":"figure out the CDF for each one of the ranges. Let\u0027s get started."},{"Start":"02:49.805 ","End":"02:51.660","Text":"Let\u0027s take a point t,"},{"Start":"02:51.660 ","End":"02:53.640","Text":"which is the value of X."},{"Start":"02:53.640 ","End":"02:55.960","Text":"Let\u0027s say it\u0027s less than 0."},{"Start":"02:55.960 ","End":"03:00.745","Text":"That\u0027s basically the first range right here where x is negative or t is negative."},{"Start":"03:00.745 ","End":"03:04.990","Text":"Well, the CDF would be 0. Why is that?"},{"Start":"03:04.990 ","End":"03:12.115","Text":"Because the density function starts at 0 and goes forward in the positive ranges of x."},{"Start":"03:12.115 ","End":"03:16.290","Text":"That means that there\u0027s no density function."},{"Start":"03:16.290 ","End":"03:20.975","Text":"There\u0027s no area to calculate where x is negative."},{"Start":"03:20.975 ","End":"03:22.910","Text":"When t is negative,"},{"Start":"03:22.910 ","End":"03:25.265","Text":"then the CDF is 0."},{"Start":"03:25.265 ","End":"03:30.950","Text":"Let\u0027s take a look at the next range between 0 and 3,"},{"Start":"03:30.950 ","End":"03:34.400","Text":"t is between 3 and 0."},{"Start":"03:34.400 ","End":"03:37.225","Text":"Well, in order to do that,"},{"Start":"03:37.225 ","End":"03:42.530","Text":"in order to calculate the cumulative distribution function in this range,"},{"Start":"03:42.530 ","End":"03:44.665","Text":"let\u0027s just pick a point,"},{"Start":"03:44.665 ","End":"03:51.230","Text":"we\u0027ll call that t. What we want to do is we want to calculate the area between"},{"Start":"03:51.230 ","End":"03:58.860","Text":"0 and t. That\u0027ll be this area right here, in yellow."},{"Start":"03:59.420 ","End":"04:01.690","Text":"In order to do that,"},{"Start":"04:01.690 ","End":"04:03.880","Text":"well, what\u0027s the area of a rectangle?"},{"Start":"04:03.880 ","End":"04:06.325","Text":"That\u0027s the base times the height."},{"Start":"04:06.325 ","End":"04:12.975","Text":"The base would be t times the height. That\u0027s 0.2."},{"Start":"04:12.975 ","End":"04:20.640","Text":"That\u0027ll be the area or the area of this rectangle right here and also"},{"Start":"04:20.640 ","End":"04:29.410","Text":"the CDF for the function of x in this range right here between 0 and 3."},{"Start":"04:29.410 ","End":"04:32.090","Text":"Let\u0027s go on to the next range."},{"Start":"04:32.090 ","End":"04:37.060","Text":"The next range is between 3 and 5."},{"Start":"04:37.370 ","End":"04:42.175","Text":"T would be between 3 and 5."},{"Start":"04:42.175 ","End":"04:48.950","Text":"What\u0027s the area underneath this rectangle right here between 3 and 5?"},{"Start":"04:48.950 ","End":"04:51.020","Text":"Well again, let\u0027s use the same logic."},{"Start":"04:51.020 ","End":"04:55.700","Text":"We\u0027ll define a point t in this range."},{"Start":"04:55.700 ","End":"04:58.670","Text":"What we\u0027d like to do is we\u0027d like to"},{"Start":"04:58.670 ","End":"05:03.620","Text":"calculate the area of this small rectangle right here."},{"Start":"05:03.620 ","End":"05:06.035","Text":"In this rectangle,"},{"Start":"05:06.035 ","End":"05:10.250","Text":"that\u0027ll be t minus 3."},{"Start":"05:10.250 ","End":"05:12.245","Text":"That\u0027s the base, times the height,"},{"Start":"05:12.245 ","End":"05:16.850","Text":"times 0.05 but we"},{"Start":"05:16.850 ","End":"05:22.040","Text":"have to remember to take into account what we\u0027ve calculated before in the first range."},{"Start":"05:22.040 ","End":"05:30.195","Text":"That\u0027ll be 0.6. That\u0027s the area of this first rectangle right here. Why is that?"},{"Start":"05:30.195 ","End":"05:32.460","Text":"Because it\u0027s 3,"},{"Start":"05:32.460 ","End":"05:37.980","Text":"that\u0027s the base times 0.2. That equals 0.6."},{"Start":"05:37.980 ","End":"05:44.540","Text":"Let\u0027s not forget to add that 0.6 plus this expression right here."},{"Start":"05:44.540 ","End":"05:50.610","Text":"That will be the CDF where x or t is between 3 and 5."},{"Start":"05:50.610 ","End":"05:54.195","Text":"Let\u0027s go on to the next range."},{"Start":"05:54.195 ","End":"05:58.005","Text":"The next range is between 5 and 6,"},{"Start":"05:58.005 ","End":"06:02.565","Text":"t here would be between 5 and 6."},{"Start":"06:02.565 ","End":"06:06.270","Text":"Again, using the same logic we\u0027ll define"},{"Start":"06:06.270 ","End":"06:13.325","Text":"a point t. We want to calculate the area in this rectangle right here."},{"Start":"06:13.325 ","End":"06:22.605","Text":"That means that\u0027s t minus 5,"},{"Start":"06:22.605 ","End":"06:24.870","Text":"that\u0027s the base times the height."},{"Start":"06:24.870 ","End":"06:30.230","Text":"The height right here is 0.15 but we still"},{"Start":"06:30.230 ","End":"06:36.080","Text":"have to know what the area of what we already calculated in the previous ranges."},{"Start":"06:36.080 ","End":"06:38.116","Text":"The first range was 0.6."},{"Start":"06:38.116 ","End":"06:39.845","Text":"What about the second range?"},{"Start":"06:39.845 ","End":"06:44.325","Text":"That\u0027s 2 times 0.05 that\u0027s 0.1."},{"Start":"06:44.325 ","End":"06:49.380","Text":"We have here 0.7 plus"},{"Start":"06:49.380 ","End":"06:56.180","Text":"this expression and that would be the CDF in this range right here."},{"Start":"06:56.180 ","End":"07:04.210","Text":"Let\u0027s take a look at the next range between 6 and 7."},{"Start":"07:05.260 ","End":"07:07.430","Text":"In this range right here,"},{"Start":"07:07.430 ","End":"07:10.610","Text":"we see that the density function is 0."},{"Start":"07:10.610 ","End":"07:15.035","Text":"That means that the CDF is a constant."},{"Start":"07:15.035 ","End":"07:20.900","Text":"Its value being the areas that we\u0027ve already calculated in the previous ranges."},{"Start":"07:20.900 ","End":"07:24.820","Text":"We have 0.6 here and 0.1 here."},{"Start":"07:24.820 ","End":"07:29.300","Text":"What\u0027s the total area between 5 and 6?"},{"Start":"07:29.300 ","End":"07:34.025","Text":"That\u0027s 1, that\u0027s the base times 0.15,"},{"Start":"07:34.025 ","End":"07:37.680","Text":"that\u0027s the height. That\u0027ll be 0.15."},{"Start":"07:39.280 ","End":"07:46.615","Text":"0.15 plus 0.1 plus 0.6 that\u0027s 0.85."},{"Start":"07:46.615 ","End":"07:51.150","Text":"That\u0027s the value,"},{"Start":"07:51.150 ","End":"07:55.765","Text":"a constant, in the range between 6 and 7."},{"Start":"07:55.765 ","End":"07:59.900","Text":"Now what about the next range between 7 and 10?"},{"Start":"07:59.900 ","End":"08:01.340","Text":"C here equals 10."},{"Start":"08:01.340 ","End":"08:04.295","Text":"We\u0027ve calculated that in the previous section."},{"Start":"08:04.295 ","End":"08:08.025","Text":"Again, let\u0027s just define"},{"Start":"08:08.025 ","End":"08:16.170","Text":"a point t. What we want to do is we want to calculate the area right here."},{"Start":"08:17.000 ","End":"08:19.220","Text":"That\u0027ll come out 2."},{"Start":"08:19.220 ","End":"08:23.435","Text":"This is in the range between 7 and 10."},{"Start":"08:23.435 ","End":"08:31.670","Text":"First of all, we have the previous areas that we have to take into account,"},{"Start":"08:31.670 ","End":"08:39.152","Text":"that\u0027s 0.85 plus the area in this range right here,"},{"Start":"08:39.152 ","End":"08:41.614","Text":"that\u0027s t minus 7,"},{"Start":"08:41.614 ","End":"08:46.980","Text":"that\u0027s the base times the height, times 0.05."},{"Start":"08:47.800 ","End":"08:53.930","Text":"The last range is where t is greater than 10."},{"Start":"08:53.930 ","End":"08:55.520","Text":"When t is greater than 10,"},{"Start":"08:55.520 ","End":"09:00.565","Text":"we\u0027ve actually accumulated all the area underneath the density function."},{"Start":"09:00.565 ","End":"09:05.785","Text":"That means that the cumulative distribution function equals to 1."},{"Start":"09:05.785 ","End":"09:13.620","Text":"There we have it. Here is the CDF or the cumulative distribution function of x."}],"ID":13091},{"Watched":false,"Name":"Exercise 4 - Part c","Duration":"2m 43s","ChapterTopicVideoID":12612,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12612.jpeg","UploadDate":"2018-07-23T15:57:22.5730000","DurationForVideoObject":"PT2M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In section c, we\u0027re asked to calculate the following probabilities."},{"Start":"00:03.750 ","End":"00:05.250","Text":"In order to do that,"},{"Start":"00:05.250 ","End":"00:10.170","Text":"let\u0027s just take a look at the cumulative distribution function."},{"Start":"00:10.170 ","End":"00:14.940","Text":"Here it is. We can start calculating these probabilities."},{"Start":"00:14.940 ","End":"00:19.815","Text":"That\u0027s start with the probability of x being greater or equal to 4."},{"Start":"00:19.815 ","End":"00:23.130","Text":"That equals to 1 minus big F at 4,"},{"Start":"00:23.130 ","End":"00:28.200","Text":"where big F is the cumulative distribution function at point 4."},{"Start":"00:28.200 ","End":"00:32.790","Text":"At point 4 means that we have to look where 4 lies,"},{"Start":"00:32.790 ","End":"00:35.520","Text":"which range the number 4 lies,"},{"Start":"00:35.520 ","End":"00:38.055","Text":"it lies in the range between 3 and 5."},{"Start":"00:38.055 ","End":"00:42.845","Text":"We\u0027ll be using this expression right here to calculate big F at 4."},{"Start":"00:42.845 ","End":"00:51.317","Text":"That equals to 1 minus 0.6 plus 4 minus 3 times 0.05,"},{"Start":"00:51.317 ","End":"00:54.515","Text":"and that equals to 0.35."},{"Start":"00:54.515 ","End":"00:58.070","Text":"What about the probability of x being"},{"Start":"00:58.070 ","End":"01:02.135","Text":"greater or equal to minus 2? Just get it right here."},{"Start":"01:02.135 ","End":"01:07.220","Text":"That equals to 1 minus big F at minus 2."},{"Start":"01:07.220 ","End":"01:15.290","Text":"Big F at minus 2 or minus 2 falls in this range right here,"},{"Start":"01:15.290 ","End":"01:17.285","Text":"where t is less than 0."},{"Start":"01:17.285 ","End":"01:21.180","Text":"The value of the CDF is 0. That means that this is 1."},{"Start":"01:21.180 ","End":"01:25.070","Text":"All this means is that there\u0027s a certainty or"},{"Start":"01:25.070 ","End":"01:30.560","Text":"probability of 1 that x will be greater or equal to minus 2."},{"Start":"01:30.560 ","End":"01:37.450","Text":"What about the probability of x being between 1 and 5?"},{"Start":"01:37.450 ","End":"01:43.515","Text":"That equals to big F at 5 minus big F at 1."},{"Start":"01:43.515 ","End":"01:47.220","Text":"Big F at the point 5,"},{"Start":"01:47.220 ","End":"01:50.660","Text":"actually, we can see that right here."},{"Start":"01:50.660 ","End":"01:52.310","Text":"It can be either here,"},{"Start":"01:52.310 ","End":"01:53.855","Text":"in this range,"},{"Start":"01:53.855 ","End":"01:56.120","Text":"or it can be here in this range."},{"Start":"01:56.120 ","End":"02:00.350","Text":"It really doesn\u0027t matter which expression we\u0027ll be using,"},{"Start":"02:00.350 ","End":"02:05.420","Text":"because x is a continuous random variable."},{"Start":"02:05.420 ","End":"02:09.890","Text":"Let\u0027s just use this guy right here, the first 1."},{"Start":"02:09.890 ","End":"02:15.950","Text":"That means that we\u0027ll be using this 0.6 plus 5"},{"Start":"02:15.950 ","End":"02:23.445","Text":"minus 3 times 0.05 minus F at 1."},{"Start":"02:23.445 ","End":"02:25.910","Text":"F at 1 is right here,"},{"Start":"02:25.910 ","End":"02:28.520","Text":"in this range right here, so we\u0027ll be using this."},{"Start":"02:28.520 ","End":"02:32.615","Text":"That\u0027s 0.2 times 1, that\u0027s 0.2."},{"Start":"02:32.615 ","End":"02:36.620","Text":"That comes out to 0.5."},{"Start":"02:36.620 ","End":"02:42.960","Text":"These are the probabilities that we were asked to calculate."}],"ID":13092},{"Watched":false,"Name":"Exercise 5","Duration":"5m 17s","ChapterTopicVideoID":12614,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12614.jpeg","UploadDate":"2018-07-23T15:58:14.6800000","DurationForVideoObject":"PT5M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.530","Text":"In this question, we\u0027re given"},{"Start":"00:01.530 ","End":"00:06.315","Text":"the following density function and we\u0027re asked what\u0027s the value of c?"},{"Start":"00:06.315 ","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 1/2 times the base times the height."},{"Start":"00:22.350 ","End":"00:24.945","Text":"Now that equals to 1/2,"},{"Start":"00:24.945 ","End":"00:27.450","Text":"the base is 10 times 10,"},{"Start":"00:27.450 ","End":"00:31.200","Text":"this is the base and the height is c"},{"Start":"00:31.200 ","End":"00:36.285","Text":"times c and 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":"00:48.430","Text":"That means that C equals to 0.2 so this here equals to 0.2."},{"Start":"00:48.740 ","End":"00:54.100","Text":"In section b, we\u0027re asked to find asymmetric interval random value 5,"},{"Start":"00:54.100 ","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":"01:09.010","Text":"That means that this line right here equals this line right here,"},{"Start":"01:09.010 ","End":"01:12.770","Text":"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 a half, 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":"02:10.130","Text":"that means that these triangles are identical and if they\u0027re identical,"},{"Start":"02:10.130 ","End":"02:17.040","Text":"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":"02:42.605","Text":"But if we plug in the value of the base and the height,"},{"Start":"02:42.605 ","End":"02:46.400","Text":"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":"03:07.675","Text":"Now, the equation of a line is y equals mx plus n."},{"Start":"03:07.675 ","End":"03:10.370","Text":"Now n equals 0 because that\u0027s the"},{"Start":"03:10.370 ","End":"03:14.990","Text":"y-intercept and this line intercepts the y-axis at points 0."},{"Start":"03:14.990 ","End":"03:17.015","Text":"Now, what about the slope?"},{"Start":"03:17.015 ","End":"03:23.270","Text":"Well the slope is equal to 0.2 minus 0,"},{"Start":"03:23.270 ","End":"03:29.270","Text":"that\u0027s y2 minus y1 divided by x2 minus x1,"},{"Start":"03:29.270 ","End":"03:31.705","Text":"that\u0027s 5 minus 0,"},{"Start":"03:31.705 ","End":"03:35.085","Text":"and 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":"03:50.990","Text":"Let\u0027s now calculate the area of this guy right here."},{"Start":"03:50.990 ","End":"03:54.470","Text":"Well, again, that\u0027ll be 1/2,"},{"Start":"03:54.470 ","End":"04:00.950","Text":"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":"04:17.355","Text":"That means that we\u0027re looking at,"},{"Start":"04:17.355 ","End":"04:23.040","Text":"5 minus a squared times 0.04,"},{"Start":"04:23.040 ","End":"04:30.360","Text":"that equals to 0.5 and that means that 5 minus a squared,"},{"Start":"04:30.360 ","End":"04:34.920","Text":"that equals to 12.5."},{"Start":"04:34.920 ","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":"05:16.920","Text":"That would be the interval under which the area of the density function equals 0.5."}],"ID":13093},{"Watched":false,"Name":"Exercise 6","Duration":"6m 31s","ChapterTopicVideoID":12615,"CourseChapterTopicPlaylistID":245048,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12615.jpeg","UploadDate":"2018-07-23T15:58:35.3670000","DurationForVideoObject":"PT6M31S","Description":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":"00:05.940","Text":"The waiting time in minutes of a customer in line at"},{"Start":"00:05.940 ","End":"00:10.080","Text":"the neighborhood supermarket has the following cumulative distribution function,"},{"Start":"00:10.080 ","End":"00:16.275","Text":"F of t equals 1 minus e^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":"00:23.115","Text":"that approximately equals to 2.718."},{"Start":"00:23.115 ","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":"00:40.974","Text":"We already have the cumulative distribution function,"},{"Start":"00:40.974 ","End":"00:42.920","Text":"that\u0027s this guy right here."},{"Start":"00:42.920 ","End":"00:51.750","Text":"That equals to 1 minus F at 15. Why 15?"},{"Start":"00:51.750 ","End":"00:57.389","Text":"Because we\u0027re looking at the units of time in minutes."},{"Start":"00:57.389 ","End":"01:03.935","Text":"All we have to do now is plug in 15 into this equation right here because that\u0027s the CDF."},{"Start":"01:03.935 ","End":"01:14.010","Text":"That\u0027s 1 minus 1 minus e^minus 0.2 times 15."},{"Start":"01:14.240 ","End":"01:24.750","Text":"That equals to e^minus 3 and that equals to 0.0498."},{"Start":"01:28.340 ","End":"01:30.860","Text":"In section B, we\u0027re asked what\u0027s"},{"Start":"01:30.860 ","End":"01:33.680","Text":"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":"01:48.080","Text":"What\u0027s given to us 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":"01:57.505","Text":"the probability that he\u0027ll wait for less than 15."},{"Start":"01:57.505 ","End":"01:59.675","Text":"We know how to deal with this."},{"Start":"01:59.675 ","End":"02:01.520","Text":"What\u0027s in the denominator?"},{"Start":"02:01.520 ","End":"02:04.070","Text":"That\u0027s the probability of what\u0027s given."},{"Start":"02:04.070 ","End":"02:08.330","Text":"X is greater than 10 and in the numerator 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":"02:22.490","Text":"We were given that the cumulative distribution function equals 1 minus e^minus"},{"Start":"02:22.490 ","End":"02:31.410","Text":"0.2 times t. Let\u0027s just plug in these numbers into this function."},{"Start":"02:31.410 ","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":"02:45.200 ","End":"02:47.250","Text":"This is, as I said,"},{"Start":"02:47.250 ","End":"02:50.610","Text":"let\u0027s just plug in these numbers into this function."},{"Start":"02:50.610 ","End":"02:56.045","Text":"F at 15, that\u0027s 1 minus e^minus"},{"Start":"02:56.045 ","End":"03:02.370","Text":"0.2 times 15 minus F at 10."},{"Start":"03:02.370 ","End":"03:07.140","Text":"F at 10 is 1 minus e^minus 0.2 times"},{"Start":"03:07.140 ","End":"03:12.765","Text":"10 divided by 1 minus f at 10."},{"Start":"03:12.765 ","End":"03:22.335","Text":"That\u0027s 1 minus. F at 10 is 1 minus e^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":"03:28.305","Text":"That 1 and 1 cancels each other out."},{"Start":"03:28.305 ","End":"03:38.450","Text":"1 minus 1 and minus here turns this guy to a plus so that\u0027s e^minus 2 minus"},{"Start":"03:38.450 ","End":"03:43.610","Text":"e^minus 3 divided by"},{"Start":"03:43.610 ","End":"03:49.910","Text":"e^minus 2 because 1"},{"Start":"03:49.910 ","End":"03:55.655","Text":"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":"04:07.280 ","End":"04:09.680","Text":"In section c we\u0027re asked,"},{"Start":"04:09.680 ","End":"04:13.640","Text":"what\u0027s the time under which 90 percent of the customers have to wait?"},{"Start":"04:13.640 ","End":"04:17.720","Text":"Here we\u0027re looking for percentile, the 90th percentile."},{"Start":"04:17.720 ","End":"04:19.805","Text":"If we recall,"},{"Start":"04:19.805 ","End":"04:24.740","Text":"the definition of the percentile is the probability of x"},{"Start":"04:24.740 ","End":"04:30.320","Text":"being less than or equal to small x_0.9,"},{"Start":"04:30.320 ","End":"04:31.850","Text":"that\u0027s the 90th percentile."},{"Start":"04:31.850 ","End":"04:35.615","Text":"The probability of X being less than or equal to"},{"Start":"04:35.615 ","End":"04:41.410","Text":"some value that would equal 0.9 when we\u0027re talking about the 90th percentile."},{"Start":"04:41.410 ","End":"04:46.720","Text":"Let\u0027s recall, the cumulative distribution function."},{"Start":"04:46.720 ","End":"04:52.860","Text":"That equal 1 minus e^minus 0.2t."},{"Start":"04:53.230 ","End":"04:58.670","Text":"What we want is for this function to equal"},{"Start":"04:58.670 ","End":"05:06.465","Text":"this probability and then we\u0027ll be able to extract t. Let\u0027s do that."},{"Start":"05:06.465 ","End":"05:18.305","Text":"That would equal to 1 minus e^minus 0.2t and that would equal to 0.9."},{"Start":"05:18.305 ","End":"05:23.650","Text":"Let\u0027s just switch things around and that equals to"},{"Start":"05:23.650 ","End":"05:32.040","Text":"e^minus 0.2t 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":"05:48.665","Text":"What we\u0027re doing is we\u0027re taking the natural logarithm of e^minus"},{"Start":"05:48.665 ","End":"05:57.390","Text":"0.2t and that will equal to the natural logarithm of 0.1."},{"Start":"05:58.130 ","End":"06:05.225","Text":"The natural logarithm of e to the power of something is that something itself;"},{"Start":"06:05.225 ","End":"06:08.953","Text":"that would be minus 0.2t,"},{"Start":"06:08.953 ","End":"06:14.070","Text":"that equals to the natural logarithm of 0.1."},{"Start":"06:14.090 ","End":"06:19.605","Text":"That means that this t right here,"},{"Start":"06:19.605 ","End":"06:23.640","Text":"which equals to the 90th percentile,"},{"Start":"06:23.640 ","End":"06:28.000","Text":"that equals to 115.13."}],"ID":13094}],"Thumbnail":null,"ID":245048},{"Name":"General Probabilities with Integrals","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"9m 11s","ChapterTopicVideoID":12616,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12616.jpeg","UploadDate":"2018-07-23T15:59:53.6370000","DurationForVideoObject":"PT9M11S","Description":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":13095},{"Watched":false,"Name":"Tutorial 2","Duration":"8m 44s","ChapterTopicVideoID":12617,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12617.jpeg","UploadDate":"2018-07-23T16:00:21.1830000","DurationForVideoObject":"PT8M44S","Description":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":13096},{"Watched":false,"Name":"Exercise 1 Parts a-b","Duration":"5m 42s","ChapterTopicVideoID":12618,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12618.jpeg","UploadDate":"2018-07-23T16:00:35.6200000","DurationForVideoObject":"PT5M42S","Description":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":13097},{"Watched":false,"Name":"Exercise 1 Parts c-d","Duration":"5m 58s","ChapterTopicVideoID":12619,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12619.jpeg","UploadDate":"2018-07-23T16:00:55.1300000","DurationForVideoObject":"PT5M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.945","Text":"In this section we\u0027re asked to calculate the following probabilities."},{"Start":"00:03.945 ","End":"00:06.840","Text":"Now, looking at these probabilities,"},{"Start":"00:06.840 ","End":"00:09.090","Text":"the best way to calculate them is to use"},{"Start":"00:09.090 ","End":"00:13.305","Text":"the cumulative distribution function and this is it."},{"Start":"00:13.305 ","End":"00:17.820","Text":"This is the CDF that we\u0027ve calculated in the previous section."},{"Start":"00:17.820 ","End":"00:19.935","Text":"Let\u0027s get to work."},{"Start":"00:19.935 ","End":"00:23.270","Text":"Now, the probability, we\u0027re doing this 1,"},{"Start":"00:23.270 ","End":"00:26.570","Text":"the probability of x being less than 4,"},{"Start":"00:26.570 ","End":"00:30.505","Text":"well, that\u0027s f at 4."},{"Start":"00:30.505 ","End":"00:34.319","Text":"Now that equals to 4."},{"Start":"00:34.319 ","End":"00:37.830","Text":"First in this race we\u0027ll be using this expression."},{"Start":"00:37.830 ","End":"00:45.605","Text":"That\u0027ll be 0.25 plus 4 minus 2 times 3/16,"},{"Start":"00:45.605 ","End":"00:50.185","Text":"and that equals to 5 divided by 8."},{"Start":"00:50.185 ","End":"00:53.000","Text":"Now, let\u0027s take a look at the next one."},{"Start":"00:53.000 ","End":"00:58.610","Text":"That\u0027s the probability of x being greater than 1.5."},{"Start":"00:58.610 ","End":"01:04.310","Text":"Well, that equals to 1 minus f at 1.5."},{"Start":"01:04.310 ","End":"01:09.965","Text":"That equals to 1 minus the 1.5 falls in this range so we\u0027ll be using this expression."},{"Start":"01:09.965 ","End":"01:16.685","Text":"That\u0027ll be 0.25 times 1.5 minus 1,"},{"Start":"01:16.685 ","End":"01:21.695","Text":"and that equals to 7/8."},{"Start":"01:21.695 ","End":"01:24.260","Text":"What about this one?"},{"Start":"01:24.260 ","End":"01:31.840","Text":"Well, that\u0027s the probability of x between 1.5 and 5."},{"Start":"01:31.840 ","End":"01:39.215","Text":"Well, that equals to f at 5 minus f at 1.5."},{"Start":"01:39.215 ","End":"01:42.425","Text":"Now that equals 2."},{"Start":"01:42.425 ","End":"01:46.610","Text":"Now f at 5, that\u0027s this expression right here."},{"Start":"01:46.610 ","End":"01:53.420","Text":"That\u0027ll be 0.25 plus 5 minus 2 times"},{"Start":"01:53.420 ","End":"02:02.360","Text":"3/16 minus f at 1.5 or f at 1.5 is this expression right here."},{"Start":"02:02.360 ","End":"02:09.330","Text":"That\u0027s 0.25 times 1.5 minus 1."},{"Start":"02:09.370 ","End":"02:16.320","Text":"Now that equals to 11/16,"},{"Start":"02:16.320 ","End":"02:18.140","Text":"and the last probability,"},{"Start":"02:18.140 ","End":"02:26.155","Text":"that\u0027s the probability of X between 5 and 10."},{"Start":"02:26.155 ","End":"02:30.814","Text":"Now, that equals to the following."},{"Start":"02:30.814 ","End":"02:37.110","Text":"That\u0027s basically F at 10 minus f at 5."},{"Start":"02:37.110 ","End":"02:41.780","Text":"Now, F at 10 that\u0027s 1."},{"Start":"02:41.780 ","End":"02:46.430","Text":"The probability of X being less than 10,"},{"Start":"02:46.430 ","End":"02:48.380","Text":"well, that\u0027s a certainty."},{"Start":"02:48.380 ","End":"02:52.250","Text":"That\u0027s 1 minus now f at 5,"},{"Start":"02:52.250 ","End":"02:55.565","Text":"f at 5, that\u0027s this expression right here."},{"Start":"02:55.565 ","End":"03:04.640","Text":"That\u0027ll be 0.25 plus 5 minus 2 times 3/16,"},{"Start":"03:04.640 ","End":"03:09.100","Text":"and that equals to 3/16."},{"Start":"03:09.100 ","End":"03:17.340","Text":"These are the probabilities right here that we were asked to calculate."},{"Start":"03:17.950 ","End":"03:21.815","Text":"In this section, we\u0027re asked to calculate the median of x."},{"Start":"03:21.815 ","End":"03:27.490","Text":"Now, the median is basically the 50th percentile."},{"Start":"03:27.490 ","End":"03:29.645","Text":"If we\u0027re dealing with percentiles,"},{"Start":"03:29.645 ","End":"03:33.995","Text":"and the definition of a percentile is a probability of"},{"Start":"03:33.995 ","End":"03:39.710","Text":"x that\u0027s less than small x sub p. Well,"},{"Start":"03:39.710 ","End":"03:43.145","Text":"that equals to p. In our case,"},{"Start":"03:43.145 ","End":"03:44.480","Text":"we want the median,"},{"Start":"03:44.480 ","End":"03:48.275","Text":"so p equals a half or 0.5."},{"Start":"03:48.275 ","End":"03:54.440","Text":"Now, that\u0027s also called the median and it\u0027s denoted as md."},{"Start":"03:54.440 ","End":"04:02.390","Text":"Now, let\u0027s see where this value falls in the range underneath the density function."},{"Start":"04:02.390 ","End":"04:04.220","Text":"Well, first of all,"},{"Start":"04:04.220 ","End":"04:09.965","Text":"we know that the area under the density function in the range between 1 and 2,"},{"Start":"04:09.965 ","End":"04:15.045","Text":"that equals to 0.25."},{"Start":"04:15.045 ","End":"04:21.500","Text":"That means that the median falls within this range between 2 and 6."},{"Start":"04:21.500 ","End":"04:25.940","Text":"When we\u0027re looking for the probability of x being less than"},{"Start":"04:25.940 ","End":"04:30.410","Text":"some value and that value falls in"},{"Start":"04:30.410 ","End":"04:35.495","Text":"this range between 2 and 6 then we\u0027ll be using this expression for the CDF."},{"Start":"04:35.495 ","End":"04:37.025","Text":"Let\u0027s do that."},{"Start":"04:37.025 ","End":"04:43.055","Text":"That\u0027ll be 0.25 plus t minus 2"},{"Start":"04:43.055 ","End":"04:50.600","Text":"times 3/16 and we want that expression to equal a half or 0.5."},{"Start":"04:50.600 ","End":"04:52.940","Text":"When we solve for t,"},{"Start":"04:52.940 ","End":"04:55.655","Text":"the value of t will be the median."},{"Start":"04:55.655 ","End":"04:58.235","Text":"Now, let\u0027s just move things around."},{"Start":"04:58.235 ","End":"05:03.470","Text":"That\u0027ll be t minus 2 times 3/16."},{"Start":"05:03.470 ","End":"05:06.745","Text":"Well, that equals to a quarter."},{"Start":"05:06.745 ","End":"05:14.510","Text":"That means that t minus 2 equals 16 over 12 and that equals to 4 over 3."},{"Start":"05:14.510 ","End":"05:21.335","Text":"That means that t equals to 2 plus 4 over 3,"},{"Start":"05:21.335 ","End":"05:26.675","Text":"and that means that t equals to 3 and 1/3."},{"Start":"05:26.675 ","End":"05:29.520","Text":"Now, that\u0027s the median."},{"Start":"05:29.880 ","End":"05:33.220","Text":"That\u0027s the value right here."},{"Start":"05:33.220 ","End":"05:41.235","Text":"X sub 0.5 and it\u0027s 3 and 1/3."},{"Start":"05:41.235 ","End":"05:44.160","Text":"That equals to 3 and 1/3,"},{"Start":"05:44.160 ","End":"05:51.905","Text":"where the area underneath the density function from 1 to 3 and 1/3,"},{"Start":"05:51.905 ","End":"05:58.050","Text":"well, that whole area here equals to 0.5 or a half."}],"ID":13098},{"Watched":false,"Name":"Exercise 2","Duration":"7m 36s","ChapterTopicVideoID":12620,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12620.jpeg","UploadDate":"2018-07-23T16:01:15.6500000","DurationForVideoObject":"PT7M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.730","Text":"In this question, we have a continuous random variable"},{"Start":"00:02.730 ","End":"00:04.920","Text":"that has the following density function."},{"Start":"00:04.920 ","End":"00:07.050","Text":"F of x equals c times x,"},{"Start":"00:07.050 ","End":"00:10.620","Text":"where x is between 0 and b and 0 otherwise."},{"Start":"00:10.620 ","End":"00:15.450","Text":"Now it\u0027s known that the probability that x is between 0 and 1,"},{"Start":"00:15.450 ","End":"00:16.995","Text":"well that equals a 1/4."},{"Start":"00:16.995 ","End":"00:20.715","Text":"We\u0027re asked to calculate the density function of X."},{"Start":"00:20.715 ","End":"00:23.310","Text":"Let\u0027s just write this down. That\u0027s done."},{"Start":"00:23.310 ","End":"00:28.350","Text":"Density function f of x that equals to c times x,"},{"Start":"00:28.350 ","End":"00:35.610","Text":"where x is between 0 and b, and 0 otherwise."},{"Start":"00:35.610 ","End":"00:41.426","Text":"Now, here we have 2 numbers that we have to figure out what the values are."},{"Start":"00:41.426 ","End":"00:46.045","Text":"Let\u0027s first see how this looks on a graph."},{"Start":"00:46.045 ","End":"00:53.450","Text":"Here it is. This is our density function where this is the x-axis,"},{"Start":"00:53.450 ","End":"00:58.610","Text":"this is the y-axis or the values of the density function."},{"Start":"00:58.610 ","End":"01:01.400","Text":"The line here, that\u0027s the density function,"},{"Start":"01:01.400 ","End":"01:03.530","Text":"that\u0027s c times X."},{"Start":"01:03.530 ","End":"01:06.305","Text":"This is a line that goes through the origin."},{"Start":"01:06.305 ","End":"01:09.305","Text":"Now, the range of X, right,"},{"Start":"01:09.305 ","End":"01:13.030","Text":"for this density function is between 0 and b."},{"Start":"01:13.030 ","End":"01:16.635","Text":"This point right here, that\u0027s b."},{"Start":"01:16.635 ","End":"01:19.970","Text":"Now, what\u0027s also known or given to us?"},{"Start":"01:19.970 ","End":"01:25.730","Text":"We\u0027re given that probability that X will be between 0 and 1,"},{"Start":"01:25.730 ","End":"01:27.545","Text":"but they\u0027ll be a 1/4."},{"Start":"01:27.545 ","End":"01:30.820","Text":"We have here the point 1,"},{"Start":"01:30.820 ","End":"01:35.150","Text":"and we also know that the area right"},{"Start":"01:35.150 ","End":"01:42.260","Text":"here is equal to a 1/4, excuse me."},{"Start":"01:42.260 ","End":"01:47.855","Text":"Now. In order to calculate what the values of c and b are,"},{"Start":"01:47.855 ","End":"01:50.165","Text":"we basically need 2 equations."},{"Start":"01:50.165 ","End":"01:52.040","Text":"Let\u0027s see what we have here."},{"Start":"01:52.040 ","End":"01:57.800","Text":"Well, first of all, we have here a big triangle right here."},{"Start":"01:57.800 ","End":"02:04.820","Text":"Now, when we figure out the area of the big triangle,"},{"Start":"02:04.820 ","End":"02:07.250","Text":"we know that the area equals to 1."},{"Start":"02:07.250 ","End":"02:15.805","Text":"The area of a triangle equals a 1/2 times b times h times the base times the height."},{"Start":"02:15.805 ","End":"02:19.080","Text":"For our first equation,"},{"Start":"02:19.080 ","End":"02:22.250","Text":"we\u0027ll take the equation of the area of the big triangle,"},{"Start":"02:22.250 ","End":"02:26.300","Text":"that\u0027s the base, times the height."},{"Start":"02:26.300 ","End":"02:28.850","Text":"Well, the height of"},{"Start":"02:28.850 ","End":"02:34.850","Text":"this triangle is basically the value of the density function at point b."},{"Start":"02:34.850 ","End":"02:38.840","Text":"We\u0027re just plug in b instead of x right here."},{"Start":"02:38.840 ","End":"02:43.845","Text":"That\u0027ll be c times b divided by 2,"},{"Start":"02:43.845 ","End":"02:45.835","Text":"and that equals to 1."},{"Start":"02:45.835 ","End":"02:51.110","Text":"The total area, or the area of the big triangle,"},{"Start":"02:51.110 ","End":"02:56.255","Text":"which is the area under the density function that has to equal to 1."},{"Start":"02:56.255 ","End":"03:02.555","Text":"That means that we have here b squared times c divided by 2,"},{"Start":"03:02.555 ","End":"03:07.880","Text":"that equals to 1, or b squared times c,"},{"Start":"03:07.880 ","End":"03:10.360","Text":"well, that equals to 2."},{"Start":"03:10.360 ","End":"03:14.120","Text":"Now this is our first equation."},{"Start":"03:14.120 ","End":"03:15.935","Text":"What\u0027s the second equation?"},{"Start":"03:15.935 ","End":"03:21.455","Text":"Well, the second equation talks about the area of the smaller triangle right here,"},{"Start":"03:21.455 ","End":"03:23.710","Text":"1 colored in yellow."},{"Start":"03:23.710 ","End":"03:26.585","Text":"Again, we\u0027ll use the,"},{"Start":"03:26.585 ","End":"03:28.415","Text":"the equation for the area."},{"Start":"03:28.415 ","End":"03:35.930","Text":"That\u0027s the base times the height divided by 2."},{"Start":"03:35.930 ","End":"03:41.570","Text":"Now, the base here is 1 times the height."},{"Start":"03:41.570 ","End":"03:46.385","Text":"Well, the height is basically the value of the density function at point 1."},{"Start":"03:46.385 ","End":"03:48.350","Text":"We\u0027ll plug in 1 instead of x."},{"Start":"03:48.350 ","End":"03:53.220","Text":"That\u0027ll be c times 1 divide by 2."},{"Start":"03:53.220 ","End":"03:57.680","Text":"That has to equal to a 1/4 that\u0027s given to us right here."},{"Start":"03:57.680 ","End":"04:05.930","Text":"That means that c equals 1/2."},{"Start":"04:05.930 ","End":"04:08.740","Text":"Now, if that\u0027s the case,"},{"Start":"04:08.740 ","End":"04:15.225","Text":"let\u0027s just plug in 1/2 back into this equation right here."},{"Start":"04:15.225 ","End":"04:23.910","Text":"When we do that, we get b squared times a 1/2 equals 2,"},{"Start":"04:23.910 ","End":"04:28.140","Text":"or b squared equals 4,"},{"Start":"04:28.140 ","End":"04:31.585","Text":"or b equals 2."},{"Start":"04:31.585 ","End":"04:33.345","Text":"If that\u0027s the case,"},{"Start":"04:33.345 ","End":"04:35.655","Text":"we have the 2 values here."},{"Start":"04:35.655 ","End":"04:39.400","Text":"Let\u0027s write down the density function as it should be."},{"Start":"04:39.400 ","End":"04:47.140","Text":"There\u0027ll be f of x and there\u0027ll be equal to c. There\u0027ll be a half times x,"},{"Start":"04:47.140 ","End":"04:55.790","Text":"where x is between 0 and 2 b, and 0 otherwise."},{"Start":"04:55.960 ","End":"04:59.975","Text":"In section B, we\u0027re asked to find the median of X."},{"Start":"04:59.975 ","End":"05:01.460","Text":"Well, if we recall,"},{"Start":"05:01.460 ","End":"05:08.555","Text":"the median of x can be denoted as md for medium or X sub 0.5."},{"Start":"05:08.555 ","End":"05:10.880","Text":"Now, what is the median?"},{"Start":"05:10.880 ","End":"05:15.890","Text":"The median is the point and let\u0027s call it t right here."},{"Start":"05:15.890 ","End":"05:22.455","Text":"Where the area under the density function from 0 until t,"},{"Start":"05:22.455 ","End":"05:26.180","Text":"well, that has to equal to 0.5 for a half."},{"Start":"05:26.180 ","End":"05:30.815","Text":"If we calculate the area of this triangle right here and solve for t,"},{"Start":"05:30.815 ","End":"05:33.860","Text":"we\u0027ll find the median. Let\u0027s do that."},{"Start":"05:33.860 ","End":"05:36.680","Text":"Well, let\u0027s recall that the area of"},{"Start":"05:36.680 ","End":"05:41.270","Text":"a triangle that equals to 1/2 times the base times the height."},{"Start":"05:41.270 ","End":"05:45.035","Text":"In our case, the base would be t."},{"Start":"05:45.035 ","End":"05:50.300","Text":"The height would be the value of the density function at point t. Well,"},{"Start":"05:50.300 ","End":"05:53.090","Text":"let\u0027s plug in t instead of x."},{"Start":"05:53.090 ","End":"05:59.695","Text":"That\u0027ll be 1/2 times t divided by a 1/2."},{"Start":"05:59.695 ","End":"06:03.465","Text":"That has to equal to a 1/2."},{"Start":"06:03.465 ","End":"06:06.030","Text":"Let\u0027s just simplify that."},{"Start":"06:06.030 ","End":"06:09.500","Text":"There\u0027ll be 1/2 t squared."},{"Start":"06:09.500 ","End":"06:14.450","Text":"That\u0027ll equal to 1, or t squared has to equal to 2."},{"Start":"06:14.450 ","End":"06:20.075","Text":"That means that t has to equal to the square root of 2,"},{"Start":"06:20.075 ","End":"06:23.465","Text":"that equals to 1.41,"},{"Start":"06:23.465 ","End":"06:26.850","Text":"and that is the median."},{"Start":"06:27.040 ","End":"06:33.780","Text":"In section C, we\u0027re asked what is the chances of x being less than 0.5."},{"Start":"06:34.070 ","End":"06:38.730","Text":"Here is x equaling 0.5,"},{"Start":"06:38.730 ","End":"06:43.595","Text":"and the chances of x being less than 0.5 is the area"},{"Start":"06:43.595 ","End":"06:48.590","Text":"under the density function between 0 and 0.5."},{"Start":"06:48.590 ","End":"06:51.415","Text":"That means this area right here."},{"Start":"06:51.415 ","End":"06:54.830","Text":"What we want to do is we want to calculate this area."},{"Start":"06:54.830 ","End":"07:00.740","Text":"Now, we know that the area of a triangle is 1/2 times the base times the height."},{"Start":"07:00.740 ","End":"07:03.950","Text":"Well, that will equal, in our case,"},{"Start":"07:03.950 ","End":"07:08.520","Text":"that\u0027ll equal to 1/2 times the base,"},{"Start":"07:08.520 ","End":"07:12.290","Text":"now the base is a 1/2 times the height."},{"Start":"07:12.290 ","End":"07:15.800","Text":"The site right here. Now the height is the value of"},{"Start":"07:15.800 ","End":"07:19.400","Text":"the density function at x equaling a 1/2."},{"Start":"07:19.400 ","End":"07:21.200","Text":"Let\u0027s just plug that in right here,"},{"Start":"07:21.200 ","End":"07:23.360","Text":"so that\u0027s a 1/2times a 1/2,"},{"Start":"07:23.360 ","End":"07:25.435","Text":"there\u0027ll be a 1/4."},{"Start":"07:25.435 ","End":"07:31.485","Text":"That equals to 1 divided by 16 or 1/16."},{"Start":"07:31.485 ","End":"07:36.810","Text":"This is the probability of x being less than a 1/2."}],"ID":13099},{"Watched":false,"Name":"Exercise 3 - Parts a-b","Duration":"9m 13s","ChapterTopicVideoID":12621,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12621.jpeg","UploadDate":"2018-07-23T16:01:45.5170000","DurationForVideoObject":"PT9M13S","Description":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":13100},{"Watched":false,"Name":"Exercise 3 - Parts c-d","Duration":"8m 55s","ChapterTopicVideoID":12622,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12622.jpeg","UploadDate":"2018-07-23T16:02:10.9030000","DurationForVideoObject":"PT8M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.340","Text":"In Section C, we\u0027re asked to calculate"},{"Start":"00:02.340 ","End":"00:06.420","Text":"the following probabilities, so let\u0027s get started."},{"Start":"00:06.420 ","End":"00:11.745","Text":"What\u0027s the probability of Y equaling 7?"},{"Start":"00:11.745 ","End":"00:13.635","Text":"That\u0027s this guy right here."},{"Start":"00:13.635 ","End":"00:16.320","Text":"Well, that equals to 0."},{"Start":"00:16.320 ","End":"00:20.610","Text":"Why is that? Well, whenever we have a continuous random"},{"Start":"00:20.610 ","End":"00:26.502","Text":"variable and we\u0027re asked the probability of this random variable equaling something,"},{"Start":"00:26.502 ","End":"00:29.050","Text":"well, it really doesn\u0027t matter what that something is."},{"Start":"00:29.050 ","End":"00:30.420","Text":"It always equals to 0,"},{"Start":"00:30.420 ","End":"00:33.010","Text":"there\u0027s nothing to calculate,"},{"Start":"00:33.320 ","End":"00:36.930","Text":"so let\u0027s get on with the next probability."},{"Start":"00:36.930 ","End":"00:42.680","Text":"The probability of Y being less than or equal to 3,"},{"Start":"00:42.680 ","End":"00:44.435","Text":"that\u0027s this guy right here."},{"Start":"00:44.435 ","End":"00:48.010","Text":"Well, that\u0027s F of 3."},{"Start":"00:48.010 ","End":"00:53.795","Text":"Now, 3 falls into this range right here between 0 and 5."},{"Start":"00:53.795 ","End":"01:00.185","Text":"Remember that we calculated the cumulative density function in the previous section."},{"Start":"01:00.185 ","End":"01:02.730","Text":"Let\u0027s just plug in 3 here."},{"Start":"01:02.730 ","End":"01:07.415","Text":"Well, that\u0027ll be 0.02 times 3 squared,"},{"Start":"01:07.415 ","End":"01:10.595","Text":"and that equals to 0.18."},{"Start":"01:10.595 ","End":"01:13.835","Text":"What about the next probability right here?"},{"Start":"01:13.835 ","End":"01:16.460","Text":"What\u0027s the probability of"},{"Start":"01:16.460 ","End":"01:24.675","Text":"Y being between 7.5 and 15.5?"},{"Start":"01:24.675 ","End":"01:33.105","Text":"Well, that\u0027s F at 15.5 minus F at 7.5."},{"Start":"01:33.105 ","End":"01:35.490","Text":"Now, F at 15.5."},{"Start":"01:35.490 ","End":"01:37.950","Text":"Well, 15.5 is greater than 10,"},{"Start":"01:37.950 ","End":"01:40.290","Text":"so that\u0027ll be 1."},{"Start":"01:40.290 ","End":"01:42.180","Text":"It\u0027d be 1 minus, now,"},{"Start":"01:42.180 ","End":"01:46.565","Text":"F at 7.5, that\u0027s right here."},{"Start":"01:46.565 ","End":"01:49.220","Text":"Let\u0027s just plug in the number here."},{"Start":"01:49.220 ","End":"01:52.520","Text":"That\u0027s 1 minus 0.02,"},{"Start":"01:52.520 ","End":"01:59.370","Text":"times 10 minus 7.5 squared,"},{"Start":"01:59.560 ","End":"02:04.780","Text":"and that comes out to 0.125."},{"Start":"02:04.780 ","End":"02:11.440","Text":"The last probability, the probability of Y being greater than 4."},{"Start":"02:11.440 ","End":"02:18.065","Text":"Well, that equals to 1 minus F at 4."},{"Start":"02:18.065 ","End":"02:19.860","Text":"That equals to 1 minus,"},{"Start":"02:19.860 ","End":"02:21.395","Text":"now 4 is right here,"},{"Start":"02:21.395 ","End":"02:22.485","Text":"in this range right here,"},{"Start":"02:22.485 ","End":"02:24.100","Text":"so we\u0027ll use this expression."},{"Start":"02:24.100 ","End":"02:29.035","Text":"It\u0027ll be 0.02 times 4 squared,"},{"Start":"02:29.035 ","End":"02:32.785","Text":"and that equals to 0.32."},{"Start":"02:32.785 ","End":"02:39.890","Text":"Here we have all 4 probabilities that we were asked to calculate."},{"Start":"02:40.050 ","End":"02:44.515","Text":"In this section, we\u0027re asked to calculate the bottom 10th percentile,"},{"Start":"02:44.515 ","End":"02:46.645","Text":"the bottom 25th percentile,"},{"Start":"02:46.645 ","End":"02:48.105","Text":"and the median of Y,"},{"Start":"02:48.105 ","End":"02:51.980","Text":"and also to infer about the upper 10th percentile."},{"Start":"02:51.980 ","End":"02:54.559","Text":"Let\u0027s start with the median."},{"Start":"02:54.559 ","End":"02:57.920","Text":"Well, first of all, when we\u0027re dealing with median,"},{"Start":"02:57.920 ","End":"02:59.330","Text":"so any types of percentile,"},{"Start":"02:59.330 ","End":"03:06.229","Text":"the definition is the probability of Y being less than or equal to some value,"},{"Start":"03:06.229 ","End":"03:07.903","Text":"in our case the median,"},{"Start":"03:07.903 ","End":"03:10.565","Text":"so it\u0027ll be Y_0.5,"},{"Start":"03:10.565 ","End":"03:16.290","Text":"we want that probability to equal 0.5."},{"Start":"03:17.120 ","End":"03:22.115","Text":"Let\u0027s take a look at our density function right here."},{"Start":"03:22.115 ","End":"03:25.235","Text":"There are several characteristics that are very important."},{"Start":"03:25.235 ","End":"03:28.675","Text":"First of all, we see that it\u0027s symmetrical."},{"Start":"03:28.675 ","End":"03:33.230","Text":"Not only that, but it\u0027s symmetrical around the value 5."},{"Start":"03:33.230 ","End":"03:34.430","Text":"Now, what does that mean?"},{"Start":"03:34.430 ","End":"03:38.735","Text":"That means that in the range between 0 and 5,"},{"Start":"03:38.735 ","End":"03:43.580","Text":"the area under the density function equals to"},{"Start":"03:43.580 ","End":"03:48.965","Text":"the area under the density function in the range between 5 and 10,"},{"Start":"03:48.965 ","End":"03:57.620","Text":"so that means that since both areas are equal and the ranges are equal as well,"},{"Start":"03:57.620 ","End":"04:07.105","Text":"then that means that the area here is 0.5 or 1/2 on both sides of 5."},{"Start":"04:07.105 ","End":"04:10.880","Text":"Now, that\u0027s exactly what we\u0027re looking for right here."},{"Start":"04:10.880 ","End":"04:12.545","Text":"What\u0027s the value?"},{"Start":"04:12.545 ","End":"04:19.100","Text":"We\u0027re looking for a value where the probability below this value equals 1/2."},{"Start":"04:19.100 ","End":"04:20.960","Text":"Well, that\u0027s 5 in our case."},{"Start":"04:20.960 ","End":"04:24.635","Text":"That means that Y_0.5,"},{"Start":"04:24.635 ","End":"04:28.900","Text":"well, that equals to 5."},{"Start":"04:28.900 ","End":"04:32.405","Text":"Let\u0027s take a look at the bottom 10th percentile."},{"Start":"04:32.405 ","End":"04:38.695","Text":"That would be the probability of Y being less than or equal to Y_0.1,"},{"Start":"04:38.695 ","End":"04:41.570","Text":"that has to equal to 0.1."},{"Start":"04:41.570 ","End":"04:44.705","Text":"Now, where does this value lie?"},{"Start":"04:44.705 ","End":"04:47.780","Text":"Well, it lies in the range between 0 and 5."},{"Start":"04:47.780 ","End":"04:51.625","Text":"It lies somewhere here, Y_0.1."},{"Start":"04:51.625 ","End":"04:55.340","Text":"Why is that? Well, if the total area under"},{"Start":"04:55.340 ","End":"04:59.050","Text":"the density function in the range between 0 and 5 equals 1/2,"},{"Start":"04:59.050 ","End":"05:02.060","Text":"and we only want 0.1,"},{"Start":"05:02.060 ","End":"05:06.455","Text":"so that means that it\u0027s somewhere here."},{"Start":"05:06.455 ","End":"05:11.330","Text":"We\u0027re looking for the value Y_0.1 where"},{"Start":"05:11.330 ","End":"05:19.170","Text":"the area under the density function between 0 and this value that has to equal to 0.1."},{"Start":"05:19.430 ","End":"05:23.804","Text":"Let\u0027s take a look at the cumulative distribution function."},{"Start":"05:23.804 ","End":"05:27.890","Text":"We\u0027re looking for F at Y_0.1,"},{"Start":"05:27.890 ","End":"05:32.650","Text":"we\u0027re looking for that to be equal to 0.1."},{"Start":"05:32.650 ","End":"05:36.230","Text":"We\u0027re looking at this expression right here."},{"Start":"05:36.230 ","End":"05:40.475","Text":"That\u0027ll be 0.02t squared,"},{"Start":"05:40.475 ","End":"05:44.105","Text":"and that has to equal to 0.1."},{"Start":"05:44.105 ","End":"05:49.280","Text":"That means that t equals to the square root of 5,"},{"Start":"05:49.280 ","End":"05:55.370","Text":"or 2.24, and that is Y_0.1,"},{"Start":"05:55.370 ","End":"05:59.970","Text":"that\u0027s the bottom 10th percentile."},{"Start":"05:59.970 ","End":"06:08.015","Text":"That\u0027s the value where the area under the density function between 0 and 2.24,"},{"Start":"06:08.015 ","End":"06:14.040","Text":"the area under this density function would equal to 0.1."},{"Start":"06:14.720 ","End":"06:18.499","Text":"Now let\u0027s do the bottom 25th percentile."},{"Start":"06:18.499 ","End":"06:25.220","Text":"That would be the probability of Y being less than or equal to Y_0.25,"},{"Start":"06:25.220 ","End":"06:28.100","Text":"and we want that to be equal to 1/4,"},{"Start":"06:28.100 ","End":"06:31.130","Text":"or 25 percent, or 0.25."},{"Start":"06:31.130 ","End":"06:36.965","Text":"That means that we\u0027re looking for some value here, Y_0.25,"},{"Start":"06:36.965 ","End":"06:42.410","Text":"where the area under the density function between 0 and this value,"},{"Start":"06:42.410 ","End":"06:46.820","Text":"that has to equal to 25 percent or 0.25."},{"Start":"06:46.820 ","End":"06:56.230","Text":"That means they were looking for F at Y_0.25 to be equal to 0.25."},{"Start":"06:56.230 ","End":"06:59.630","Text":"Again, when in this range we\u0027ll use this expression,"},{"Start":"06:59.630 ","End":"07:05.669","Text":"and that\u0027ll equal to 0.02t squared,"},{"Start":"07:05.669 ","End":"07:08.660","Text":"and we want that to equal 0.25."},{"Start":"07:08.660 ","End":"07:13.100","Text":"That means that t now equals to the square root of 12.5,"},{"Start":"07:13.100 ","End":"07:17.540","Text":"and that equals to 3.54,"},{"Start":"07:17.540 ","End":"07:23.640","Text":"and that will equal to Y_0.25."},{"Start":"07:24.500 ","End":"07:27.690","Text":"Let\u0027s take a look at the last thing that we have to do."},{"Start":"07:27.690 ","End":"07:33.335","Text":"We have to infer about the upper 10th percentile, or Y_0.9."},{"Start":"07:33.335 ","End":"07:35.299","Text":"Now, we can calculate this,"},{"Start":"07:35.299 ","End":"07:37.615","Text":"or we can do something a little bit more elegant."},{"Start":"07:37.615 ","End":"07:39.510","Text":"Now, here,"},{"Start":"07:39.510 ","End":"07:42.780","Text":"we\u0027ve calculated the bottom 10th percentile,"},{"Start":"07:42.780 ","End":"07:49.795","Text":"or Y_0.1, where this area here was 0.1."},{"Start":"07:49.795 ","End":"07:54.049","Text":"Now, when we\u0027re asked about the 90th percentile,"},{"Start":"07:54.049 ","End":"07:58.580","Text":"we\u0027re asking what\u0027s the value here Y_0.9,"},{"Start":"07:58.580 ","End":"08:03.575","Text":"where this area right here that also equals to 0.1?"},{"Start":"08:03.575 ","End":"08:08.735","Text":"Now, because this is a symmetrical density function,"},{"Start":"08:08.735 ","End":"08:11.510","Text":"then we can say the following."},{"Start":"08:11.510 ","End":"08:17.385","Text":"We can say that Y_0.9, well,"},{"Start":"08:17.385 ","End":"08:23.870","Text":"that equals to 10 minus Y_0.1, and why is that?"},{"Start":"08:23.870 ","End":"08:28.400","Text":"Because this range right here has to"},{"Start":"08:28.400 ","End":"08:34.440","Text":"equal to this range right here in order for the areas to be the same."},{"Start":"08:35.080 ","End":"08:39.690","Text":"We\u0027ve calculated Y_0.1, that\u0027s 2.24,"},{"Start":"08:39.690 ","End":"08:43.155","Text":"so that\u0027s 10 minus 2.24,"},{"Start":"08:43.155 ","End":"08:46.980","Text":"that equals to 7.76."},{"Start":"08:46.980 ","End":"08:56.050","Text":"This would be the upper 10th percentile, or Y_0.9."}],"ID":13101},{"Watched":false,"Name":"Exercise 4","Duration":"8m 33s","ChapterTopicVideoID":12623,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12623.jpeg","UploadDate":"2018-07-23T16:02:32.9830000","DurationForVideoObject":"PT8M33S","Description":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":13102},{"Watched":false,"Name":"Exercise 5","Duration":"4m 47s","ChapterTopicVideoID":12624,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12624.jpeg","UploadDate":"2018-07-23T16:02:45.4670000","DurationForVideoObject":"PT4M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.314","Text":"In this question, we\u0027re given the following density function,"},{"Start":"00:03.314 ","End":"00:07.020","Text":"a triangle, and we\u0027re asked what\u0027s the value of c?"},{"Start":"00:07.020 ","End":"00:11.417","Text":"Well, we know that the total area under a density function has to equal to 1,"},{"Start":"00:11.417 ","End":"00:15.765","Text":"so let\u0027s calculate the area of this triangle."},{"Start":"00:15.765 ","End":"00:20.805","Text":"Area of this triangle equals to 1/2 times the base times the height."},{"Start":"00:20.805 ","End":"00:22.950","Text":"Now, in our case,"},{"Start":"00:22.950 ","End":"00:29.715","Text":"the base is 10 and the height is c. That has to equal to 1,"},{"Start":"00:29.715 ","End":"00:34.335","Text":"so that means that 5c equals 1,"},{"Start":"00:34.335 ","End":"00:39.850","Text":"or c equals a 1/5, or 0.2."},{"Start":"00:40.210 ","End":"00:44.240","Text":"In this section, we\u0027re asked to find a symmetric interval around"},{"Start":"00:44.240 ","End":"00:48.715","Text":"the value 5 in which the probability equals 0.5."},{"Start":"00:48.715 ","End":"00:51.500","Text":"What\u0027s a symmetric interval?"},{"Start":"00:51.500 ","End":"00:55.340","Text":"That\u0027s basically an interval where on both sides of 5,"},{"Start":"00:55.340 ","End":"00:57.185","Text":"we have the same distance."},{"Start":"00:57.185 ","End":"01:02.208","Text":"Let\u0027s call this distance a on both sides of 5,"},{"Start":"01:02.208 ","End":"01:05.180","Text":"so this part will be 5 minus a,"},{"Start":"01:05.180 ","End":"01:07.895","Text":"and this point would be 5 plus a."},{"Start":"01:07.895 ","End":"01:10.775","Text":"Now, in this range right here,"},{"Start":"01:10.775 ","End":"01:17.885","Text":"we want the area underneath the density function to equal 1/2."},{"Start":"01:17.885 ","End":"01:22.750","Text":"This area here in yellow equals 1/2."},{"Start":"01:22.750 ","End":"01:29.270","Text":"Now, if this area equals 1/2 then the area outside of this range also equals 1/2."},{"Start":"01:29.270 ","End":"01:38.245","Text":"That means that each of these triangles has an area of 0.25 or a quarter."},{"Start":"01:38.245 ","End":"01:40.685","Text":"Now how does that help us?"},{"Start":"01:40.685 ","End":"01:43.010","Text":"Well, we want to solve for a,"},{"Start":"01:43.010 ","End":"01:45.050","Text":"we want to know what the value of a is."},{"Start":"01:45.050 ","End":"01:53.020","Text":"Let\u0027s use the equation for the area of a triangle to isolate and solve for a."},{"Start":"01:53.020 ","End":"01:55.565","Text":"Let\u0027s take a look at this triangle right here."},{"Start":"01:55.565 ","End":"01:57.410","Text":"Really doesn\u0027t matter which one."},{"Start":"01:57.410 ","End":"02:00.350","Text":"Now, the area of a triangle, well,"},{"Start":"02:00.350 ","End":"02:03.965","Text":"that equals to 1/2 times the base times the height."},{"Start":"02:03.965 ","End":"02:06.250","Text":"Now, our base here,"},{"Start":"02:06.250 ","End":"02:10.265","Text":"that\u0027s 5 minus a. What\u0027s the height?"},{"Start":"02:10.265 ","End":"02:15.305","Text":"Well, the height is the value of this line at 0.5 minus a."},{"Start":"02:15.305 ","End":"02:20.120","Text":"Let\u0027s just calculate the equation for this line right here."},{"Start":"02:20.120 ","End":"02:27.110","Text":"Now, the equation for the line is y equals mx plus n. Now,"},{"Start":"02:27.110 ","End":"02:31.370","Text":"in our case, n equals 0 because this line goes through the origins."},{"Start":"02:31.370 ","End":"02:33.515","Text":"What about m, the slope?"},{"Start":"02:33.515 ","End":"02:39.895","Text":"Well, let\u0027s take a look at these 2 points that will help us to calculate the slope."},{"Start":"02:39.895 ","End":"02:43.070","Text":"We know that c equals 0.2."},{"Start":"02:43.070 ","End":"02:45.575","Text":"We\u0027ve calculated that in the previous section,"},{"Start":"02:45.575 ","End":"02:50.570","Text":"so that\u0027ll be 0.2 minus 0 divided by 5 minus 0."},{"Start":"02:50.570 ","End":"02:52.960","Text":"That equals to 0.04."},{"Start":"02:52.960 ","End":"03:02.250","Text":"That means that the equation for this [inaudible] equals Y equals 0.04 times x."},{"Start":"03:02.320 ","End":"03:08.360","Text":"Let\u0027s now get back to the height of this triangle."},{"Start":"03:08.360 ","End":"03:10.340","Text":"The height, as we said,"},{"Start":"03:10.340 ","End":"03:14.180","Text":"equals to the value of this line at this point."},{"Start":"03:14.180 ","End":"03:20.200","Text":"So all we have to do now is plug in 5 minus a instead of x to get the height."},{"Start":"03:20.200 ","End":"03:26.457","Text":"The height becomes 0.04 times 5 minus a,"},{"Start":"03:26.457 ","End":"03:32.500","Text":"so let\u0027s just plug these values into this equation."},{"Start":"03:32.500 ","End":"03:37.730","Text":"We know that the area of this triangle equals 0.25."},{"Start":"03:37.730 ","End":"03:41.480","Text":"So we can now isolate for a. Let\u0027s do that."},{"Start":"03:41.480 ","End":"03:45.065","Text":"That\u0027ll be 1/2 times the base,"},{"Start":"03:45.065 ","End":"03:48.590","Text":"5 minus a times the height."},{"Start":"03:48.590 ","End":"03:52.355","Text":"Well, that\u0027s 0.04 times 5 minus a."},{"Start":"03:52.355 ","End":"03:56.880","Text":"We know that this equals to a quarter."},{"Start":"03:57.110 ","End":"03:59.310","Text":"Let\u0027s simplify that."},{"Start":"03:59.310 ","End":"04:08.640","Text":"There will be 5 minus a squared times 0.02 and that equals a quarter,"},{"Start":"04:08.640 ","End":"04:12.045","Text":"and that equals to 0.25."},{"Start":"04:12.045 ","End":"04:17.655","Text":"That means that 5 minus a squared equals"},{"Start":"04:17.655 ","End":"04:24.870","Text":"12.5 and 5 minus a equals 3.54,"},{"Start":"04:24.870 ","End":"04:29.055","Text":"and a then equals to 1.46."},{"Start":"04:29.055 ","End":"04:32.010","Text":"This is range,"},{"Start":"04:32.010 ","End":"04:36.195","Text":"5 plus minus 1.46."},{"Start":"04:36.195 ","End":"04:47.220","Text":"That\u0027s the range under which the area of the density function equals 1/2."}],"ID":13103},{"Watched":false,"Name":"Exercise 6 - Parts a-b","Duration":"6m 23s","ChapterTopicVideoID":12625,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12625.jpeg","UploadDate":"2018-07-23T16:03:04.6470000","DurationForVideoObject":"PT6M23S","Description":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":13104},{"Watched":false,"Name":"Exercise 6 - Parts c-e","Duration":"3m 54s","ChapterTopicVideoID":12626,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12626.jpeg","UploadDate":"2018-07-23T16:03:17.3930000","DurationForVideoObject":"PT3M54S","Description":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":13105},{"Watched":false,"Name":"Exercise 7 - Part c","Duration":"4m 24s","ChapterTopicVideoID":12627,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12627.jpeg","UploadDate":"2018-07-23T16:03:27.6970000","DurationForVideoObject":"PT4M24S","Description":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":13106},{"Watched":false,"Name":"Exercise 7 - Parts a-b","Duration":"9m 21s","ChapterTopicVideoID":12628,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12628.jpeg","UploadDate":"2018-07-23T16:03:54.1700000","DurationForVideoObject":"PT9M21S","Description":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":13107},{"Watched":false,"Name":"Exercise 8 - Parts a-b","Duration":"6m 34s","ChapterTopicVideoID":12629,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12629.jpeg","UploadDate":"2018-07-23T16:04:11.7600000","DurationForVideoObject":"PT6M34S","Description":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":13108},{"Watched":false,"Name":"Exercise 8 - Parts c-d","Duration":"3m 38s","ChapterTopicVideoID":12630,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12630.jpeg","UploadDate":"2018-07-23T16:04:20.6130000","DurationForVideoObject":"PT3M38S","Description":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":13109},{"Watched":false,"Name":"Exercise 9 - Parts a-b","Duration":"5m 19s","ChapterTopicVideoID":12631,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12631.jpeg","UploadDate":"2018-07-23T16:04:36.7130000","DurationForVideoObject":"PT5M19S","Description":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":13110},{"Watched":false,"Name":"Exercise 9 - Parts c-e","Duration":"9m 1s","ChapterTopicVideoID":12632,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12632.jpeg","UploadDate":"2018-07-23T16:04:59.8530000","DurationForVideoObject":"PT9M1S","Description":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":13111},{"Watched":false,"Name":"Exercise 10","Duration":"7m 38s","ChapterTopicVideoID":12633,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12633.jpeg","UploadDate":"2018-07-23T16:05:28.1800000","DurationForVideoObject":"PT7M38S","Description":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":13112},{"Watched":false,"Name":"Exercise 11","Duration":"7m 32s","ChapterTopicVideoID":12634,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12634.jpeg","UploadDate":"2018-07-23T16:05:52.9200000","DurationForVideoObject":"PT7M32S","Description":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":13113},{"Watched":false,"Name":"Exercise 12 - Part c","Duration":"3m 16s","ChapterTopicVideoID":12635,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12635.jpeg","UploadDate":"2018-07-23T16:06:27.2100000","DurationForVideoObject":"PT3M16S","Description":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":13114},{"Watched":false,"Name":"Exercise 12 - Parts a-b","Duration":"6m 56s","ChapterTopicVideoID":12636,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12636.jpeg","UploadDate":"2018-07-23T16:06:50.5300000","DurationForVideoObject":"PT6M56S","Description":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":13115},{"Watched":false,"Name":"Exercise 13 - Part a","Duration":"1m 54s","ChapterTopicVideoID":12637,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12637.jpeg","UploadDate":"2018-07-23T16:06:55.7070000","DurationForVideoObject":"PT1M54S","Description":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":13116},{"Watched":false,"Name":"Exercise 13 - Parts b-c","Duration":"7m 7s","ChapterTopicVideoID":12638,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12638.jpeg","UploadDate":"2018-07-23T16:07:12.1170000","DurationForVideoObject":"PT7M7S","Description":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":13117},{"Watched":false,"Name":"Exercise 14","Duration":"8m 55s","ChapterTopicVideoID":12639,"CourseChapterTopicPlaylistID":245049,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12639.jpeg","UploadDate":"2018-07-23T16:07:40.1730000","DurationForVideoObject":"PT8M55S","Description":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":13118}],"Thumbnail":null,"ID":245049},{"Name":"Exponential Probability","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"3m 44s","ChapterTopicVideoID":12640,"CourseChapterTopicPlaylistID":245050,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12640.jpeg","UploadDate":"2018-07-23T16:09:12.2170000","DurationForVideoObject":"PT3M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this chapter, we\u0027ll be talking about special continuous probability,"},{"Start":"00:03.720 ","End":"00:06.915","Text":"specifically the exponential probability."},{"Start":"00:06.915 ","End":"00:09.330","Text":"Now, this is a continuous probability that"},{"Start":"00:09.330 ","End":"00:13.005","Text":"characterizes the time until a given event occurs."},{"Start":"00:13.005 ","End":"00:17.130","Text":"We define Lambda as the average number of events occurring in"},{"Start":"00:17.130 ","End":"00:21.720","Text":"a unit of time and this is the same parameter as in the Poisson probability."},{"Start":"00:21.720 ","End":"00:25.530","Text":"Whenever we have a continuous random variable,"},{"Start":"00:25.530 ","End":"00:29.350","Text":"say X, and it\u0027s distributed with an exponential distribution,"},{"Start":"00:29.350 ","End":"00:30.435","Text":"we write it like this."},{"Start":"00:30.435 ","End":"00:34.965","Text":"X is distributed with an exponential distribution with parameter lambda,"},{"Start":"00:34.965 ","End":"00:37.125","Text":"where Lambda is greater than 0."},{"Start":"00:37.125 ","End":"00:41.535","Text":"Now, Lambda is always greater than 0 because we\u0027re dealing with time."},{"Start":"00:41.535 ","End":"00:45.575","Text":"Sometimes we can see that instead of lambda,"},{"Start":"00:45.575 ","End":"00:48.455","Text":"we can see the letter theta."},{"Start":"00:48.455 ","End":"00:54.410","Text":"Now, this probability distribution must be specified in the exercise or it"},{"Start":"00:54.410 ","End":"00:56.960","Text":"must be stated that the number of events in"},{"Start":"00:56.960 ","End":"01:00.140","Text":"a given unit of time has a Poisson probability,"},{"Start":"01:00.140 ","End":"01:05.480","Text":"in which case, the time until the next event has an exponential probability."},{"Start":"01:05.480 ","End":"01:12.050","Text":"Now, this is a connection between the Poisson and the exponential probabilities."},{"Start":"01:12.050 ","End":"01:16.385","Text":"The density function of the probability distribution is this."},{"Start":"01:16.385 ","End":"01:19.580","Text":"This is small f of x and that equals to Lambda times"},{"Start":"01:19.580 ","End":"01:24.140","Text":"e^minus Lambda x for x being greater or equal to 0."},{"Start":"01:24.140 ","End":"01:27.979","Text":"Now, what does this function look like?"},{"Start":"01:27.979 ","End":"01:33.195","Text":"Here we have x and here we have f of x."},{"Start":"01:33.195 ","End":"01:36.135","Text":"Now, when x equals 0,"},{"Start":"01:36.135 ","End":"01:38.895","Text":"we\u0027ll just plug in 0 here,"},{"Start":"01:38.895 ","End":"01:41.940","Text":"that\u0027ll be Lambda times e^0."},{"Start":"01:41.940 ","End":"01:43.595","Text":"Now, e to the power 0 is 1,"},{"Start":"01:43.595 ","End":"01:46.410","Text":"so that\u0027ll equal to lambda."},{"Start":"01:46.580 ","End":"01:51.510","Text":"Now, what happens when x goes to infinity?"},{"Start":"01:51.510 ","End":"01:57.670","Text":"Well, e^minus Lambda x goes to 0."},{"Start":"01:57.670 ","End":"02:07.160","Text":"That means that this is a monotonously decreasing function until infinity,"},{"Start":"02:07.160 ","End":"02:09.410","Text":"where the infinity it equals 0."},{"Start":"02:09.410 ","End":"02:12.890","Text":"Now, what about the cumulative distribution function?"},{"Start":"02:12.890 ","End":"02:16.415","Text":"Well, that\u0027s big F of t and that equals to the probability of"},{"Start":"02:16.415 ","End":"02:20.285","Text":"x being less than or equal to t. In our case,"},{"Start":"02:20.285 ","End":"02:28.060","Text":"that equals to 1 minus e^minus Lambda t. What does this function look like?"},{"Start":"02:28.060 ","End":"02:31.440","Text":"Well, again, we have here x,"},{"Start":"02:31.440 ","End":"02:34.200","Text":"we have here now big F of x."},{"Start":"02:34.200 ","End":"02:38.450","Text":"Now, when x equals 0, well,"},{"Start":"02:38.450 ","End":"02:43.805","Text":"that means that this expression equals 1."},{"Start":"02:43.805 ","End":"02:47.495","Text":"We have 1 minus 1 that equals to 0."},{"Start":"02:47.495 ","End":"02:50.330","Text":"What happens at infinity?"},{"Start":"02:50.330 ","End":"02:54.890","Text":"That means that this expression right here goes to 0."},{"Start":"02:54.890 ","End":"02:57.980","Text":"So at infinity, big F of t equals 1."},{"Start":"02:57.980 ","End":"03:06.260","Text":"This is a monotonously increasing function and at infinity,"},{"Start":"03:06.260 ","End":"03:08.405","Text":"it\u0027ll be equal to 1."},{"Start":"03:08.405 ","End":"03:13.100","Text":"Now, the expectation of x is 1 divided by"},{"Start":"03:13.100 ","End":"03:18.170","Text":"Lambda and the variance of x is 1 divided by Lambda squared."},{"Start":"03:18.170 ","End":"03:23.585","Text":"Now, this distribution has another special characteristic."},{"Start":"03:23.585 ","End":"03:27.800","Text":"This is the memory-less feature here,"},{"Start":"03:27.800 ","End":"03:34.650","Text":"which means the probability of X being greater than a plus b given a,"},{"Start":"03:34.650 ","End":"03:39.325","Text":"well, that equals to the probability of X being greater than b."},{"Start":"03:39.325 ","End":"03:41.015","Text":"Enough theory."},{"Start":"03:41.015 ","End":"03:44.099","Text":"Let\u0027s look at an example."}],"ID":13119},{"Watched":false,"Name":"Example","Duration":"7m 25s","ChapterTopicVideoID":12641,"CourseChapterTopicPlaylistID":245050,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12641.jpeg","UploadDate":"2018-07-23T16:09:31.0730000","DurationForVideoObject":"PT7M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.390","Text":"In our example, we\u0027re given that the lifespan of a battery is"},{"Start":"00:03.390 ","End":"00:06.490","Text":"exponentially distributed with an expectation of 8 hours,"},{"Start":"00:06.490 ","End":"00:12.225","Text":"and we\u0027re asked, what\u0027s the probability that the battery will last less than 9 hours?"},{"Start":"00:12.225 ","End":"00:17.820","Text":"Well, first of all, let\u0027s take a look at the exponential distribution."},{"Start":"00:17.820 ","End":"00:28.090","Text":"We\u0027re given a random variable x and that equals to the lifespan of the battery."},{"Start":"00:28.340 ","End":"00:34.610","Text":"Now, we\u0027re also given that x is distributed with an exponential distribution,"},{"Start":"00:34.610 ","End":"00:37.030","Text":"but we\u0027re not given the parameter."},{"Start":"00:37.030 ","End":"00:40.965","Text":"Now, given that the expectation of x,"},{"Start":"00:40.965 ","End":"00:43.320","Text":"well that equals to 8 hours."},{"Start":"00:43.320 ","End":"00:48.970","Text":"Now, we know that since x is distributed with an exponential distribution,"},{"Start":"00:48.970 ","End":"00:51.330","Text":"the expectation of x,"},{"Start":"00:51.330 ","End":"00:54.615","Text":"well that equals to 1 divided by Lambda."},{"Start":"00:54.615 ","End":"00:59.495","Text":"That means that 1 divided by Lambda,"},{"Start":"00:59.495 ","End":"01:01.265","Text":"well that equals to 8,"},{"Start":"01:01.265 ","End":"01:08.345","Text":"that means that Lambda here equals to 1/8."},{"Start":"01:08.345 ","End":"01:13.310","Text":"This is the parameter of the exponential distribution."},{"Start":"01:13.310 ","End":"01:15.725","Text":"Now that we have this,"},{"Start":"01:15.725 ","End":"01:17.300","Text":"we can calculate A,"},{"Start":"01:17.300 ","End":"01:21.816","Text":"the probability that the battery will last less than 9 hours."},{"Start":"01:21.816 ","End":"01:28.260","Text":"We\u0027re looking for the probability of x being less than 9."},{"Start":"01:28.260 ","End":"01:36.635","Text":"Now, we know the probability of x being less than or equal to any value of t. Well,"},{"Start":"01:36.635 ","End":"01:40.025","Text":"that\u0027s big F of t,"},{"Start":"01:40.025 ","End":"01:46.340","Text":"that equals to 1 minus e to the power of minus Lambda t. That\u0027s"},{"Start":"01:46.340 ","End":"01:53.195","Text":"the definition of the cumulative distribution function of an exponential distribution."},{"Start":"01:53.195 ","End":"01:57.690","Text":"In our case, that will equal to"},{"Start":"01:57.690 ","End":"02:05.170","Text":"1 minus e to the power of minus 1/8th times 9,"},{"Start":"02:05.170 ","End":"02:09.910","Text":"and that equals to 0.675."},{"Start":"02:12.270 ","End":"02:17.530","Text":"In this section, we\u0027re asked what\u0027s the standard deviation of the battery lifespan?"},{"Start":"02:17.530 ","End":"02:20.230","Text":"Well, we know that the variance of x,"},{"Start":"02:20.230 ","End":"02:24.375","Text":"that equals to 1 divided by Lambda squared."},{"Start":"02:24.375 ","End":"02:26.770","Text":"The standard deviation of x,"},{"Start":"02:26.770 ","End":"02:31.120","Text":"well that equals to the square root of the variance,"},{"Start":"02:31.120 ","End":"02:37.455","Text":"or 1 divided by Lambda squared and that equals to 1 divided by Lambda."},{"Start":"02:37.455 ","End":"02:39.955","Text":"Now, 1 divided by Lambda,"},{"Start":"02:39.955 ","End":"02:44.140","Text":"we\u0027re given that\u0027s the expectation of x also,"},{"Start":"02:44.140 ","End":"02:46.900","Text":"and that equals to 8."},{"Start":"02:46.900 ","End":"02:49.360","Text":"In section C, we\u0027re asked,"},{"Start":"02:49.360 ","End":"02:51.430","Text":"if a battery lasts more than 2 hours,"},{"Start":"02:51.430 ","End":"02:55.110","Text":"what is the chances that they will last a total of more than s7 hours?"},{"Start":"02:55.110 ","End":"02:58.820","Text":"Well, this looks like a conditional probability, so let\u0027s set it up."},{"Start":"02:58.820 ","End":"03:01.070","Text":"Let be the probability of,"},{"Start":"03:01.070 ","End":"03:02.825","Text":"now what\u0027s given to us?"},{"Start":"03:02.825 ","End":"03:07.150","Text":"Well, we\u0027re given that x is greater than 2, right?"},{"Start":"03:07.150 ","End":"03:10.580","Text":"The battery lasts more than 2 hours and what are we asked?"},{"Start":"03:10.580 ","End":"03:14.450","Text":"Were asked about the chances that would last more than 7 hours,"},{"Start":"03:14.450 ","End":"03:17.755","Text":"that means x has to be greater than 7."},{"Start":"03:17.755 ","End":"03:20.810","Text":"Let\u0027s just write this out."},{"Start":"03:20.810 ","End":"03:28.085","Text":"In the denominator we have the probability of what\u0027s given and in the numerator, well,"},{"Start":"03:28.085 ","End":"03:30.080","Text":"that\u0027s the probability of the intersect,"},{"Start":"03:30.080 ","End":"03:34.175","Text":"that\u0027s the probability of x being greater than 7 intersect,"},{"Start":"03:34.175 ","End":"03:36.200","Text":"x being greater than 2."},{"Start":"03:36.200 ","End":"03:39.530","Text":"Now, the intersect of these guys,"},{"Start":"03:39.530 ","End":"03:41.940","Text":"well, that\u0027s x greater than 7,"},{"Start":"03:41.940 ","End":"03:45.965","Text":"so that\u0027ll be the probability of x being greater than 7"},{"Start":"03:45.965 ","End":"03:52.760","Text":"divided by the probability of x being greater than 2."},{"Start":"03:52.760 ","End":"03:54.680","Text":"Now, what do we know?"},{"Start":"03:54.680 ","End":"04:03.045","Text":"We know that the probability of x being less than or equal to t. That\u0027s big F at t,"},{"Start":"04:03.045 ","End":"04:06.665","Text":"that equals to 1 minus e to the power of minus Lambda"},{"Start":"04:06.665 ","End":"04:10.520","Text":"times t in an exponential distribution."},{"Start":"04:10.520 ","End":"04:18.710","Text":"In our case, that will be equal to 1 minus p. The probability of x"},{"Start":"04:18.710 ","End":"04:22.840","Text":"being less than or equal to 7 divided"},{"Start":"04:22.840 ","End":"04:27.260","Text":"by 1 minus the probability of x being less than or equal to 2."},{"Start":"04:27.260 ","End":"04:37.470","Text":"Well, that equals to 1 minus F at 7 divided by 1 minus F at 2."},{"Start":"04:37.470 ","End":"04:39.150","Text":"Let\u0027s just write that out."},{"Start":"04:39.150 ","End":"04:40.860","Text":"That will be 1 minus."},{"Start":"04:40.860 ","End":"04:42.330","Text":"Now, what\u0027s Fat 7?"},{"Start":"04:42.330 ","End":"04:48.515","Text":"F at 7 is 1 minus e to the power of minus 1/8 times 7 minus an"},{"Start":"04:48.515 ","End":"04:55.500","Text":"1/8 and 1/8 is the value of Lambda divided by 1 minus."},{"Start":"04:55.500 ","End":"04:57.030","Text":"Now, F at 2,"},{"Start":"04:57.030 ","End":"05:01.580","Text":"that\u0027ll be 1 minus e to the power of minus 1/8 times 2."},{"Start":"05:01.580 ","End":"05:06.560","Text":"Now, that equals to e to the power of minus 1/8 times"},{"Start":"05:06.560 ","End":"05:11.990","Text":"7 divided by e to the power of minus 1/8 times 2."},{"Start":"05:11.990 ","End":"05:13.955","Text":"Now, let\u0027s just simplify that."},{"Start":"05:13.955 ","End":"05:22.745","Text":"That\u0027ll be e to the power of minus 1/8 times 7 plus 1/8 times 2."},{"Start":"05:22.745 ","End":"05:30.910","Text":"Now, that equals to e to the power of minus 1/8 times 5."},{"Start":"05:30.910 ","End":"05:35.990","Text":"This then is the probability of a battery lasting"},{"Start":"05:35.990 ","End":"05:43.595","Text":"more than 7 hours given that it lasted more than 2 hours."},{"Start":"05:43.595 ","End":"05:46.850","Text":"Now we can also solve this problem by"},{"Start":"05:46.850 ","End":"05:50.420","Text":"using the special characteristic of this distribution,"},{"Start":"05:50.420 ","End":"05:53.720","Text":"which has the memoryless characteristic."},{"Start":"05:53.720 ","End":"05:56.785","Text":"Let\u0027s just see how this works."},{"Start":"05:56.785 ","End":"05:59.900","Text":"The feature goes like this."},{"Start":"05:59.900 ","End":"06:04.970","Text":"We\u0027re looking at the probability of x being greater than a plus b,"},{"Start":"06:04.970 ","End":"06:08.180","Text":"given that x is greater than a,"},{"Start":"06:08.180 ","End":"06:14.040","Text":"well that equals to the probability of x being greater than b."},{"Start":"06:14.360 ","End":"06:24.620","Text":"In our case, that\u0027s the probability of x being greater than 7 given x is greater than 2."},{"Start":"06:24.620 ","End":"06:28.950","Text":"Now, 7 minus 2,"},{"Start":"06:28.950 ","End":"06:31.770","Text":"that\u0027s 5, so that means that b equals 5."},{"Start":"06:31.770 ","End":"06:36.155","Text":"We\u0027re looking at the probability of x being greater than 5."},{"Start":"06:36.155 ","End":"06:42.365","Text":"Now, that means that this is 1 minus the probability of x being"},{"Start":"06:42.365 ","End":"06:49.680","Text":"less than or equal to 5 and that equals to 1 minus F at 5."},{"Start":"06:50.180 ","End":"06:54.720","Text":"That equals to 1 minus. Now what\u0027s F at 5?"},{"Start":"06:54.720 ","End":"07:00.330","Text":"Well, that\u0027s 1 minus e to the power of minus 8 times 5."},{"Start":"07:00.350 ","End":"07:07.150","Text":"Others expression becomes e to the power of minus 1/8 times 5."},{"Start":"07:07.150 ","End":"07:09.110","Text":"Now, as you can see,"},{"Start":"07:09.110 ","End":"07:11.135","Text":"this is a much shorter way to do this,"},{"Start":"07:11.135 ","End":"07:15.545","Text":"but I don\u0027t recommend using this feature right here."},{"Start":"07:15.545 ","End":"07:19.220","Text":"I\u0027d prefer you guys using the long way of doing"},{"Start":"07:19.220 ","End":"07:24.510","Text":"things using the standard conditional probabilities equations."}],"ID":13120},{"Watched":false,"Name":"Exercise 1","Duration":"5m 44s","ChapterTopicVideoID":12642,"CourseChapterTopicPlaylistID":245050,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12642.jpeg","UploadDate":"2018-07-23T16:09:45.8670000","DurationForVideoObject":"PT5M44S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.575","Text":"In this question, we\u0027re given that the time until a malfunction occurs in the system"},{"Start":"00:04.575 ","End":"00:09.735","Text":"has an exponential probability distribution with an expectation of 0.5 hours,"},{"Start":"00:09.735 ","End":"00:12.360","Text":"and we\u0027re asked what\u0027s the probability that"},{"Start":"00:12.360 ","End":"00:16.960","Text":"the next malfunction will occur in more than 1,2 an hour?"},{"Start":"00:16.970 ","End":"00:20.495","Text":"First of all, let\u0027s define a random variable."},{"Start":"00:20.495 ","End":"00:29.520","Text":"X, that will be the time until a malfunction."},{"Start":"00:30.910 ","End":"00:36.960","Text":"Now, we know that x is distributed with an exponential distribution,"},{"Start":"00:36.960 ","End":"00:40.170","Text":"but we don\u0027t know what Lambda is right now."},{"Start":"00:40.170 ","End":"00:46.660","Text":"But we\u0027re given that the expectation of x,"},{"Start":"00:47.500 ","End":"00:51.395","Text":"well, that equals to 0.5."},{"Start":"00:51.395 ","End":"00:55.130","Text":"Now that equals also to 1 divided by Lambda."},{"Start":"00:55.130 ","End":"00:59.240","Text":"That means that Lambda here equals to 2."},{"Start":"00:59.240 ","End":"01:02.885","Text":"Now, that means we can plug that in right here."},{"Start":"01:02.885 ","End":"01:04.610","Text":"Once we have that,"},{"Start":"01:04.610 ","End":"01:06.440","Text":"we can answer this question."},{"Start":"01:06.440 ","End":"01:13.551","Text":"We\u0027re looking then for the probability of x being greater than 0.5."},{"Start":"01:13.551 ","End":"01:19.415","Text":"Now we know that the probability of x being less than or equal to t,"},{"Start":"01:19.415 ","End":"01:22.735","Text":"any value that equals to F at t,"},{"Start":"01:22.735 ","End":"01:30.050","Text":"that equals to 1 minus e to the power of minus Lambda times t, that\u0027s the definition."},{"Start":"01:30.050 ","End":"01:37.630","Text":"In our case, that equals to 1 minus F at 0.5."},{"Start":"01:37.630 ","End":"01:42.390","Text":"Let\u0027s just plug in the equation for big F,"},{"Start":"01:42.390 ","End":"01:49.295","Text":"that\u0027ll be 1 minus 1 minus e to the power of,"},{"Start":"01:49.295 ","End":"01:51.110","Text":"now Lambda is 2,"},{"Start":"01:51.110 ","End":"01:53.780","Text":"so it\u0027ll be minus 2 times t,"},{"Start":"01:53.780 ","End":"01:57.370","Text":"where t is 0.5."},{"Start":"01:57.370 ","End":"02:01.060","Text":"That equals to e to the power of minus 1,"},{"Start":"02:01.060 ","End":"02:06.750","Text":"so that equals to 0.368."},{"Start":"02:06.750 ","End":"02:14.165","Text":"That\u0027s the probability that the next malfunction will occur in more than a 1/2 an hour."},{"Start":"02:14.165 ","End":"02:17.390","Text":"In this section, we\u0027re asked what\u0027s the probability that"},{"Start":"02:17.390 ","End":"02:20.915","Text":"the next malfunction will occur in less than an hour?"},{"Start":"02:20.915 ","End":"02:26.645","Text":"We\u0027re looking for the probability of x being less than 1."},{"Start":"02:26.645 ","End":"02:31.850","Text":"That means that we\u0027re looking for F at 1."},{"Start":"02:31.850 ","End":"02:39.655","Text":"Now F at 1 is 1 minus e to the power of minus 2."},{"Start":"02:39.655 ","End":"02:42.825","Text":"That\u0027s Lambda times 1."},{"Start":"02:42.825 ","End":"02:48.815","Text":"That equals to 1 minus e to the power of minus 2,"},{"Start":"02:48.815 ","End":"02:53.015","Text":"and that equals to 0.865."},{"Start":"02:53.015 ","End":"02:59.470","Text":"There is a probability that the next malfunction will occur in less than an hour."},{"Start":"02:59.470 ","End":"03:02.450","Text":"In this section, we\u0027re asked to find the median time"},{"Start":"03:02.450 ","End":"03:05.660","Text":"for the occurrence of a malfunction in the system."},{"Start":"03:05.660 ","End":"03:10.345","Text":"Let\u0027s define m as the median."},{"Start":"03:10.345 ","End":"03:12.975","Text":"Now, let\u0027s recall what a median is."},{"Start":"03:12.975 ","End":"03:15.825","Text":"The median is the value of x,"},{"Start":"03:15.825 ","End":"03:18.710","Text":"where 1/2 of the probability is above"},{"Start":"03:18.710 ","End":"03:22.415","Text":"this value and 1/2 the probability is below this value."},{"Start":"03:22.415 ","End":"03:24.320","Text":"Now in more formal terms,"},{"Start":"03:24.320 ","End":"03:28.340","Text":"that\u0027s the probability of x being less than or equal to"},{"Start":"03:28.340 ","End":"03:33.830","Text":"m and we want that probability to be equal to a 1/2."},{"Start":"03:33.830 ","End":"03:35.570","Text":"Now, what\u0027s this?"},{"Start":"03:35.570 ","End":"03:38.750","Text":"This is basically F at m,"},{"Start":"03:38.750 ","End":"03:42.365","Text":"and we want that to be equal to 0.5."},{"Start":"03:42.365 ","End":"03:47.360","Text":"In our case, there\u0027ll be 1 minus e to"},{"Start":"03:47.360 ","End":"03:53.015","Text":"the power of minus 2m and we want that to be equal to 0.5."},{"Start":"03:53.015 ","End":"03:54.965","Text":"Now, the minute we have this,"},{"Start":"03:54.965 ","End":"03:56.780","Text":"then the rest is algebra."},{"Start":"03:56.780 ","End":"04:00.070","Text":"All we need to do is solve for m. Let\u0027s do that."},{"Start":"04:00.070 ","End":"04:03.885","Text":"Well, that\u0027ll be e to the power of minus 2m,"},{"Start":"04:03.885 ","End":"04:05.467","Text":"that equals to 0.5."},{"Start":"04:05.467 ","End":"04:08.252","Text":"We\u0027ll take ln from both sides,"},{"Start":"04:08.252 ","End":"04:13.595","Text":"that\u0027ll be ln of e to the power of minus 2m,"},{"Start":"04:13.595 ","End":"04:18.185","Text":"that equals to ln of 0.5."},{"Start":"04:18.185 ","End":"04:24.525","Text":"Now, we can write this as minus 2m ln e,"},{"Start":"04:24.525 ","End":"04:28.280","Text":"and that equals to ln of 0.5."},{"Start":"04:28.280 ","End":"04:29.930","Text":"Now, why did we do this?"},{"Start":"04:29.930 ","End":"04:32.270","Text":"Well, we can use,"},{"Start":"04:32.270 ","End":"04:34.730","Text":"or we use this characteristics of"},{"Start":"04:34.730 ","End":"04:42.905","Text":"the logarithmic operator where we say that ln of x to the power of a,"},{"Start":"04:42.905 ","End":"04:47.470","Text":"well that equals to a times ln of x."},{"Start":"04:47.470 ","End":"04:52.625","Text":"This is a characteristic of the ln operator, the log operator."},{"Start":"04:52.625 ","End":"04:56.090","Text":"Now, once we have that, well,"},{"Start":"04:56.090 ","End":"04:58.385","Text":"ln of e equals 1,"},{"Start":"04:58.385 ","End":"05:00.905","Text":"so that\u0027ll be minus 2m,"},{"Start":"05:00.905 ","End":"05:05.820","Text":"that equals to ln of 0.5."},{"Start":"05:06.860 ","End":"05:09.510","Text":"Let\u0027s continue over here,"},{"Start":"05:09.510 ","End":"05:16.780","Text":"m equals ln of 0.5 divided by minus 2."},{"Start":"05:16.780 ","End":"05:25.925","Text":"Now, this equals to 0.347 and we\u0027re dealing in hours."},{"Start":"05:25.925 ","End":"05:31.370","Text":"This is a value of"},{"Start":"05:31.370 ","End":"05:34.520","Text":"x where 1/2 of the probability is above"},{"Start":"05:34.520 ","End":"05:37.922","Text":"this value and a 1/2 of the probability is below this value."},{"Start":"05:37.922 ","End":"05:44.160","Text":"That means that this is a median time for the occurrence of a malfunction in the system."}],"ID":13121},{"Watched":false,"Name":"Exercise 2","Duration":"4m 11s","ChapterTopicVideoID":12643,"CourseChapterTopicPlaylistID":245050,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12643.jpeg","UploadDate":"2018-07-23T16:09:57.2770000","DurationForVideoObject":"PT4M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.535","Text":"In this question we will be dealing with accidents."},{"Start":"00:02.535 ","End":"00:03.840","Text":"Now in a given highway,"},{"Start":"00:03.840 ","End":"00:06.510","Text":"the duration between accidents is distributed"},{"Start":"00:06.510 ","End":"00:11.220","Text":"exponentially with an expectation time of 24 hours and we\u0027re asked,"},{"Start":"00:11.220 ","End":"00:15.675","Text":"what\u0027s the standard deviation of the time until the next accident?"},{"Start":"00:15.675 ","End":"00:18.210","Text":"Let\u0027s define a random variable,"},{"Start":"00:18.210 ","End":"00:19.838","Text":"that\u0027ll be x,"},{"Start":"00:19.838 ","End":"00:30.390","Text":"and that\u0027s defined as the time until an accident."},{"Start":"00:31.790 ","End":"00:35.040","Text":"Now x is defined with"},{"Start":"00:35.040 ","End":"00:38.010","Text":"an exponential distribution where"},{"Start":"00:38.010 ","End":"00:42.205","Text":"Lambda is a variant that we still don\u0027t know what it is."},{"Start":"00:42.205 ","End":"00:46.760","Text":"Now we\u0027re given that the expectation time is"},{"Start":"00:46.760 ","End":"00:52.070","Text":"24 hours that means that the expectation of x is 24 hours."},{"Start":"00:52.070 ","End":"00:56.510","Text":"Now we know that since x is distributed with an exponential distribution,"},{"Start":"00:56.510 ","End":"00:59.560","Text":"the expectation is 1 divided by Lambda."},{"Start":"00:59.560 ","End":"01:06.450","Text":"That means that Lambda is 1 divided by 24 so let\u0027s just plug that in here."},{"Start":"01:06.450 ","End":"01:09.015","Text":"That\u0027ll be 1 divided by 24."},{"Start":"01:09.015 ","End":"01:10.800","Text":"Once we have that,"},{"Start":"01:10.800 ","End":"01:14.035","Text":"we can answer section A."},{"Start":"01:14.035 ","End":"01:18.450","Text":"Now we know that the variance of x,"},{"Start":"01:18.450 ","End":"01:22.545","Text":"well, that equals to 1 divided by Lambda squared."},{"Start":"01:22.545 ","End":"01:25.685","Text":"The standard deviation of x, well,"},{"Start":"01:25.685 ","End":"01:27.860","Text":"that\u0027s the square root of the variance and"},{"Start":"01:27.860 ","End":"01:30.200","Text":"that\u0027s the square root of this expression right"},{"Start":"01:30.200 ","End":"01:35.165","Text":"here so that would be the square root of 1 divided by Lambda squared."},{"Start":"01:35.165 ","End":"01:38.920","Text":"That equals to 1 divided by Lambda,"},{"Start":"01:38.920 ","End":"01:42.930","Text":"now we\u0027re given that 1 divided by Lambda,"},{"Start":"01:42.930 ","End":"01:45.485","Text":"well that\u0027s 24 hours."},{"Start":"01:45.485 ","End":"01:50.605","Text":"So the standard deviation is 24 hours."},{"Start":"01:50.605 ","End":"01:52.460","Text":"In this section we\u0027re asked,"},{"Start":"01:52.460 ","End":"01:57.380","Text":"what\u0027s the probability that the next accident will occur in less than 24 hours?"},{"Start":"01:57.380 ","End":"02:03.410","Text":"That means we\u0027re looking for the probability of x being less than or equal to 24."},{"Start":"02:03.410 ","End":"02:08.945","Text":"Now we know that the probability of x being less than or equal to any value t,"},{"Start":"02:08.945 ","End":"02:13.460","Text":"that equals to f of t and that equals to 1 minus e"},{"Start":"02:13.460 ","End":"02:18.620","Text":"to the power of minus Lambda times t for an exponential distribution."},{"Start":"02:18.620 ","End":"02:21.965","Text":"In our case, we\u0027re looking at"},{"Start":"02:21.965 ","End":"02:30.080","Text":"F at 24 and that equals to 1 minus e to the power of,"},{"Start":"02:30.080 ","End":"02:32.525","Text":"now what\u0027s Lambda, well,"},{"Start":"02:32.525 ","End":"02:40.370","Text":"that\u0027s 1 divided by 24 times t, where t is 24."},{"Start":"02:40.370 ","End":"02:45.035","Text":"Now that equals to 1 minus e to the power of minus 1,"},{"Start":"02:45.035 ","End":"02:49.565","Text":"and that equals to 0.632."},{"Start":"02:49.565 ","End":"02:56.705","Text":"That\u0027s the probability that the next accident will occur in less than 24 hours."},{"Start":"02:56.705 ","End":"02:58.520","Text":"In this section we\u0027re asked,"},{"Start":"02:58.520 ","End":"03:03.095","Text":"what\u0027s the probability that the next accident will occur within at least 2 days?"},{"Start":"03:03.095 ","End":"03:10.580","Text":"We\u0027re looking for the probability of x being greater than 48. Now why is that?"},{"Start":"03:10.580 ","End":"03:13.610","Text":"Well, 48 here, well that\u0027s 2 days."},{"Start":"03:13.610 ","End":"03:17.780","Text":"Don\u0027t forget that the units for extra hours and not days."},{"Start":"03:17.780 ","End":"03:20.420","Text":"So when we\u0027re asked about 2 days, well,"},{"Start":"03:20.420 ","End":"03:29.235","Text":"that\u0027s 48 hours and we\u0027re looking at at least something that\u0027s greater than."},{"Start":"03:29.235 ","End":"03:33.800","Text":"We\u0027re looking for the probability of x being greater than 48."},{"Start":"03:33.800 ","End":"03:40.775","Text":"Well, that equals to 1 minus the probability of x being less than or equal to 48."},{"Start":"03:40.775 ","End":"03:47.350","Text":"That means that we\u0027re looking at 1 minus f at 48."},{"Start":"03:47.350 ","End":"03:49.860","Text":"That\u0027s 1 minus,"},{"Start":"03:49.860 ","End":"03:51.600","Text":"now f at 48,"},{"Start":"03:51.600 ","End":"03:58.010","Text":"that\u0027s 1 minus e to the power of minus 1 divided by 24 times 48."},{"Start":"03:58.010 ","End":"04:05.300","Text":"That equals to e to the power of minus 2 and that equals to 0.135."},{"Start":"04:05.300 ","End":"04:12.240","Text":"That\u0027s the probability that the next accident will occur within at least 2 days."}],"ID":13122},{"Watched":false,"Name":"Exercise 3 - Parts a-b","Duration":"3m 14s","ChapterTopicVideoID":12644,"CourseChapterTopicPlaylistID":245050,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12644.jpeg","UploadDate":"2018-07-23T16:10:11.7400000","DurationForVideoObject":"PT3M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.435","Text":"The time that students work continuously on a computer"},{"Start":"00:03.435 ","End":"00:07.140","Text":"is exponentially distributed with an expectation of 30 minutes,"},{"Start":"00:07.140 ","End":"00:10.470","Text":"and we\u0027re asked, what are the chances that a student will work"},{"Start":"00:10.470 ","End":"00:14.775","Text":"continuously on the computer for less than 15 minutes?"},{"Start":"00:14.775 ","End":"00:18.540","Text":"First let\u0027s define our random variable that\u0027s X,"},{"Start":"00:18.540 ","End":"00:23.790","Text":"and that\u0027ll be equal to the time that a student"},{"Start":"00:23.790 ","End":"00:33.370","Text":"works continuously on a computer."},{"Start":"00:35.930 ","End":"00:39.285","Text":"This is in minutes,"},{"Start":"00:39.285 ","End":"00:44.420","Text":"and we\u0027re also given that X is distributed exponentially,"},{"Start":"00:44.420 ","End":"00:47.945","Text":"with Lambda being equal to something, we don\u0027t know what,"},{"Start":"00:47.945 ","End":"00:52.085","Text":"but we\u0027re given that the expectation of X,"},{"Start":"00:52.085 ","End":"00:55.255","Text":"that equals to 30 minutes."},{"Start":"00:55.255 ","End":"01:01.250","Text":"That means that if the expectation is 30 minutes on one hand and on the other hand,"},{"Start":"01:01.250 ","End":"01:06.335","Text":"the expectation of a random variable with an exponential distribution,"},{"Start":"01:06.335 ","End":"01:09.440","Text":"that equals to 1 divided by Lambda,"},{"Start":"01:09.440 ","End":"01:15.630","Text":"that means that Lambda then equals 1 divided by 30."},{"Start":"01:15.910 ","End":"01:18.665","Text":"We can put that here,"},{"Start":"01:18.665 ","End":"01:21.845","Text":"and we can solve for a."},{"Start":"01:21.845 ","End":"01:30.920","Text":"We\u0027re looking for the probability that X would be less than 15 minutes."},{"Start":"01:30.920 ","End":"01:35.645","Text":"That equals to F at 15."},{"Start":"01:35.645 ","End":"01:45.640","Text":"That equals to 1 minus e to the power of minus 1 divided by 30 times 15."},{"Start":"01:45.640 ","End":"01:53.030","Text":"That equals to 1 minus e to the power of minus 1/2,"},{"Start":"01:53.030 ","End":"02:00.030","Text":"and that equals to 0.393."},{"Start":"02:00.030 ","End":"02:01.930","Text":"This is the probability,"},{"Start":"02:01.930 ","End":"02:04.400","Text":"or the chance that the student will work continuously on"},{"Start":"02:04.400 ","End":"02:07.985","Text":"a computer for less than 15 minutes."},{"Start":"02:07.985 ","End":"02:11.660","Text":"In this section, we\u0027re asked what is the chances that the students work"},{"Start":"02:11.660 ","End":"02:15.185","Text":"on the computer will last between 15 and 30 minutes."},{"Start":"02:15.185 ","End":"02:23.495","Text":"We\u0027re looking for the probability of X being between 15 and 30 minutes."},{"Start":"02:23.495 ","End":"02:30.540","Text":"That equals to F at 30 minus F at 15."},{"Start":"02:30.540 ","End":"02:38.340","Text":"F of 30 is 1 minus e to the power of minus 1 divided by 30 times 30,"},{"Start":"02:38.390 ","End":"02:44.705","Text":"minus F at 15 we\u0027ve solved that in the last section,"},{"Start":"02:44.705 ","End":"02:47.975","Text":"that equals to 0.393."},{"Start":"02:47.975 ","End":"02:52.645","Text":"That will equal to"},{"Start":"02:52.645 ","End":"03:00.045","Text":"1 minus e to the power of minus 1 minus 0.393,"},{"Start":"03:00.045 ","End":"03:06.285","Text":"and that equals to 0.239."},{"Start":"03:06.285 ","End":"03:14.130","Text":"That\u0027s the probability of X being between 15 and 30 minutes."}],"ID":13123},{"Watched":false,"Name":"Exercise 3 - Parts c-d","Duration":"7m 13s","ChapterTopicVideoID":12645,"CourseChapterTopicPlaylistID":245050,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12645.jpeg","UploadDate":"2018-07-23T16:10:34.1600000","DurationForVideoObject":"PT7M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.565","Text":"In this section, we\u0027re asked that if a student has"},{"Start":"00:02.565 ","End":"00:05.700","Text":"already been working on the computer for more than 10 minutes,"},{"Start":"00:05.700 ","End":"00:10.410","Text":"what\u0027s the probability that his overall work duration will exceed 30 minutes?"},{"Start":"00:10.410 ","End":"00:13.905","Text":"Well, this looks like a conditional probability, so let\u0027s set it up."},{"Start":"00:13.905 ","End":"00:15.690","Text":"This is a probability,"},{"Start":"00:15.690 ","End":"00:17.340","Text":"now, what\u0027s given to us?"},{"Start":"00:17.340 ","End":"00:21.000","Text":"We\u0027re given that the student worked for more than 10 minutes,"},{"Start":"00:21.000 ","End":"00:24.435","Text":"so x here is greater than 10."},{"Start":"00:24.435 ","End":"00:26.190","Text":"What are we looking for?"},{"Start":"00:26.190 ","End":"00:31.460","Text":"We\u0027re looking for the probability that he\u0027ll work more than 30 minutes."},{"Start":"00:31.460 ","End":"00:35.035","Text":"That means that x is greater than 30."},{"Start":"00:35.035 ","End":"00:38.390","Text":"This is the probability that we have to calculate."},{"Start":"00:38.390 ","End":"00:41.075","Text":"Now, instead of using the conventional methods,"},{"Start":"00:41.075 ","End":"00:46.175","Text":"let\u0027s use 1 of the characteristics of the exponential distribution."},{"Start":"00:46.175 ","End":"00:49.415","Text":"That being that the distribution is memory less."},{"Start":"00:49.415 ","End":"00:51.380","Text":"Now, how is that defined?"},{"Start":"00:51.380 ","End":"00:57.043","Text":"Well, that\u0027s the probability of x being greater than a plus b,"},{"Start":"00:57.043 ","End":"01:01.010","Text":"given that x is greater than a,"},{"Start":"01:01.010 ","End":"01:06.330","Text":"well, that equals to the probability of x being greater than b."},{"Start":"01:06.580 ","End":"01:12.240","Text":"In our case, a here equals 10."},{"Start":"01:12.240 ","End":"01:14.380","Text":"What does b equal to?"},{"Start":"01:14.380 ","End":"01:19.185","Text":"Well, b equals to 30 minus 10, well that\u0027s 20."},{"Start":"01:19.185 ","End":"01:22.345","Text":"Let\u0027s plug these guys back into here."},{"Start":"01:22.345 ","End":"01:29.685","Text":"Well, that would be equal to the probability of x being greater than 10 plus 20,"},{"Start":"01:29.685 ","End":"01:34.540","Text":"that\u0027s 30, given that x is greater than 10, well,"},{"Start":"01:34.540 ","End":"01:40.059","Text":"that has to equal to the probability of x being greater than 20."},{"Start":"01:40.059 ","End":"01:46.360","Text":"Now, this probability equals to 1 minus the probability of x being"},{"Start":"01:46.360 ","End":"01:54.320","Text":"less than or equal to 20 or it equals to 1 minus F at 20."},{"Start":"01:57.380 ","End":"02:00.090","Text":"What\u0027s F of 20?"},{"Start":"02:00.090 ","End":"02:06.535","Text":"Well that\u0027s 1 minus e to the power of minus 1/30 times 20."},{"Start":"02:06.535 ","End":"02:12.338","Text":"All this expression equals to e"},{"Start":"02:12.338 ","End":"02:19.620","Text":"to the power of minus 2/3 and that equals to 0.513."},{"Start":"02:20.360 ","End":"02:24.125","Text":"Too easy. If that\u0027s the case,"},{"Start":"02:24.125 ","End":"02:29.045","Text":"let\u0027s now solve this using the conventional methods."},{"Start":"02:29.045 ","End":"02:32.060","Text":"Well, the convention method goes like this."},{"Start":"02:32.060 ","End":"02:36.955","Text":"In the denominator, we have the probability of what\u0027s given to us."},{"Start":"02:36.955 ","End":"02:41.545","Text":"That\u0027s the probability of x being greater than 10."},{"Start":"02:41.545 ","End":"02:43.100","Text":"What about the numerator?"},{"Start":"02:43.100 ","End":"02:46.355","Text":"Well, that\u0027s the probability of the intersect,"},{"Start":"02:46.355 ","End":"02:49.370","Text":"or that will be x being greater than 30,"},{"Start":"02:49.370 ","End":"02:53.465","Text":"intersect x being greater than 10."},{"Start":"02:53.465 ","End":"02:56.360","Text":"The intersect of these 2 guys,"},{"Start":"02:56.360 ","End":"03:03.440","Text":"so that\u0027s the probability of x being greater than 30 divided by this,"},{"Start":"03:03.440 ","End":"03:08.395","Text":"that\u0027s the probability of x being greater than 10."},{"Start":"03:08.395 ","End":"03:16.745","Text":"That equals to 1 minus the probability of x less than or equal to 30,"},{"Start":"03:16.745 ","End":"03:24.640","Text":"divided by 1 minus the probability of x being less than or equal to 10."},{"Start":"03:24.640 ","End":"03:29.115","Text":"That\u0027s 10. Let\u0027s continue on from here."},{"Start":"03:29.115 ","End":"03:40.060","Text":"That\u0027s 1 minus F at 30 divided by 1 minus F at 10."},{"Start":"03:41.960 ","End":"03:44.430","Text":"Now, what\u0027s F at 30?"},{"Start":"03:44.430 ","End":"03:51.340","Text":"That\u0027s 1 minus e to the power of minus 1/30 times 30."},{"Start":"03:51.340 ","End":"03:54.880","Text":"That\u0027s this guy right here."},{"Start":"03:55.130 ","End":"03:57.600","Text":"Now F at 10,"},{"Start":"03:57.600 ","End":"04:05.460","Text":"that\u0027s 1 minus e to the power of minus 1/30 times 10."},{"Start":"04:05.570 ","End":"04:16.935","Text":"That equals to e to the power of minus 1 divided by e to the power of minus 1/3."},{"Start":"04:16.935 ","End":"04:25.655","Text":"That equals to e to the power of minus 1 plus 1/3,"},{"Start":"04:25.655 ","End":"04:30.770","Text":"and that equals to e to the power of minus 2/3."},{"Start":"04:30.770 ","End":"04:33.185","Text":"As we saw right up here,"},{"Start":"04:33.185 ","End":"04:40.235","Text":"that equals to 0.513. Here we have it."},{"Start":"04:40.235 ","End":"04:44.090","Text":"We can use the characteristic of"},{"Start":"04:44.090 ","End":"04:48.830","Text":"the distribution of that being memoryless or we can use the conventional method."},{"Start":"04:48.830 ","End":"04:55.175","Text":"I personally prefer the conventional method because it\u0027s much more straightforward."},{"Start":"04:55.175 ","End":"04:58.460","Text":"In this section, we\u0027re asked what\u0027s the time under"},{"Start":"04:58.460 ","End":"05:01.625","Text":"which the student will complete 90 percent of his work."},{"Start":"05:01.625 ","End":"05:07.355","Text":"We\u0027re looking for the probability that x will be less than, let\u0027s say,"},{"Start":"05:07.355 ","End":"05:15.305","Text":"a specific value m and we want that probability to be equal to 0.9."},{"Start":"05:15.305 ","End":"05:19.460","Text":"Now, this is the definition of a percentile,"},{"Start":"05:19.460 ","End":"05:22.670","Text":"specifically the 90th percentile."},{"Start":"05:22.670 ","End":"05:29.620","Text":"M then would be the 90th percentile."},{"Start":"05:30.740 ","End":"05:33.254","Text":"If that\u0027s the case,"},{"Start":"05:33.254 ","End":"05:35.420","Text":"let\u0027s just see what\u0027s going on here."},{"Start":"05:35.420 ","End":"05:42.410","Text":"Well, that\u0027s the definition of F at point m and we want that to be equal to 0.9."},{"Start":"05:42.410 ","End":"05:45.330","Text":"Now, F of t,"},{"Start":"05:45.330 ","End":"05:49.755","Text":"if we recall that equals to 1 minus e to the power of"},{"Start":"05:49.755 ","End":"05:54.500","Text":"minus 1/30 t. Let\u0027s just plug in m here,"},{"Start":"05:54.500 ","End":"06:02.900","Text":"and that equals to 1 minus e to the power of minus 1/30 m and that has to equal to 0.9."},{"Start":"06:02.900 ","End":"06:05.825","Text":"Now, let\u0027s just switch things around here."},{"Start":"06:05.825 ","End":"06:12.245","Text":"That means that e to the power of minus 1/30 m and that equals to 0.1."},{"Start":"06:12.245 ","End":"06:14.615","Text":"Let\u0027s take ln of both sides."},{"Start":"06:14.615 ","End":"06:20.420","Text":"That\u0027ll be ln of e to the power of minus 1/30 m,"},{"Start":"06:20.420 ","End":"06:24.515","Text":"and that equals to ln of 0.1."},{"Start":"06:24.515 ","End":"06:26.960","Text":"Now, let\u0027s continue up here."},{"Start":"06:26.960 ","End":"06:32.290","Text":"That will be minus 1/30 m"},{"Start":"06:32.290 ","End":"06:38.250","Text":"times ln of e and that equals to ln of 0.1."},{"Start":"06:38.250 ","End":"06:40.620","Text":"In ln times ln of e,"},{"Start":"06:40.620 ","End":"06:42.150","Text":"well that equals to 1,"},{"Start":"06:42.150 ","End":"06:46.020","Text":"so that will be minus 1/30 m,"},{"Start":"06:46.020 ","End":"06:49.845","Text":"and that equals to ln of 0.1."},{"Start":"06:49.845 ","End":"06:56.605","Text":"M here would be equal to minus 30 times ln of 0.1."},{"Start":"06:56.605 ","End":"07:03.410","Text":"That means that n will be equal to 69.08."},{"Start":"07:03.410 ","End":"07:06.955","Text":"Now, this is the time in"},{"Start":"07:06.955 ","End":"07:12.850","Text":"minutes under which the student will complete 90 percent of his work."}],"ID":13124},{"Watched":false,"Name":"Exercise 4","Duration":"7m 57s","ChapterTopicVideoID":12646,"CourseChapterTopicPlaylistID":245050,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12646.jpeg","UploadDate":"2018-07-23T16:11:07.7270000","DurationForVideoObject":"PT7M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"In this question, we\u0027re given that an average of 4 patients per"},{"Start":"00:03.240 ","End":"00:06.840","Text":"hour arrive at the emergency room in a Poisson flow."},{"Start":"00:06.840 ","End":"00:10.935","Text":"Now, Betty the secretary comes to the emergency room and then we\u0027re asked,"},{"Start":"00:10.935 ","End":"00:13.590","Text":"what\u0027s the probability that the time that she"},{"Start":"00:13.590 ","End":"00:17.370","Text":"waits for the next patient is more than 20 minutes?"},{"Start":"00:17.370 ","End":"00:19.005","Text":"What\u0027s given to us?"},{"Start":"00:19.005 ","End":"00:24.270","Text":"First of all, we\u0027re given the rate of people or patients coming into the emergency room."},{"Start":"00:24.270 ","End":"00:27.945","Text":"That\u0027s 4 patients per hour."},{"Start":"00:27.945 ","End":"00:30.465","Text":"Now they come in at a Poisson flow,"},{"Start":"00:30.465 ","End":"00:36.090","Text":"but we\u0027re asked about the time between patients coming in,"},{"Start":"00:36.090 ","End":"00:40.385","Text":"that the time that she waits for the next patient."},{"Start":"00:40.385 ","End":"00:49.620","Text":"X, then we\u0027ll define this random variable X as the time between patients."},{"Start":"00:50.950 ","End":"00:54.710","Text":"Now, what do we know about the relationship between"},{"Start":"00:54.710 ","End":"00:58.550","Text":"the Poisson distribution and the exponential distribution?"},{"Start":"00:58.550 ","End":"01:04.310","Text":"We say that if we have events coming in at a Poisson flow then"},{"Start":"01:04.310 ","End":"01:07.460","Text":"the time between the events are distributed"},{"Start":"01:07.460 ","End":"01:11.060","Text":"with an exponential distribution with the same Lambda."},{"Start":"01:11.060 ","End":"01:14.840","Text":"That means that X here is distributed with"},{"Start":"01:14.840 ","End":"01:19.590","Text":"an exponential distribution with the same Lambda."},{"Start":"01:19.630 ","End":"01:25.270","Text":"What are we asked? We\u0027re asked for the probability of x,"},{"Start":"01:25.270 ","End":"01:29.725","Text":"the waiting time, that has to be greater than 20 minutes."},{"Start":"01:29.725 ","End":"01:32.020","Text":"Now, Lambda here is in hours,"},{"Start":"01:32.020 ","End":"01:33.610","Text":"it\u0027s not in minutes."},{"Start":"01:33.610 ","End":"01:37.870","Text":"Let\u0027s just convert the 20 minutes into units of hours."},{"Start":"01:37.870 ","End":"01:40.595","Text":"That would be 20 divided by 60."},{"Start":"01:40.595 ","End":"01:46.860","Text":"Now we have to take care about the units and convert where necessary."},{"Start":"01:46.860 ","End":"01:52.610","Text":"That means that this is the probability of x being greater than 1/3,"},{"Start":"01:52.610 ","End":"01:58.120","Text":"or 1 minus the probability of x being less than or equal to 1/3,"},{"Start":"01:58.120 ","End":"02:03.790","Text":"or that would equal to 1 minus F at 1/3."},{"Start":"02:03.790 ","End":"02:08.730","Text":"Now, since x is distributed with an exponential distribution,"},{"Start":"02:12.280 ","End":"02:14.915","Text":"now F at 1/3,"},{"Start":"02:14.915 ","End":"02:20.270","Text":"that equals to 1 minus e to the power minus Lambda."},{"Start":"02:20.270 ","End":"02:26.925","Text":"Lambda is 4, times this value right here. That\u0027s 1/3."},{"Start":"02:26.925 ","End":"02:31.895","Text":"Now that equals to e to the power of minus 4/3,"},{"Start":"02:31.895 ","End":"02:35.400","Text":"and that equals to 0.264."},{"Start":"02:37.780 ","End":"02:43.430","Text":"In this section, we\u0027re asked if Betty waited more than 15 minutes for the next patient,"},{"Start":"02:43.430 ","End":"02:48.490","Text":"what\u0027s the probability that she\u0027ll have to wait a total of more than 30 minutes?"},{"Start":"02:48.490 ","End":"02:52.100","Text":"This is a conditional probability. Let\u0027s set it up."},{"Start":"02:52.100 ","End":"02:54.215","Text":"We\u0027re looking for the probability."},{"Start":"02:54.215 ","End":"02:55.745","Text":"Now, what\u0027s given to us?"},{"Start":"02:55.745 ","End":"02:58.805","Text":"We\u0027re given that she waited more than 15 minutes."},{"Start":"02:58.805 ","End":"03:03.665","Text":"X here is greater than 15 divided by 60,"},{"Start":"03:03.665 ","End":"03:07.505","Text":"we\u0027re dealing in units of hours, not minutes."},{"Start":"03:07.505 ","End":"03:12.485","Text":"We\u0027re looking for the probability that she\u0027ll wait a total of more than 30 minutes."},{"Start":"03:12.485 ","End":"03:17.455","Text":"That means that x is greater than 30 divided by 60."},{"Start":"03:17.455 ","End":"03:21.410","Text":"That equals to the probability of x being greater than"},{"Start":"03:21.410 ","End":"03:26.675","Text":"1/2 given that x is greater than 1/4."},{"Start":"03:26.675 ","End":"03:31.250","Text":"Now, we can use 1 of the characteristics of"},{"Start":"03:31.250 ","End":"03:37.505","Text":"the exponential distribution that the distribution is memoryless."},{"Start":"03:37.505 ","End":"03:39.095","Text":"Now if we remember,"},{"Start":"03:39.095 ","End":"03:44.135","Text":"that\u0027s the probability of x being greater than a plus b,"},{"Start":"03:44.135 ","End":"03:47.270","Text":"given that x is greater than a,"},{"Start":"03:47.270 ","End":"03:52.240","Text":"that equals to the probability of x being greater than b."},{"Start":"03:52.240 ","End":"03:54.275","Text":"Now, in our case,"},{"Start":"03:54.275 ","End":"03:57.950","Text":"a here is 1/4 and a plus b is 1/2."},{"Start":"03:57.950 ","End":"04:01.130","Text":"That means that b is also 1/4."},{"Start":"04:01.130 ","End":"04:04.685","Text":"This probability right here,"},{"Start":"04:04.685 ","End":"04:11.640","Text":"that equals to the probability of x being greater than 1/4."},{"Start":"04:12.610 ","End":"04:16.700","Text":"That means that this equals to 1 minus"},{"Start":"04:16.700 ","End":"04:21.600","Text":"the probability of x being less than or equal to 1/4."},{"Start":"04:21.600 ","End":"04:27.195","Text":"Or 1 minus F at 1/4."},{"Start":"04:27.195 ","End":"04:32.580","Text":"Now, what\u0027s F at 1/4?"},{"Start":"04:32.580 ","End":"04:38.865","Text":"That\u0027s 1 minus e to the power of minus 4 times 1/4,"},{"Start":"04:38.865 ","End":"04:42.200","Text":"and that equals to e to the power of minus 1,"},{"Start":"04:42.200 ","End":"04:47.375","Text":"and that equals to 0.368."},{"Start":"04:47.375 ","End":"04:52.580","Text":"This is a probability that Betty will have to wait a total of"},{"Start":"04:52.580 ","End":"04:59.180","Text":"more than 30 minutes if she\u0027s already waited a total of 15 minutes."},{"Start":"04:59.180 ","End":"05:01.550","Text":"In section c, we\u0027re asked what\u0027s"},{"Start":"05:01.550 ","End":"05:04.850","Text":"the probability that more than 15 minutes will have passed between"},{"Start":"05:04.850 ","End":"05:07.190","Text":"the first and second patient and"},{"Start":"05:07.190 ","End":"05:10.475","Text":"less than 15 minutes between the second and third patients?"},{"Start":"05:10.475 ","End":"05:16.220","Text":"Now, the first thing that we need to understand is that in an exponential distribution,"},{"Start":"05:16.220 ","End":"05:19.700","Text":"the waiting time between patients is independent."},{"Start":"05:19.700 ","End":"05:22.015","Text":"Now, we\u0027ll use that later on."},{"Start":"05:22.015 ","End":"05:25.070","Text":"Let\u0027s define our random variables."},{"Start":"05:25.070 ","End":"05:29.545","Text":"Let\u0027s say X_1 is the waiting time"},{"Start":"05:29.545 ","End":"05:36.839","Text":"between the first and second patient,"},{"Start":"05:36.839 ","End":"05:46.470","Text":"and X_2 that\u0027s the waiting time between the second patient and the third patient."},{"Start":"05:47.180 ","End":"05:57.270","Text":"We know that X_1 is distributed with an exponential distribution where Lambda equals 4."},{"Start":"05:57.270 ","End":"05:59.070","Text":"Same thing with X_2."},{"Start":"05:59.070 ","End":"06:06.460","Text":"X_2 is distributed with an exponential distribution where Lambda equals 4."},{"Start":"06:06.500 ","End":"06:10.850","Text":"Here we\u0027re asked, what\u0027s the probability that"},{"Start":"06:10.850 ","End":"06:15.170","Text":"more than 15 minutes will have passed between the first and second patients."},{"Start":"06:15.170 ","End":"06:21.150","Text":"We\u0027re looking for the probability that X_1 is greater than 1/4,"},{"Start":"06:21.150 ","End":"06:23.850","Text":"15 minutes, that\u0027s 1/4 of an hour."},{"Start":"06:23.850 ","End":"06:30.905","Text":"Now, we\u0027ve calculated that in the last section, that equals 0.368."},{"Start":"06:30.905 ","End":"06:33.710","Text":"What about this guy right here?"},{"Start":"06:33.710 ","End":"06:39.885","Text":"Well, the probability that X_2 will be less than 1/4,"},{"Start":"06:39.885 ","End":"06:41.759","Text":"that\u0027s what we\u0027re asked for here,"},{"Start":"06:41.759 ","End":"06:46.130","Text":"and less than 50 minutes between the second and third patient,"},{"Start":"06:46.130 ","End":"06:50.250","Text":"that equals to 1 minus this guy right here,"},{"Start":"06:50.250 ","End":"06:58.210","Text":"that\u0027s 1 minus 0.368 and that equals to 0.632."},{"Start":"06:58.370 ","End":"07:01.220","Text":"Let\u0027s just put everything together."},{"Start":"07:01.220 ","End":"07:08.150","Text":"We\u0027re asked now for the probability that X_1 would be greater than"},{"Start":"07:08.150 ","End":"07:17.100","Text":"1/4 and that X_2 will be less than 1/4."},{"Start":"07:17.100 ","End":"07:20.340","Text":"Because X_1 and X_2 are independent,"},{"Start":"07:20.340 ","End":"07:25.130","Text":"this becomes the probability of X_1 being greater than"},{"Start":"07:25.130 ","End":"07:31.435","Text":"1/4 times the probability of X_2 being less than 1/4."},{"Start":"07:31.435 ","End":"07:36.965","Text":"That means that we have here 0.368"},{"Start":"07:36.965 ","End":"07:44.370","Text":"times 0.632 and that equals to 0.232."},{"Start":"07:45.230 ","End":"07:50.630","Text":"This is a probability that more than 15 minutes will have passed between"},{"Start":"07:50.630 ","End":"07:57.900","Text":"the first and second patient and less than 15 minutes between the 2nd and 3rd patients."}],"ID":13125},{"Watched":false,"Name":"Exercise 5 - Part a","Duration":"5m 5s","ChapterTopicVideoID":12647,"CourseChapterTopicPlaylistID":245050,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12647.jpeg","UploadDate":"2018-07-23T16:11:47.3970000","DurationForVideoObject":"PT5M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"In this question, the following diagram illustrates an electrical system that"},{"Start":"00:04.200 ","End":"00:08.415","Text":"has 4 identical electronic components that are operating simultaneously."},{"Start":"00:08.415 ","End":"00:10.530","Text":"This is component 1, component 2,"},{"Start":"00:10.530 ","End":"00:13.170","Text":"component 3, and component 4."},{"Start":"00:13.170 ","End":"00:18.105","Text":"The system will work properly when at least 1 component works properly."},{"Start":"00:18.105 ","End":"00:20.340","Text":"The lifespan of each component has"},{"Start":"00:20.340 ","End":"00:24.780","Text":"an exponential probability distribution with an average of 100 hours."},{"Start":"00:24.780 ","End":"00:27.540","Text":"We\u0027re asked, what\u0027s the probability of"},{"Start":"00:27.540 ","End":"00:31.890","Text":"the system operating properly for at least 100 hours?"},{"Start":"00:31.890 ","End":"00:34.590","Text":"Let\u0027s get started."},{"Start":"00:34.590 ","End":"00:41.015","Text":"First of all, we\u0027re asked for the probability that the system"},{"Start":"00:41.015 ","End":"00:49.040","Text":"will work for at least 100 hours."},{"Start":"00:49.040 ","End":"00:52.445","Text":"Now, instead of calculating this probability,"},{"Start":"00:52.445 ","End":"00:57.020","Text":"it\u0027ll be much easier for us to calculate the following probability."},{"Start":"00:57.020 ","End":"00:58.910","Text":"That\u0027s the complimentary set right here."},{"Start":"00:58.910 ","End":"01:05.090","Text":"That\u0027s 1 minus the probability that the system will"},{"Start":"01:05.090 ","End":"01:12.529","Text":"work for less than 100 hours."},{"Start":"01:12.529 ","End":"01:16.429","Text":"This is what we\u0027re going to calculate."},{"Start":"01:16.429 ","End":"01:18.455","Text":"Now, what are we given?"},{"Start":"01:18.455 ","End":"01:20.950","Text":"Well, we\u0027re given X_i."},{"Start":"01:20.950 ","End":"01:28.055","Text":"Now, X_i is the lifespan of component i."},{"Start":"01:28.055 ","End":"01:29.840","Text":"We\u0027re given that that has"},{"Start":"01:29.840 ","End":"01:32.915","Text":"an exponential distribution with"},{"Start":"01:32.915 ","End":"01:36.110","Text":"Lambda equals something and we don\u0027t know what that something is,"},{"Start":"01:36.110 ","End":"01:37.940","Text":"so let\u0027s calculate that right now."},{"Start":"01:37.940 ","End":"01:41.030","Text":"What else are we given? We\u0027re given that"},{"Start":"01:41.030 ","End":"01:46.700","Text":"the expectation of X_i or the lifespan of each 1 of the components,"},{"Start":"01:46.700 ","End":"01:49.550","Text":"that equals to 100 hours."},{"Start":"01:49.550 ","End":"01:51.995","Text":"Now, we know that the expectation of"},{"Start":"01:51.995 ","End":"01:55.550","Text":"any random variable that has an exponential distribution,"},{"Start":"01:55.550 ","End":"01:58.985","Text":"that equals to 1 divided by Lambda and in our case,"},{"Start":"01:58.985 ","End":"02:00.485","Text":"1 divided by Lambda,"},{"Start":"02:00.485 ","End":"02:01.865","Text":"that equals to 100."},{"Start":"02:01.865 ","End":"02:07.280","Text":"So Lambda then equals to 1/100,"},{"Start":"02:07.280 ","End":"02:09.080","Text":"that equals to 0.01."},{"Start":"02:09.080 ","End":"02:13.310","Text":"Let\u0027s just plug that in right here, that\u0027s 0.01."},{"Start":"02:13.310 ","End":"02:17.240","Text":"Now, what else are we given?"},{"Start":"02:17.240 ","End":"02:25.185","Text":"Well, we want to assume that X_i are independent."},{"Start":"02:25.185 ","End":"02:27.035","Text":"Now, if X_i are independent,"},{"Start":"02:27.035 ","End":"02:28.880","Text":"and why are we going to assume that?"},{"Start":"02:28.880 ","End":"02:30.319","Text":"Well the components are independent."},{"Start":"02:30.319 ","End":"02:34.535","Text":"Therefore, the lifespan of each component would be independent of each other."},{"Start":"02:34.535 ","End":"02:37.895","Text":"Let\u0027s now calculate this guy right here."},{"Start":"02:37.895 ","End":"02:39.800","Text":"Well, that\u0027ll be 1 minus,"},{"Start":"02:39.800 ","End":"02:43.970","Text":"now what\u0027s the probability of the system working less than 100 hours?"},{"Start":"02:43.970 ","End":"02:50.690","Text":"That means that each 1 of the components then has to work less than 100 hours."},{"Start":"02:50.690 ","End":"02:56.150","Text":"We\u0027re looking for the probability of X_1 being less than 100,"},{"Start":"02:56.150 ","End":"02:59.690","Text":"and X_2 being less than 100,"},{"Start":"02:59.690 ","End":"03:03.245","Text":"and X_3 being less than 100,"},{"Start":"03:03.245 ","End":"03:07.280","Text":"and X_4 being less than 100."},{"Start":"03:07.280 ","End":"03:14.580","Text":"Now, since all the X_is are independent of each other,"},{"Start":"03:14.580 ","End":"03:17.449","Text":"then this becomes the following."},{"Start":"03:17.449 ","End":"03:22.640","Text":"That is 1 minus the probability of X_1 being less than 100,"},{"Start":"03:22.640 ","End":"03:27.710","Text":"times the probability of X_2 being less than 100,"},{"Start":"03:27.710 ","End":"03:30.320","Text":"times the probability of X_3 being"},{"Start":"03:30.320 ","End":"03:37.115","Text":"less than 100 times the probability of X_4 being less than 100."},{"Start":"03:37.115 ","End":"03:42.560","Text":"What is the probability of X,"},{"Start":"03:42.560 ","End":"03:44.300","Text":"and it doesn\u0027t matter which 1,"},{"Start":"03:44.300 ","End":"03:46.175","Text":"being less than 100?"},{"Start":"03:46.175 ","End":"03:47.300","Text":"Why doesn\u0027t it matter?"},{"Start":"03:47.300 ","End":"03:51.815","Text":"Because each 1 of the X_i has the same distribution with the same Lambda."},{"Start":"03:51.815 ","End":"03:56.045","Text":"Let\u0027s calculate this probability right here."},{"Start":"03:56.045 ","End":"04:03.530","Text":"The probability of X being less than 100, in our case,"},{"Start":"04:03.530 ","End":"04:08.720","Text":"that equals to 1 minus e to the power of minus 0.01,"},{"Start":"04:08.720 ","End":"04:11.690","Text":"that\u0027s our Lambda, times 100."},{"Start":"04:11.690 ","End":"04:16.085","Text":"That equals to 1 minus e to the power of minus 1,"},{"Start":"04:16.085 ","End":"04:21.170","Text":"and that equals to 0.6321."},{"Start":"04:21.170 ","End":"04:27.620","Text":"Now, let\u0027s plug this value back into this long thing right here."},{"Start":"04:27.620 ","End":"04:30.905","Text":"That\u0027ll be equal to 1 minus, now,"},{"Start":"04:30.905 ","End":"04:33.710","Text":"the probability of X_1 being less than 100,"},{"Start":"04:33.710 ","End":"04:35.185","Text":"that\u0027s this guy right here,"},{"Start":"04:35.185 ","End":"04:40.580","Text":"0.6321, and the probability of X_2 being less than 100,"},{"Start":"04:40.580 ","End":"04:44.600","Text":"that also equals to 0.6321 and so on and so forth."},{"Start":"04:44.600 ","End":"04:51.095","Text":"That means that all of this equals to 1 minus 0.6321 to the fourth,"},{"Start":"04:51.095 ","End":"04:57.455","Text":"and that equals to 0.8403."},{"Start":"04:57.455 ","End":"05:05.790","Text":"Now, this then is the probability of the system working for at least 100 hours."}],"ID":13126},{"Watched":false,"Name":"Exercise 5 - Part b","Duration":"5m 43s","ChapterTopicVideoID":12648,"CourseChapterTopicPlaylistID":245050,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12648.jpeg","UploadDate":"2018-07-23T16:12:23.3530000","DurationForVideoObject":"PT5M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"In this section, the designers of the system are"},{"Start":"00:02.310 ","End":"00:05.100","Text":"considering adding another component to the system."},{"Start":"00:05.100 ","End":"00:08.415","Text":"The cost of the extra component is K dollars."},{"Start":"00:08.415 ","End":"00:10.590","Text":"If the system works less than 100 hours,"},{"Start":"00:10.590 ","End":"00:13.230","Text":"it causes damages totaling $A."},{"Start":"00:13.230 ","End":"00:18.525","Text":"We\u0027re asked what\u0027s the condition whereby adding another component is feasible."},{"Start":"00:18.525 ","End":"00:21.345","Text":"How do we approach this type of question?"},{"Start":"00:21.345 ","End":"00:24.735","Text":"The first thing that we need to do is we need to define"},{"Start":"00:24.735 ","End":"00:29.535","Text":"the expected loss of the system when the system has 4 components and 5 components,"},{"Start":"00:29.535 ","End":"00:36.085","Text":"and compare these losses to the cost of the extra component that we\u0027ll add."},{"Start":"00:36.085 ","End":"00:38.465","Text":"Let\u0027s get started."},{"Start":"00:38.465 ","End":"00:43.820","Text":"Let\u0027s define y as the loss of the system."},{"Start":"00:43.820 ","End":"00:47.930","Text":"Now, the loss of the system can have 2 values."},{"Start":"00:47.930 ","End":"00:51.995","Text":"It can either be 0 or it can be A."},{"Start":"00:51.995 ","End":"00:53.885","Text":"Now, why is that?"},{"Start":"00:53.885 ","End":"01:02.745","Text":"Well, it\u0027s 0 if the system works for at least 100 hours then there are no damages."},{"Start":"01:02.745 ","End":"01:06.090","Text":"Now, if the system works for less than 100 hours,"},{"Start":"01:06.090 ","End":"01:09.975","Text":"then we cause damages totaling A."},{"Start":"01:09.975 ","End":"01:13.295","Text":"Now, what\u0027s the probability of y?"},{"Start":"01:13.295 ","End":"01:16.620","Text":"Well, when n equals 4,"},{"Start":"01:16.620 ","End":"01:20.375","Text":"the probability of y equaling 0,"},{"Start":"01:20.375 ","End":"01:30.670","Text":"or the probability of the system working for at least 100 hours, that\u0027s 0.8403."},{"Start":"01:30.670 ","End":"01:36.945","Text":"What\u0027s the probability of the system incurring damages totaling A?"},{"Start":"01:36.945 ","End":"01:39.030","Text":"Well, that\u0027s 1 minus this,"},{"Start":"01:39.030 ","End":"01:44.190","Text":"so that\u0027ll be 0.1597."},{"Start":"01:44.190 ","End":"01:47.355","Text":"Now, what\u0027s the expectation of y?"},{"Start":"01:47.355 ","End":"01:52.950","Text":"Well, that equals to 0 times 0.8403,"},{"Start":"01:52.950 ","End":"01:56.880","Text":"plus 8 times 0.1597,"},{"Start":"01:56.880 ","End":"02:04.095","Text":"that equals to 0.1597 times A."},{"Start":"02:04.095 ","End":"02:09.510","Text":"Now let\u0027s take a look at the system where we have 5 components,"},{"Start":"02:09.510 ","End":"02:10.950","Text":"where n equals 5."},{"Start":"02:10.950 ","End":"02:13.440","Text":"Again, we still have y,"},{"Start":"02:13.440 ","End":"02:16.200","Text":"and we still have 2 values for y,"},{"Start":"02:16.200 ","End":"02:18.795","Text":"either 0 or A."},{"Start":"02:18.795 ","End":"02:23.410","Text":"We want to know what the probability of y for each 1 of the 2 values."},{"Start":"02:23.410 ","End":"02:26.270","Text":"Let\u0027s take a look at this value right here,"},{"Start":"02:26.270 ","End":"02:28.070","Text":"where y equals 0."},{"Start":"02:28.070 ","End":"02:30.005","Text":"Where y equals 0,"},{"Start":"02:30.005 ","End":"02:35.050","Text":"that\u0027s the probability of the system working for"},{"Start":"02:35.050 ","End":"02:40.685","Text":"at least 100 hours."},{"Start":"02:40.685 ","End":"02:44.870","Text":"Now, that equals to 1 minus the probability of"},{"Start":"02:44.870 ","End":"02:51.124","Text":"the system working for less than 100 hours."},{"Start":"02:51.124 ","End":"02:58.245","Text":"Now, what\u0027s the probability of 1 component working for less than 100 hours?"},{"Start":"02:58.245 ","End":"03:01.635","Text":"Well, that\u0027s 0.6321."},{"Start":"03:01.635 ","End":"03:05.250","Text":"Now we have 5 independent components."},{"Start":"03:05.250 ","End":"03:09.860","Text":"The probability of the whole system working for less than 100 hours then,"},{"Start":"03:09.860 ","End":"03:15.225","Text":"it\u0027ll be 0.6321^5."},{"Start":"03:15.225 ","End":"03:22.395","Text":"All this, 1 minus this expression equals to 0.8991."},{"Start":"03:22.395 ","End":"03:27.830","Text":"This is the probability of the system working for at least 100 hours."},{"Start":"03:27.830 ","End":"03:32.600","Text":"Let\u0027s just plug that in here, that\u0027s 0.8991."},{"Start":"03:32.600 ","End":"03:35.255","Text":"The probability of y equaling A,"},{"Start":"03:35.255 ","End":"03:43.880","Text":"or that the system will work less than 100 hours and cause damages in the order of A,"},{"Start":"03:43.880 ","End":"03:46.910","Text":"that\u0027ll be equal to, again,1 minus that."},{"Start":"03:46.910 ","End":"03:50.460","Text":"That equals to 0.1009."},{"Start":"03:50.720 ","End":"03:55.500","Text":"The expectation of y where we have 5 components,"},{"Start":"03:55.500 ","End":"03:57.810","Text":"that equals to 0 times"},{"Start":"03:57.810 ","End":"04:04.500","Text":"0.8991 plus"},{"Start":"04:04.500 ","End":"04:09.915","Text":"A times 0.1009,"},{"Start":"04:09.915 ","End":"04:15.900","Text":"and that equals to 0.1009A."},{"Start":"04:15.900 ","End":"04:21.680","Text":"Here we have the 2 expectations of the loss of the system."},{"Start":"04:21.680 ","End":"04:25.470","Text":"The 1 expectation where n equals 4,"},{"Start":"04:25.470 ","End":"04:28.355","Text":"and the other expectation when n equals 5."},{"Start":"04:28.355 ","End":"04:34.695","Text":"Now let\u0027s see how we can compare the 2 and check for feasibility."},{"Start":"04:34.695 ","End":"04:39.545","Text":"What we want to do is we want to take now the cost of the fifth component,"},{"Start":"04:39.545 ","End":"04:48.965","Text":"that will be k plus the expectation that the system will work less than 100 hours,"},{"Start":"04:48.965 ","End":"04:54.835","Text":"that\u0027s this expectation right here, plus 0.1009A."},{"Start":"04:54.835 ","End":"05:00.390","Text":"This has to be less than the expectation of"},{"Start":"05:00.390 ","End":"05:06.075","Text":"the system working less than 100 hours when we have 4 components."},{"Start":"05:06.075 ","End":"05:12.515","Text":"That means that it has to be less than 0.1597A."},{"Start":"05:12.515 ","End":"05:16.155","Text":"Now, let\u0027s just shuffle things around."},{"Start":"05:16.155 ","End":"05:26.325","Text":"That means that k has to be less than 0.0588A."},{"Start":"05:26.325 ","End":"05:28.505","Text":"This is a criteria."},{"Start":"05:28.505 ","End":"05:30.710","Text":"The cost of the extra component,"},{"Start":"05:30.710 ","End":"05:36.200","Text":"the fifth component has to be less than 0.0588 times the"},{"Start":"05:36.200 ","End":"05:43.530","Text":"damages that are caused when the system works for less than 100 hours."}],"ID":13127}],"Thumbnail":null,"ID":245050},{"Name":"Normal Probability","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"8m 1s","ChapterTopicVideoID":12649,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12649.jpeg","UploadDate":"2018-07-23T16:12:06.1430000","DurationForVideoObject":"PT8M1S","Description":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":"00:05.490","Text":"specifically the normal distribution."},{"Start":"00:05.490 ","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 180 and 181 centimeter,"},{"Start":"00:29.280 ","End":"00:32.835","Text":"well, within that 1-centimeter range,"},{"Start":"00:32.835 ","End":"00:35.980","Text":"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":"00:41.495","Text":"this is a common distribution and"},{"Start":"00:41.495 ","End":"00:44.210","Text":"the variables that have a normal distribution can"},{"Start":"00:44.210 ","End":"00:47.150","Text":"describe many things: production times,"},{"Start":"00:47.150 ","End":"00:52.290","Text":"baby\u0027s weight, 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":"01:18.230","Text":"As we can see here if we have"},{"Start":"01:18.230 ","End":"01:24.455","Text":"our random variable x and that\u0027s the x-axis and we can see here various bell curves."},{"Start":"01:24.455 ","End":"01:27.410","Text":"We have tall, thin ones, we have short,"},{"Start":"01:27.410 ","End":"01:36.365","Text":"fat ones and they\u0027re all defined by their averages Mu and standard deviation Sigma."},{"Start":"01:36.365 ","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":"01:46.040","Text":"a normal distribution with Mu and Sigma squared, where again,"},{"Start":"01:46.040 ","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":"02:19.505","Text":"Now, how did we do that?"},{"Start":"02:19.505 ","End":"02:24.770","Text":"Well, we took the density function of x and we\u0027ve calculated"},{"Start":"02:24.770 ","End":"02:32.180","Text":"the area under the density function from minus infinity till this value a."},{"Start":"02:32.180 ","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":"02:42.080","Text":"we need to transform any normal distribution to a standard normal distribution."},{"Start":"02:42.080 ","End":"02:44.300","Text":"Why is that? Well, again,"},{"Start":"02:44.300 ","End":"02:49.970","Text":"we\u0027ve seen that the bell curves can come in many shapes and sizes."},{"Start":"02:49.970 ","End":"02:51.680","Text":"Tall, thin ones, short,"},{"Start":"02:51.680 ","End":"02:53.810","Text":"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":"and 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":"03:35.320","Text":"the exceptions that the average is 0 and the standard deviation is 1."},{"Start":"03:35.320 ","End":"03:37.485","Text":"Now, it\u0027ll be denoted as Z."},{"Start":"03:37.485 ","End":"03:44.405","Text":"That means that if we take x and after the transformation or the standardization,"},{"Start":"03:44.405 ","End":"03:50.270","Text":"it will become Z and Z will be distributed with"},{"Start":"03:50.270 ","End":"03:58.292","Text":"a normal distribution where Mu equals 0 and Sigma squared equals 1 squared."},{"Start":"03:58.292 ","End":"04:00.800","Text":"We talked about the standardization process,"},{"Start":"04:00.800 ","End":"04:02.720","Text":"but how do we actually do it?"},{"Start":"04:02.720 ","End":"04:07.775","Text":"Well, the standardization process is carried out using the following formula."},{"Start":"04:07.775 ","End":"04:10.490","Text":"We have X, a random variable."},{"Start":"04:10.490 ","End":"04:12.725","Text":"We subtract Mu,"},{"Start":"04:12.725 ","End":"04:16.489","Text":"our average, and we divide by Sigma."},{"Start":"04:16.489 ","End":"04:18.905","Text":"That\u0027s the standard deviation."},{"Start":"04:18.905 ","End":"04:22.175","Text":"This transformation, we\u0027ll call that Z."},{"Start":"04:22.175 ","End":"04:30.495","Text":"Some schools will teach you that Z equals to x minus x bar divided by S,"},{"Start":"04:30.495 ","End":"04:36.570","Text":"where X-bar is comparable to Mu and S is comparable to Sigma,"},{"Start":"04:36.570 ","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":"04:47.769","Text":"I prefer to use Mu and Sigma."},{"Start":"04:47.769 ","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":"05:10.890","Text":"the value x deviates from the average 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":"05:29.870","Text":"Because once I\u0027ve transformed x that"},{"Start":"05:29.870 ","End":"05:34.820","Text":"has any normal distribution with any Mu and any Sigma into"},{"Start":"05:34.820 ","End":"05:38.405","Text":"a standardized normal distribution where"},{"Start":"05:38.405 ","End":"05:44.330","Text":"the normal distribution it has Mu equaling 0 and Sigma equaling 1,"},{"Start":"05:44.330 ","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 peek 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 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":"06:25.420","Text":"The grayed-out area right here, well,"},{"Start":"06:25.420 ","End":"06:27.624","Text":"that\u0027s the area underneath"},{"Start":"06:27.624 ","End":"06:33.130","Text":"the density function from minus infinity until Z or standard score."},{"Start":"06:33.130 ","End":"06:36.490","Text":"Well, this grayed-out area right here, well,"},{"Start":"06:36.490 ","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":"06:51.810","Text":"This is the table and we\u0027ll use it in a bit."},{"Start":"06:51.810 ","End":"06:56.825","Text":"Now, all this process might seem a little bit complicated, it 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":"07:05.060","Text":"In general, we describe the standardization process in the following schematic diagram."},{"Start":"07:05.060 ","End":"07:08.180","Text":"But 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":"07:17.685","Text":"we want to transform this."},{"Start":"07:17.685 ","End":"07:21.645","Text":"Or we want to standardize x into Z. How do we do that?"},{"Start":"07:21.645 ","End":"07:23.655","Text":"Well, we have X,"},{"Start":"07:23.655 ","End":"07:29.165","Text":"we subtract Mu and we divide by Sigma, and getting Z."},{"Start":"07:29.165 ","End":"07:33.320","Text":"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":"07:54.020","Text":"look up the value of the area underneath the density function."},{"Start":"07:54.020 ","End":"07:56.900","Text":"Well, that\u0027s our probability."},{"Start":"07:56.900 ","End":"08:01.500","Text":"Enough theory. Let\u0027s get to the exercise."}],"ID":13128},{"Watched":false,"Name":"Example - Part a","Duration":"8m 12s","ChapterTopicVideoID":12650,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12650.jpeg","UploadDate":"2018-07-23T16:12:47.3400000","DurationForVideoObject":"PT8M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.300","Text":"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":"00:12.490","Text":"100 grams and a standard deviation of 8 grams."},{"Start":"00:12.490 ","End":"00:17.850","Text":"We\u0027re asked what proportion of chocolate bars weigh less than 110 grams?"},{"Start":"00:17.850 ","End":"00:21.930","Text":"Well, the first thing that we need to know or to make sure,"},{"Start":"00:21.930 ","End":"00:25.350","Text":"is that we\u0027re dealing with a normal probability distribution,"},{"Start":"00:25.350 ","End":"00:27.750","Text":"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":"00:38.215","Text":"We don\u0027t know what we\u0027re dealing with."},{"Start":"00:38.215 ","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":"00:50.750","Text":"Well, let\u0027s call a random variable X,"},{"Start":"00:50.750 ","End":"00:58.500","Text":"and that\u0027ll be defined as the weight of the chocolate bar,"},{"Start":"00:59.150 ","End":"01:07.955","Text":"and we\u0027re given that that\u0027s distributed with a normal distribution where Mu equals 100,"},{"Start":"01:07.955 ","End":"01:10.670","Text":"that\u0027s 100 grams right here, that\u0027s the average,"},{"Start":"01:10.670 ","End":"01:16.350","Text":"and Sigma, the standard deviation, is 8."},{"Start":"01:16.350 ","End":"01:19.035","Text":"That\u0027s given right here."},{"Start":"01:19.035 ","End":"01:21.583","Text":"This is information that we\u0027re given,"},{"Start":"01:21.583 ","End":"01:25.970","Text":"and we\u0027re asked what proportion of chocolate bars weigh less than 110 grams?"},{"Start":"01:25.970 ","End":"01:30.620","Text":"The next steps that we need to do is we need to draw"},{"Start":"01:30.620 ","End":"01:36.235","Text":"the density function and map this information onto the density function."},{"Start":"01:36.235 ","End":"01:42.469","Text":"Let\u0027s do that. This is our density function right here,"},{"Start":"01:42.469 ","End":"01:44.570","Text":"and in the middle right here, well,"},{"Start":"01:44.570 ","End":"01:46.775","Text":"that\u0027s our average, that\u0027s Mu."},{"Start":"01:46.775 ","End":"01:51.240","Text":"Let\u0027s write this Mu then would be equal to 100."},{"Start":"01:51.240 ","End":"01:56.390","Text":"We\u0027re looking for the proportion of chocolate bars that weigh less than 110 grams,"},{"Start":"01:56.390 ","End":"02:01.705","Text":"so let\u0027s write here 110 grams."},{"Start":"02:01.705 ","End":"02:06.620","Text":"What does that mean? What proportion of chocolate bars weigh less than 110 grams?"},{"Start":"02:06.620 ","End":"02:15.730","Text":"We\u0027re looking for the area underneath the density function from minus infinity to 110,"},{"Start":"02:15.730 ","End":"02:20.790","Text":"and that would be defined as the proportion."},{"Start":"02:20.790 ","End":"02:23.715","Text":"Let\u0027s just draw this."},{"Start":"02:23.715 ","End":"02:27.200","Text":"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":"02:33.680","Text":"which equals to 1,"},{"Start":"02:33.680 ","End":"02:35.585","Text":"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":"02:52.115","Text":"we had X,"},{"Start":"02:52.115 ","End":"02:56.540","Text":"and we needed to standardize it into Z,"},{"Start":"02:56.540 ","End":"03:01.715","Text":"and only then could we figure out or calculate the probability."},{"Start":"03:01.715 ","End":"03:03.950","Text":"Now, Z, if we recall,"},{"Start":"03:03.950 ","End":"03:10.100","Text":"that\u0027s x minus Mu divided by Sigma, that\u0027s the transformation."},{"Start":"03:10.100 ","End":"03:13.160","Text":"When we\u0027re dealing with this probability,"},{"Start":"03:13.160 ","End":"03:19.460","Text":"we\u0027re looking for the probability of X minus Mu divided by Sigma,"},{"Start":"03:19.460 ","End":"03:25.700","Text":"that has to be less than 110 minus Mu divided by Sigma."},{"Start":"03:25.700 ","End":"03:33.440","Text":"Well, that equals to the probability now X minus Mu divided by Sigma. Well that\u0027s Z."},{"Start":"03:33.440 ","End":"03:36.870","Text":"That will be Z, which is less than."},{"Start":"03:36.870 ","End":"03:39.713","Text":"Now, Mu is 100,"},{"Start":"03:39.713 ","End":"03:41.300","Text":"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":"04:05.330","Text":"the probability of X being less than 110 is comparable,"},{"Start":"04:05.330 ","End":"04:09.970","Text":"or 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":"04:20.720","Text":"How does this look on this density function?"},{"Start":"04:20.720 ","End":"04:23.430","Text":"Well, let\u0027s draw this out."},{"Start":"04:24.620 ","End":"04:27.630","Text":"This then is the Z axis,"},{"Start":"04:27.630 ","End":"04:30.090","Text":"which is comparable to the X axis,"},{"Start":"04:30.090 ","End":"04:35.240","Text":"and we can see in the middle of the Z axis we have 0 which"},{"Start":"04:35.240 ","End":"04:40.790","Text":"is comparable to the 100 on the X axis."},{"Start":"04:40.790 ","End":"04:48.420","Text":"We\u0027ve calculated that Z being equal to 1.25,"},{"Start":"04:48.880 ","End":"04:55.265","Text":"and that\u0027s comparable to the 110 on the x-axis."},{"Start":"04:55.265 ","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":"05:13.870","Text":"Now, once we\u0027ve calculated this,"},{"Start":"05:13.870 ","End":"05:15.350","Text":"the Z value,"},{"Start":"05:15.350 ","End":"05:25.210","Text":"only then can we go to the standard normal distribution table and figure"},{"Start":"05:25.210 ","End":"05:35.855","Text":"out what the area is under the density function from minus infinity until Z."},{"Start":"05:35.855 ","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":"05:51.695","Text":"Now, let\u0027s look at the structure of this table before we can use this."},{"Start":"05:51.695 ","End":"05:54.265","Text":"Well, we have here,"},{"Start":"05:54.265 ","End":"05:59.330","Text":"as row names and column names, our Z values."},{"Start":"05:59.330 ","End":"06:05.240","Text":"In the middle we have what\u0027s called Phi of Z. Phi is"},{"Start":"06:05.240 ","End":"06:11.345","Text":"the area under the density function that we\u0027re looking for."},{"Start":"06:11.345 ","End":"06:14.255","Text":"In our case, Z or standard score,"},{"Start":"06:14.255 ","End":"06:17.740","Text":"was equal to 1.25."},{"Start":"06:17.740 ","End":"06:21.980","Text":"Let\u0027s try to find this value right here."},{"Start":"06:21.980 ","End":"06:29.425","Text":"Well, 1.25 means that we\u0027re looking for Z in the rows here, and we have 1.2."},{"Start":"06:29.425 ","End":"06:33.435","Text":"Now, we need that extra 0.05,"},{"Start":"06:33.435 ","End":"06:36.405","Text":"because we\u0027re at 1.25,"},{"Start":"06:36.405 ","End":"06:39.900","Text":"and that\u0027s this column right here."},{"Start":"06:39.900 ","End":"06:46.475","Text":"Let\u0027s take a look now at the intersection of this column and this row."},{"Start":"06:46.475 ","End":"06:49.985","Text":"Well, that\u0027s this value right here."},{"Start":"06:49.985 ","End":"06:54.210","Text":"That means that the area under"},{"Start":"06:54.210 ","End":"07:03.425","Text":"the standard normal density function from minus infinity until Z or standard score,"},{"Start":"07:03.425 ","End":"07:05.705","Text":"that equals to 1.25,"},{"Start":"07:05.705 ","End":"07:12.775","Text":"that area equals to 0.8944."},{"Start":"07:12.775 ","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":"07:25.490","Text":"We last left off right here,"},{"Start":"07:25.490 ","End":"07:31.645","Text":"where we\u0027re looking for the probability of Z being less than 1.25."},{"Start":"07:31.645 ","End":"07:37.140","Text":"Now we know that that equals to Phi of Z,"},{"Start":"07:37.140 ","End":"07:39.630","Text":"which is 1.25,"},{"Start":"07:39.630 ","End":"07:44.530","Text":"and we know that that equals to 0.8944."},{"Start":"07:45.470 ","End":"07:49.820","Text":"That is the proportion of"},{"Start":"07:49.820 ","End":"07:56.285","Text":"this yellow area in comparison with the total area under the curve."},{"Start":"07:56.285 ","End":"08:02.600","Text":"We\u0027re looking at the proportion of chocolate bars that weigh less than 110 grams."},{"Start":"08:02.600 ","End":"08:08.040","Text":"That\u0027s 89.44 percent,"},{"Start":"08:08.040 ","End":"08:12.360","Text":"and this is the answer for this question."}],"ID":13129},{"Watched":false,"Name":"Example - Parts b-c","Duration":"6m 27s","ChapterTopicVideoID":12652,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12652.jpeg","UploadDate":"2018-07-23T16:13:44.3070000","DurationForVideoObject":"PT6M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.275","Text":"In this section we\u0027re asked,"},{"Start":"00:01.275 ","End":"00:05.745","Text":"what proportion of 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 were asked the same question,"},{"Start":"00:09.615 ","End":"00:14.895","Text":"but we were 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":"00:28.365","Text":"Well, that\u0027s comparable,"},{"Start":"00:28.365 ","End":"00:35.055","Text":"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":"00:43.190","Text":"Well, if we take a look at the density function,"},{"Start":"00:43.190 ","End":"00:47.435","Text":"right now 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":"00:53.810","Text":"Now we know that the total area under the density function equals 1,"},{"Start":"00:53.810 ","End":"00:59.690","Text":"so we\u0027re looking at 1 minus the area shaded in yellow."},{"Start":"00:59.690 ","End":"01:06.380","Text":"Well, the area shaded in yellow is the probability of z being less than 1.25."},{"Start":"01:06.380 ","End":"01:11.990","Text":"As we said, we\u0027ve calculated then the last section that equals to 1 minus"},{"Start":"01:11.990 ","End":"01:19.967","Text":"0.8944 and that equals to 0.1056."},{"Start":"01:19.967 ","End":"01:25.080","Text":"This is the answer for this question."},{"Start":"01:25.080 ","End":"01:32.330","Text":"This is basically the proportion of chocolate bars that weigh more than 110 grams."},{"Start":"01:32.330 ","End":"01:35.180","Text":"Let\u0027s just sum this point up."},{"Start":"01:35.180 ","End":"01:42.575","Text":"Let\u0027s first of all draw the standardized normal density function, and here it is."},{"Start":"01:42.575 ","End":"01:49.415","Text":"Now, this is our z-axis and we\u0027re looking at a value of z, let\u0027s say a,"},{"Start":"01:49.415 ","End":"01:56.720","Text":"positive number, and 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":"02:20.915","Text":"then the area in white then is 1 minus Phi of a."},{"Start":"02:20.915 ","End":"02:27.092","Text":"Let\u0027s just take a look at the table to make sure that we\u0027re right."},{"Start":"02:27.092 ","End":"02:33.395","Text":"Here we can see the standard normal density function,"},{"Start":"02:33.395 ","End":"02:35.740","Text":"here we have our standard score z,"},{"Start":"02:35.740 ","End":"02:37.645","Text":"and here we have Phi of z,"},{"Start":"02:37.645 ","End":"02:39.455","Text":"the grayed-out area right here,"},{"Start":"02:39.455 ","End":"02:43.790","Text":"which is the area that\u0027s given to us by the table,"},{"Start":"02:43.790 ","End":"02:47.679","Text":"these values within the table right here."},{"Start":"02:47.679 ","End":"02:51.780","Text":"This then is the summation."},{"Start":"02:51.780 ","End":"02:53.320","Text":"To the left of a,"},{"Start":"02:53.320 ","End":"02:54.730","Text":"what\u0027s given in the table, well,"},{"Start":"02:54.730 ","End":"02:59.470","Text":"that\u0027s Phi of a and 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":"03:07.735","Text":"In this section we\u0027re asked what proportion of chocolate bars weigh less than 92 grams?"},{"Start":"03:07.735 ","End":"03:09.640","Text":"The first thing we want to do is,"},{"Start":"03:09.640 ","End":"03:12.976","Text":"we want to draw the density function."},{"Start":"03:12.976 ","End":"03:19.449","Text":"Here\u0027s our density function and we have here a value of 100,"},{"Start":"03:19.449 ","End":"03:21.325","Text":"that\u0027s our average,"},{"Start":"03:21.325 ","End":"03:28.750","Text":"and we\u0027re looking for the proportion of chocolate bars that weigh less than 92 grams."},{"Start":"03:28.880 ","End":"03:33.639","Text":"We\u0027re looking to calculate the area colored in red right here."},{"Start":"03:33.639 ","End":"03:40.384","Text":"That\u0027s the probability of x being less than 92 grams."},{"Start":"03:40.384 ","End":"03:43.310","Text":"Now if we recall the process, we have x,"},{"Start":"03:43.310 ","End":"03:45.788","Text":"we have to standardize this into z,"},{"Start":"03:45.788 ","End":"03:49.700","Text":"and only then can we calculate the probability."},{"Start":"03:49.700 ","End":"03:54.213","Text":"Now z is x minus Mu divided by Sigma."},{"Start":"03:54.213 ","End":"03:55.880","Text":"Let\u0027s do that right here."},{"Start":"03:55.880 ","End":"04:01.770","Text":"That\u0027s the probability of x minus Mu divided by Sigma,"},{"Start":"04:01.770 ","End":"04:05.585","Text":"well that has to be less than 92 minus, now,"},{"Start":"04:05.585 ","End":"04:10.390","Text":"Mu is 100 and Sigma is 8."},{"Start":"04:10.390 ","End":"04:17.745","Text":"That equals to the probability of z being less than minus 1."},{"Start":"04:17.745 ","End":"04:22.442","Text":"Well, that\u0027s Phi of minus 1."},{"Start":"04:22.442 ","End":"04:28.340","Text":"Now we can go to the standard table and see if we"},{"Start":"04:28.340 ","End":"04:34.945","Text":"can calculate the area under the curve where our standard score is minus 1."},{"Start":"04:34.945 ","End":"04:37.050","Text":"Let\u0027s sum up this point."},{"Start":"04:37.050 ","End":"04:41.290","Text":"Now, assume you have a standard normal density function,"},{"Start":"04:41.290 ","End":"04:47.019","Text":"that means that the axis here is the z axis,"},{"Start":"04:47.019 ","End":"04:49.505","Text":"and when we\u0027re talking about the z axis,"},{"Start":"04:49.505 ","End":"04:53.150","Text":"then 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":"05:15.900","Text":"You want to calculate Phi of minus a."},{"Start":"05:15.900 ","End":"05:17.870","Text":"So you go to the standard table,"},{"Start":"05:17.870 ","End":"05:23.560","Text":"but you can\u0027t find minus a because the table supports only positive values of z."},{"Start":"05:23.560 ","End":"05:27.070","Text":"Now, in order to resolve this issue,"},{"Start":"05:27.070 ","End":"05:30.745","Text":"you\u0027re going to have to use 1 of the characteristics of the density function,"},{"Start":"05:30.745 ","End":"05:35.950","Text":"and that is, that the density function is symmetrical around the average."},{"Start":"05:35.950 ","End":"05:37.795","Text":"What does that mean?"},{"Start":"05:37.795 ","End":"05:43.585","Text":"That means that this area right here from minus infinity to minus"},{"Start":"05:43.585 ","End":"05:50.860","Text":"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":"06:05.760","Text":"that equals to 1 minus Phi of a,"},{"Start":"06:05.760 ","End":"06:11.270","Text":"and this white area equals Phi of a."},{"Start":"06:11.270 ","End":"06:19.800","Text":"We know that Phi of minus a then equals 1 minus Phi of a,"},{"Start":"06:19.800 ","End":"06:27.540","Text":"and that also means that this white area also equals Phi of a."}],"ID":13130},{"Watched":false,"Name":"Example - Part d","Duration":"6m 52s","ChapterTopicVideoID":12651,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12651.jpeg","UploadDate":"2018-07-23T16:13:09.4000000","DurationForVideoObject":"PT6M52S","Description":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":"00:20.370","Text":"we were given the weight and we wanted to calculate the probability or the proportion,"},{"Start":"00:20.370 ","End":"00:22.800","Text":"now we have to work backwards."},{"Start":"00:22.800 ","End":"00:24.780","Text":"Why is that? Because we\u0027re given"},{"Start":"00:24.780 ","End":"00:31.425","Text":"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":"00:37.935","Text":"the production line way less than what weight?"},{"Start":"00:37.935 ","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":"01:14.900","Text":"Then we had to go to the standard table and"},{"Start":"01:14.900 ","End":"01:19.790","Text":"calculate or find the probability or the proportion."},{"Start":"01:19.790 ","End":"01:22.310","Text":"Now here, we\u0027re not doing that."},{"Start":"01:22.310 ","End":"01:24.155","Text":"Here we have to work backwards."},{"Start":"01:24.155 ","End":"01:26.825","Text":"Here we\u0027re given the proportion,"},{"Start":"01:26.825 ","End":"01:30.125","Text":"we have to go to the table and find"},{"Start":"01:30.125 ","End":"01:38.465","Text":"the standard score Z 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":"01:48.555","Text":"our average was 100,"},{"Start":"01:48.555 ","End":"01:54.645","Text":"and we\u0027re trying to find out X_0.9, the 90th percentile."},{"Start":"01:54.645 ","End":"01:59.870","Text":"Again, the 90th percentile is the value of x,"},{"Start":"01:59.870 ","End":"02:05.240","Text":"where 90 percent of the area underneath the density function,"},{"Start":"02:05.240 ","End":"02:11.455","Text":"we can find that from minus infinity to X_0.9,"},{"Start":"02:11.455 ","End":"02:15.570","Text":"that\u0027s this shaded area in yellow."},{"Start":"02:15.570 ","End":"02:21.435","Text":"Now, when we go to the z-axis,"},{"Start":"02:21.435 ","End":"02:26.270","Text":"we know that 0 is comparable to 100 on"},{"Start":"02:26.270 ","End":"02:31.100","Text":"the x-axis and what would try 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":"02:35.915","Text":"What\u0027s the standard score here?"},{"Start":"02:35.915 ","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":"02:54.330","Text":"Let\u0027s go to the standard table and see how we can find this value right here for Z."},{"Start":"02:54.950 ","End":"03:00.365","Text":"Here we see our standard normal probability distribution table."},{"Start":"03:00.365 ","End":"03:03.395","Text":"Now, we were given if we recall,"},{"Start":"03:03.395 ","End":"03:08.180","Text":"Phi of Z 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":"03:23.640","Text":"Now, where can we find that value?"},{"Start":"03:23.640 ","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":"03:35.270","Text":"Now, we have to find a value that\u0027s closest to 0.9."},{"Start":"03:35.270 ","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":"03:43.440","Text":"This is the value 0.8997,"},{"Start":"03:43.440 ","End":"03:46.805","Text":"that\u0027s the closest we can come to 0.9."},{"Start":"03:46.805 ","End":"03:51.305","Text":"So 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":"04:05.795","Text":"0.8997 is in the intersection of this row right here and this column."},{"Start":"04:05.795 ","End":"04:13.430","Text":"Z_0.9, that would be equal to 1.28."},{"Start":"04:13.430 ","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":"04:53.510","Text":"which is a little bit more exactly what we found out within the table,"},{"Start":"04:53.510 ","End":"04:56.570","Text":"so let\u0027s use this value right here."},{"Start":"04:56.570 ","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":"05:07.790","Text":"that\u0027s equal to 1.282."},{"Start":"05:07.790 ","End":"05:13.235","Text":"Again, let\u0027s just write this out, Z_0.9,"},{"Start":"05:13.235 ","End":"05:21.390","Text":"that\u0027s equal to 1.282 and we\u0027ll put that right here on the z-axis as well,"},{"Start":"05:21.390 ","End":"05:24.825","Text":"that equals to 1.282."},{"Start":"05:24.825 ","End":"05:32.090","Text":"Now, what\u0027s the corresponding x value to Z_0.9?"},{"Start":"05:32.090 ","End":"05:34.585","Text":"Well, if we recall,"},{"Start":"05:34.585 ","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":"05:54.034","Text":"now Mu is 100,"},{"Start":"05:54.034 ","End":"05:57.250","Text":"divided by 8, that\u0027s our sigma."},{"Start":"05:57.410 ","End":"06:01.415","Text":"Now we have an equation with 1 variable."},{"Start":"06:01.415 ","End":"06:02.855","Text":"It\u0027s very easy to solve,"},{"Start":"06:02.855 ","End":"06:04.670","Text":"basic math, let\u0027s do it."},{"Start":"06:04.670 ","End":"06:11.328","Text":"We\u0027ll divide both sides by 8 and we\u0027ll get 10.256,"},{"Start":"06:11.328 ","End":"06:18.920","Text":"that equals to X_0.9 minus 100 and that means that X_0.9, well,"},{"Start":"06:18.920 ","End":"06:24.745","Text":"that equals to 110.256,"},{"Start":"06:24.745 ","End":"06:26.850","Text":"and 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":"06:52.230","Text":"meaning that 90 percent of the chocolate bars weigh less than 110.256 grams."}],"ID":13131},{"Watched":false,"Name":"Exercise 1 - Part a","Duration":"6m 4s","ChapterTopicVideoID":12653,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12653.jpeg","UploadDate":"2018-07-23T16:14:27.3200000","DurationForVideoObject":"PT6M4S","Description":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":"00:07.215","Text":"a normal probability distribution with an average of"},{"Start":"00:07.215 ","End":"00:11.520","Text":"a 107 centimeters and a standard deviation of 10 centimeters."},{"Start":"00:11.520 ","End":"00:18.196","Text":"We\u0027re asked, what\u0027s the proportion of people who are shorter than 182.4 centimeters?"},{"Start":"00:18.196 ","End":"00:21.290","Text":"Well, the first thing that we have to make sure is that we\u0027re"},{"Start":"00:21.290 ","End":"00:25.100","Text":"dealing with a normal probability distribution and we are."},{"Start":"00:25.100 ","End":"00:27.200","Text":"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":"00:47.450","Text":"Now, let\u0027s call that X. X is defined as the height of people in centimeters."},{"Start":"00:47.450 ","End":"00:55.620","Text":"We\u0027re given that that\u0027s distributed with a normal distribution where Mu, the average,"},{"Start":"00:55.620 ","End":"01:00.260","Text":"is a 170 centimeters and the standard deviation,"},{"Start":"01:00.260 ","End":"01:04.470","Text":"well, that equals to 10 centimeters."},{"Start":"01:04.580 ","End":"01:09.005","Text":"Since this is a normal probability distribution,"},{"Start":"01:09.005 ","End":"01:12.130","Text":"let\u0027s just draw the density function."},{"Start":"01:12.130 ","End":"01:14.660","Text":"Here we have the density functions."},{"Start":"01:14.660 ","End":"01:21.635","Text":"Let\u0027s just put the data onto the chart."},{"Start":"01:21.635 ","End":"01:24.630","Text":"Mu is 170, so right here,"},{"Start":"01:24.630 ","End":"01:29.175","Text":"that\u0027s the average, that\u0027s 170 centimeters."},{"Start":"01:29.175 ","End":"01:35.615","Text":"Now, we\u0027re looking at the proportion of people who are shorter than a 182.4 centimeters."},{"Start":"01:35.615 ","End":"01:37.640","Text":"That\u0027s larger than the average."},{"Start":"01:37.640 ","End":"01:40.036","Text":"We\u0027ll put that here,"},{"Start":"01:40.036 ","End":"01:46.185","Text":"and that will be 182.4 centimeters, that\u0027s right here."},{"Start":"01:46.185 ","End":"01:52.640","Text":"We\u0027re looking also for the area under the density function from"},{"Start":"01:52.640 ","End":"02:00.190","Text":"minus infinity until 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":"02:10.130","Text":"What are the steps that we have to go through in order to figure out this area?"},{"Start":"02:10.130 ","End":"02:14.300","Text":"First of all, 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":"02:22.543","Text":"Then we go to the standard normal table,"},{"Start":"02:22.543 ","End":"02:25.850","Text":"and then find or calculate what p was,"},{"Start":"02:25.850 ","End":"02:28.265","Text":"what the proportion was."},{"Start":"02:28.265 ","End":"02:36.840","Text":"We know that x is 182.4 centimeters. What\u0027s z?"},{"Start":"02:36.840 ","End":"02:46.910","Text":"We\u0027re looking for the probability of x being less than 182.4 centimeters."},{"Start":"02:46.910 ","End":"02:54.690","Text":"That equals, remember that z is x minus Mu divided by Sigma,"},{"Start":"02:54.690 ","End":"03:00.950","Text":"so we\u0027re looking here for the probability of x minus Mu divided by"},{"Start":"03:00.950 ","End":"03:08.550","Text":"Sigma that has to be less than 182.4 minus, what\u0027s Mu?"},{"Start":"03:08.550 ","End":"03:09.990","Text":"That\u0027s a 170."},{"Start":"03:09.990 ","End":"03:15.665","Text":"170 divided by Sigma, where Sigma is 10."},{"Start":"03:15.665 ","End":"03:21.220","Text":"So we\u0027re looking for the probability then of z,"},{"Start":"03:21.220 ","End":"03:24.400","Text":"x minus Mu divided by Sigma, that\u0027s here,"},{"Start":"03:24.400 ","End":"03:28.580","Text":"that\u0027s z, has to be less than this expression right here."},{"Start":"03:28.580 ","End":"03:31.445","Text":"That\u0027s 182.4 minus 170."},{"Start":"03:31.445 ","End":"03:38.840","Text":"That\u0027s 12.4 divided by 10, so that\u0027s 1.24."},{"Start":"03:39.590 ","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":"03:57.880","Text":"So 182.4, that\u0027s comparable to this value which is 1.24."},{"Start":"03:58.700 ","End":"04:01.220","Text":"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":"04:10.871","Text":"that\u0027s 1.24 standard deviations away from 170."},{"Start":"04:10.871 ","End":"04:16.370","Text":"Now, we have to go to"},{"Start":"04:16.370 ","End":"04:24.450","Text":"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":"04:35.265","Text":"Let\u0027s go to the table."},{"Start":"04:35.265 ","End":"04:38.115","Text":"Here we are and this is the table,"},{"Start":"04:38.115 ","End":"04:40.320","Text":"the standard normal table,"},{"Start":"04:40.320 ","End":"04:43.710","Text":"and we\u0027re looking for z,"},{"Start":"04:43.710 ","End":"04:48.400","Text":"the standard score that equals 1.24."},{"Start":"04:50.510 ","End":"04:54.540","Text":"Let\u0027s just look right here."},{"Start":"04:54.540 ","End":"04:56.970","Text":"On the side, that\u0027s 1.2,"},{"Start":"04:56.970 ","End":"05:01.685","Text":"that\u0027s this line right here, and we need 1.24."},{"Start":"05:01.685 ","End":"05:04.130","Text":"That means that we\u0027re looking at this column."},{"Start":"05:04.130 ","End":"05:09.675","Text":"The intersection of this row and this column,"},{"Start":"05:09.675 ","End":"05:14.718","Text":"well, that equals to 0.8925."},{"Start":"05:14.718 ","End":"05:19.917","Text":"Phi of 1.24, well,"},{"Start":"05:19.917 ","End":"05:25.285","Text":"that equals to 0.8925."},{"Start":"05:25.285 ","End":"05:31.751","Text":"Let\u0027s now go back to our question. Here we go."},{"Start":"05:31.751 ","End":"05:35.970","Text":"The probability of z being less than 1.24, well,"},{"Start":"05:35.970 ","End":"05:40.755","Text":"that equals to Phi of 1.24,"},{"Start":"05:40.755 ","End":"05:44.980","Text":"and that equals to 0.8925."},{"Start":"05:47.510 ","End":"05:50.187","Text":"That\u0027s our answer."},{"Start":"05:50.187 ","End":"05:53.690","Text":"The proportion of people who are"},{"Start":"05:53.690 ","End":"05:57.880","Text":"shorter than 182.4 centimeters or if you wanted in percent,"},{"Start":"05:57.880 ","End":"06:03.120","Text":"that equals to 89.25 percent."}],"ID":13132},{"Watched":false,"Name":"Exercise 1 - Parts b-c","Duration":"5m 59s","ChapterTopicVideoID":12654,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12654.jpeg","UploadDate":"2018-07-23T16:14:44.7270000","DurationForVideoObject":"PT5M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.135","Text":"In this section, we\u0027re asked what\u0027s the proportion of people who are"},{"Start":"00:03.135 ","End":"00:06.450","Text":"taller than 190 centimeters?"},{"Start":"00:06.450 ","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":"00:23.880","Text":"Here we have our density function, that\u0027s our x-axis."},{"Start":"00:23.880 ","End":"00:33.370","Text":"The average is 170 and we\u0027re looking at 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":"00:54.650","Text":"We have to standardize x into a standard score, that\u0027ll be z."},{"Start":"00:54.650 ","End":"00:59.630","Text":"Then we have to go to our table and see what the proportion is."},{"Start":"00:59.630 ","End":"01:04.885","Text":"Well, here, our x is 190."},{"Start":"01:04.885 ","End":"01:10.555","Text":"The transformation of z is x minus Mu divided by Sigma."},{"Start":"01:10.555 ","End":"01:18.470","Text":"That equals then to the probability of x minus Mu divided by Sigma,"},{"Start":"01:18.470 ","End":"01:23.870","Text":"that has to be greater than 190 minus 170,"},{"Start":"01:23.870 ","End":"01:26.705","Text":"that\u0027s our Mu, divided by 10."},{"Start":"01:26.705 ","End":"01:34.325","Text":"That\u0027s our Sigma. 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, that\u0027s 190 minus 170,"},{"Start":"01:38.630 ","End":"01:42.065","Text":"that\u0027s 20 divided by 10, that\u0027s 2."},{"Start":"01:42.065 ","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":"02:06.705","Text":"It gives us the proportion of the area where Z is less than our value."},{"Start":"02:06.705 ","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":"02:24.710","Text":"that means that this equals to 1 minus Phi of 2."},{"Start":"02:24.710 ","End":"02:26.645","Text":"Excellent. So now,"},{"Start":"02:26.645 ","End":"02:30.030","Text":"we\u0027re ready to go to our table."},{"Start":"02:30.700 ","End":"02:40.715","Text":"Here\u0027s our table and we\u0027re looking for Z being equal to 2."},{"Start":"02:40.715 ","End":"02:45.505","Text":"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":"so Phi of 2 is this value right here."},{"Start":"03:01.430 ","End":"03:06.730","Text":"That equals to 0.9772."},{"Start":"03:06.730 ","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 190 centimeters,"},{"Start":"03:31.085 ","End":"03:33.770","Text":"or if you want this in percentages,"},{"Start":"03:33.770 ","End":"03:38.255","Text":"well, that\u0027s 2.28 percent."},{"Start":"03:38.255 ","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 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":"03:58.895","Text":"There\u0027s nothing to calculate here and why is that?"},{"Start":"03:58.895 ","End":"04:01.345","Text":"Well, the answer is 0."},{"Start":"04:01.345 ","End":"04:05.725","Text":"Again, let\u0027s just take a look at our density function."},{"Start":"04:05.725 ","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":"04:26.840","Text":"Well, the average is 170 and we\u0027re looking at 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 173.6 centimeters?"},{"Start":"04:33.860 ","End":"04:36.800","Text":"Then I would have asked about this area"},{"Start":"04:36.800 ","End":"04:40.175","Text":"right here 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":"04:46.849","Text":"greater than 173.6 centimeters,"},{"Start":"04:46.849 ","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 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 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 they\u0027re are around a 173.6."},{"Start":"05:23.900 ","End":"05:28.940","Text":"They\u0027re not exactly 173.6 centimeters tall."},{"Start":"05:28.940 ","End":"05:34.835","Text":"There\u0027s no person on earth that\u0027s exactly 173.6 centimeters tall."},{"Start":"05:34.835 ","End":"05:36.710","Text":"They\u0027re around 173,"},{"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 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":13133},{"Watched":false,"Name":"Exercise 1 - Parts d-e","Duration":"4m 13s","ChapterTopicVideoID":12655,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12655.jpeg","UploadDate":"2018-07-23T16:15:11.2900000","DurationForVideoObject":"PT4M13S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.350","Text":"In this section we\u0027re asked,"},{"Start":"00:01.350 ","End":"00:04.940","Text":"what\u0027s the proportion of people who are shorter than a 170 centimeters?"},{"Start":"00:04.940 ","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. We can see that a 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":"00:36.615","Text":"We can easily see that this equals to a 0.5."},{"Start":"00:36.615 ","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":"00:46.370","Text":"Where we have x, we have to standardize x into z."},{"Start":"00:46.370 ","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 a 170."},{"Start":"00:54.630 ","End":"00:56.565","Text":"Now, z,"},{"Start":"00:56.565 ","End":"01:00.680","Text":"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":"The probability of x being less than a 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":"01:23.575","Text":"Now, Mu is a 170 divided by 10."},{"Start":"01:23.575 ","End":"01:27.425","Text":"That means that we\u0027re looking for the probability of z,"},{"Start":"01:27.425 ","End":"01:30.560","Text":"x minus Mu divided by Sigma, that\u0027s z."},{"Start":"01:30.560 ","End":"01:35.850","Text":"It has to be less than 0."},{"Start":"01:35.850 ","End":"01:41.735","Text":"Now, why 0? Well, that\u0027s a 170 minus a 170 divided by 10, that\u0027s 0."},{"Start":"01:41.735 ","End":"01:46.065","Text":"We\u0027re looking at Phi of 0."},{"Start":"01:46.065 ","End":"01:50.875","Text":"Let\u0027s go now to the standard table to see what Phi of 0 is."},{"Start":"01:50.875 ","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 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 methods where we had x and then we standardized it,"},{"Start":"02:22.910 ","End":"02:27.850","Text":"we went to the table and looked up Phi of 0,"},{"Start":"02:27.850 ","End":"02:35.470","Text":"or we use the characteristic of symmetry around the average of the normal distribution."},{"Start":"02:35.470 ","End":"02:40.550","Text":"Since we\u0027re looking for the probability of x being less than the average,"},{"Start":"02:40.550 ","End":"02:45.630","Text":"well, it was easily seen that that was 0.5."},{"Start":"02:45.640 ","End":"02:47.870","Text":"In this section, we\u0027re asked,"},{"Start":"02:47.870 ","End":"02:51.740","Text":"what\u0027s the proportion of people who are at most a 170 centimeters tall?"},{"Start":"02:51.740 ","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 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":"03:22.295","Text":"we\u0027re looking at areas under the density function and the area"},{"Start":"03:22.295 ","End":"03:28.680","Text":"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":"03:35.945","Text":"That means that these 2 probabilities are the same with"},{"Start":"03:35.945 ","End":"03:40.460","Text":"or without the inclusion of 170 centers,"},{"Start":"03:40.460 ","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":"03:54.170","Text":"or the number of pieces of candy on our kitchen table or whatnot,"},{"Start":"03:54.170 ","End":"03:56.255","Text":"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":13134},{"Watched":false,"Name":"Exercise 2 - Part a","Duration":"5m 8s","ChapterTopicVideoID":12656,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12656.jpeg","UploadDate":"2018-07-23T16:15:44.1900000","DurationForVideoObject":"PT5M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.430","Text":"In this question, we assume that the time it takes for"},{"Start":"00:02.430 ","End":"00:04.800","Text":"a certain medication to take effect has"},{"Start":"00:04.800 ","End":"00:07.380","Text":"a normal probability distribution with an average of"},{"Start":"00:07.380 ","End":"00:10.210","Text":"30 minutes and a variance of 9 minutes."},{"Start":"00:10.210 ","End":"00:13.200","Text":"We\u0027re asked, what\u0027s the proportion of cases where"},{"Start":"00:13.200 ","End":"00:16.740","Text":"the medication takes longer than an hour to work?"},{"Start":"00:16.740 ","End":"00:20.775","Text":"The first thing that we need to do is define a random variable."},{"Start":"00:20.775 ","End":"00:22.545","Text":"That\u0027ll be x,"},{"Start":"00:22.545 ","End":"00:29.715","Text":"and x would be defined as the time it takes for the medication to take effect."},{"Start":"00:29.715 ","End":"00:31.395","Text":"Let\u0027s just write this out,"},{"Start":"00:31.395 ","End":"00:37.780","Text":"that\u0027s the time to take effect."},{"Start":"00:38.780 ","End":"00:43.050","Text":"That has a normal distribution,"},{"Start":"00:43.050 ","End":"00:47.115","Text":"where Mu equals to 30 minutes."},{"Start":"00:47.115 ","End":"00:49.995","Text":"What about Sigma squared?"},{"Start":"00:49.995 ","End":"00:52.350","Text":"Sigma squared is the variance,"},{"Start":"00:52.350 ","End":"00:57.165","Text":"Sigma squared, that equals to 9 minutes."},{"Start":"00:57.165 ","End":"01:04.085","Text":"Sigma then equals to the square root of 9, which equals to 3."},{"Start":"01:04.085 ","End":"01:06.860","Text":"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":"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":"01:23.885","Text":"Let\u0027s just draw the density function and plug in the numbers on the density function."},{"Start":"01:23.885 ","End":"01:26.135","Text":"Here\u0027s our density function."},{"Start":"01:26.135 ","End":"01:29.180","Text":"The average, Mu equals 30,"},{"Start":"01:29.180 ","End":"01:31.730","Text":"that\u0027s this thing right here."},{"Start":"01:31.730 ","End":"01:34.640","Text":"That\u0027s 30 on the x-axis,"},{"Start":"01:34.640 ","End":"01:37.670","Text":"and we\u0027re looking for the proportion of"},{"Start":"01:37.670 ","End":"01:40.890","Text":"cases where the medication takes longer than an hour to work."},{"Start":"01:40.890 ","End":"01:42.930","Text":"An hour is 60 minutes."},{"Start":"01:42.930 ","End":"01:45.330","Text":"Let\u0027s write this thing right here."},{"Start":"01:45.330 ","End":"01:46.515","Text":"That\u0027s 60 minutes,"},{"Start":"01:46.515 ","End":"01:53.720","Text":"and we\u0027re looking for the probability where x is greater than 60 minutes."},{"Start":"01:53.720 ","End":"01:56.760","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":"02:05.600","Text":"We have x, we have to standardize it into the standard score,"},{"Start":"02:05.600 ","End":"02:11.010","Text":"that\u0027s z, and then we have to go to the table and find the proportion."},{"Start":"02:11.140 ","End":"02:16.030","Text":"X here, that equals to 60,"},{"Start":"02:16.030 ","End":"02:21.000","Text":"and z, that\u0027s x minus Mu divided by Sigma."},{"Start":"02:21.000 ","End":"02:22.575","Text":"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":"02:33.240","Text":"that equals to the probability of x minus Mu divided by Sigma,"},{"Start":"02:33.240 ","End":"02:36.300","Text":"that\u0027s greater than 16 minus,"},{"Start":"02:36.300 ","End":"02:40.425","Text":"Mu equals 30 divided by Sigma."},{"Start":"02:40.425 ","End":"02:43.000","Text":"Sigma equals 3."},{"Start":"02:43.310 ","End":"02:48.330","Text":"That equals to the probability of z,"},{"Start":"02:48.330 ","End":"02:51.135","Text":"x minus Mu divided by Sigma, that\u0027s z,"},{"Start":"02:51.135 ","End":"02:54.420","Text":"has to be greater than 60 minus 30,"},{"Start":"02:54.420 ","End":"02:57.630","Text":"that\u0027s 30 divided by 3, that\u0027s 10."},{"Start":"02:57.630 ","End":"03:01.620","Text":"On the z-axis right here,"},{"Start":"03:01.620 ","End":"03:08.200","Text":"60 on the x-axis would be comparable to 10 on the z-axis."},{"Start":"03:08.200 ","End":"03:18.625","Text":"Since our standard table shows us the probability of z being less than a specific value,"},{"Start":"03:18.625 ","End":"03:22.510","Text":"then let\u0027s just convert this probability and that\u0027ll 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":"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":"03:55.610 ","End":"04:00.915","Text":"Here, the highest value z is 3.4,"},{"Start":"04:00.915 ","End":"04:05.540","Text":"and in the whole of the table, that\u0027s 3.49."},{"Start":"04:05.540 ","End":"04:08.610","Text":"That\u0027s this value right here."},{"Start":"04:08.680 ","End":"04:12.870","Text":"Anything above this value,"},{"Start":"04:12.870 ","End":"04:15.480","Text":"3.49 or 3.5,"},{"Start":"04:15.480 ","End":"04:19.980","Text":"it\u0027s so close to 1, it\u0027s insignificant."},{"Start":"04:19.980 ","End":"04:23.480","Text":"That means that if we\u0027re looking for Phi of 10,"},{"Start":"04:23.480 ","End":"04:27.775","Text":"which is way higher than 3.5,"},{"Start":"04:27.775 ","End":"04:31.710","Text":"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":"04:38.705","Text":"let\u0027s go back to our question. Here we go."},{"Start":"04:38.705 ","End":"04:42.960","Text":"This then equals to 1 minus,"},{"Start":"04:42.960 ","End":"04:45.610","Text":"Phi of 10 as we said,"},{"Start":"04:45.610 ","End":"04:48.200","Text":"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":"04:55.754","Text":"that equals to 0."},{"Start":"04:55.754 ","End":"04:59.300","Text":"What we\u0027re looking at is the proportion of"},{"Start":"04:59.300 ","End":"05:02.720","Text":"cases where the medication takes longer than an hour to work,"},{"Start":"05:02.720 ","End":"05:06.390","Text":"that\u0027s 0 or 0 percent."}],"ID":13135},{"Watched":false,"Name":"Exercise 2 - Parts b-c","Duration":"6m 2s","ChapterTopicVideoID":12658,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12658.jpeg","UploadDate":"2018-07-23T16:16:29.0330000","DurationForVideoObject":"PT6M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.880","Text":"In this section, we\u0027re asked what\u0027s the proportion of cases with"},{"Start":"00:02.880 ","End":"00:06.750","Text":"the medication takes between 35 and 37 minutes to work."},{"Start":"00:06.750 ","End":"00:16.875","Text":"We\u0027re looking for the probability where x is between 35 and 37 minutes."},{"Start":"00:16.875 ","End":"00:22.915","Text":"Let\u0027s just draw our density function and plug in these numbers."},{"Start":"00:22.915 ","End":"00:26.010","Text":"Here we have our density function."},{"Start":"00:26.010 ","End":"00:27.720","Text":"The average is 30."},{"Start":"00:27.720 ","End":"00:31.695","Text":"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":"Now, 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":"00:49.745","Text":"we have to standardize it into standard scores, that\u0027s z."},{"Start":"00:49.745 ","End":"00:56.425","Text":"Then we have to go to the standard table to calculate the proportions."},{"Start":"00:56.425 ","End":"01:03.755","Text":"We have to standardize x for both values for 35 and 37. Let\u0027s do that."},{"Start":"01:03.755 ","End":"01:07.160","Text":"z for x equals 35,"},{"Start":"01:07.160 ","End":"01:15.025","Text":"well 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":"01:21.788","Text":"That equals to 5 divided by 3,"},{"Start":"01:21.788 ","End":"01:25.580","Text":"well that\u0027s 1.67 approximately."},{"Start":"01:25.580 ","End":"01:30.015","Text":"Now, z for 37,"},{"Start":"01:30.015 ","End":"01:35.120","Text":"well that equals to 37 minus 30 divided by 3."},{"Start":"01:35.120 ","End":"01:42.295","Text":"Again, that\u0027s 7 divided by 3, that\u0027s 2.33."},{"Start":"01:42.295 ","End":"01:49.370","Text":"What\u0027s the correlation here or what\u0027s comparable to these numbers?"},{"Start":"01:49.370 ","End":"01:53.210","Text":"What 35 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":"02:01.070","Text":"Now, how do we calculate this area right here?"},{"Start":"02:01.070 ","End":"02:08.205","Text":"Well, we\u0027re going to have to take the area from minus infinity to"},{"Start":"02:08.205 ","End":"02:17.000","Text":"2.33 and we have to subtract the area from minus infinity to 1.67."},{"Start":"02:17.000 ","End":"02:22.600","Text":"That\u0027ll give us this area right here. Let\u0027s do that."},{"Start":"02:22.600 ","End":"02:27.225","Text":"We have probability of 35,"},{"Start":"02:27.225 ","End":"02:29.940","Text":"being less than x being less than 37,"},{"Start":"02:29.940 ","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":"02:44.850","Text":"z being between 1.67 and 2.33."},{"Start":"02:45.730 ","End":"02:53.195","Text":"As we said, that equals to Phi of 2.33."},{"Start":"02:53.195 ","End":"02:58.385","Text":"That\u0027s this area right here from minus infinity to 2.33."},{"Start":"02:58.385 ","End":"03:05.270","Text":"We have to subtract Phi of 1.67."},{"Start":"03:05.270 ","End":"03:09.080","Text":"That\u0027s this area from minus infinity to 1.67."},{"Start":"03:09.080 ","End":"03:13.175","Text":"What\u0027s left is this area right here."},{"Start":"03:13.175 ","End":"03:16.650","Text":"Now we\u0027re ready to go to our table."},{"Start":"03:17.050 ","End":"03:27.690","Text":"For z equaling 1.67,"},{"Start":"03:27.690 ","End":"03:32.295","Text":"Phi of 1.67 equals, well,"},{"Start":"03:32.295 ","End":"03:38.010","Text":"here\u0027s 1.6 and here\u0027s the 7."},{"Start":"03:38.010 ","End":"03:42.260","Text":"The intersection right here is less 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":"03:59.165","Text":"Well, Phi of 2.33, well that equals,"},{"Start":"03:59.165 ","End":"04:07.545","Text":"let\u0027s just scroll down here 2.33 and that\u0027s 2.3 right here."},{"Start":"04:07.545 ","End":"04:13.155","Text":"That\u0027s 0123, that\u0027s this value right here."},{"Start":"04:13.155 ","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":"04:33.335","Text":"that 0.9901 minus Phi of 1.67,"},{"Start":"04:33.335 ","End":"04:43.095","Text":"well that\u0027s 0.9525 and that equals to 0.0376."},{"Start":"04:43.095 ","End":"04:48.710","Text":"The proportion of cases where the medication takes between 35 and 37 minutes is"},{"Start":"04:48.710 ","End":"04:56.400","Text":"0.0376 or 3.76 percent."},{"Start":"04:56.400 ","End":"04:59.450","Text":"In this section we\u0027re asked what are the chances that"},{"Start":"04:59.450 ","End":"05:02.440","Text":"the medication will help after exactly 36 minutes."},{"Start":"05:02.440 ","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":"Well, we don\u0027t have to do a lot of calculations here."},{"Start":"05:12.050 ","End":"05:15.500","Text":"We know that since this is a continuous variable,"},{"Start":"05:15.500 ","End":"05:18.185","Text":"we\u0027re dealing with a continuous distribution, well,"},{"Start":"05:18.185 ","End":"05:24.655","Text":"the probability of x being equal to something that\u0027s always equals to 0."},{"Start":"05:24.655 ","End":"05:27.560","Text":"Again, let\u0027s just take a look why."},{"Start":"05:27.560 ","End":"05:30.380","Text":"We have our density function here."},{"Start":"05:30.380 ","End":"05:32.900","Text":"This is our average 30."},{"Start":"05:32.900 ","End":"05:36.380","Text":"We\u0027re looking at 36."},{"Start":"05:36.380 ","End":"05:39.680","Text":"Now, what\u0027s the probability of x being equal 36?"},{"Start":"05:39.680 ","End":"05:44.300","Text":"What\u0027s the area under the density function at exactly 36?"},{"Start":"05:44.300 ","End":"05:45.320","Text":"The point 36, well,"},{"Start":"05:45.320 ","End":"05:47.110","Text":"that equals to 0."},{"Start":"05:47.110 ","End":"05:50.660","Text":"Again, let\u0027s remember whenever we have a continuous"},{"Start":"05:50.660 ","End":"05:54.020","Text":"random variable and we\u0027re asked for the probability that,"},{"Start":"05:54.020 ","End":"05:58.910","Text":"that variable equals something that always equals to 0."},{"Start":"05:58.910 ","End":"06:02.430","Text":"The probability always equals 0."}],"ID":13136},{"Watched":false,"Name":"Exercise 2 - Part d","Duration":"5m 46s","ChapterTopicVideoID":12657,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12657.jpeg","UploadDate":"2018-07-23T16:16:09.9500000","DurationForVideoObject":"PT5M46S","Description":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":"00:19.395","Text":"Here\u0027s our density function."},{"Start":"00:19.395 ","End":"00:25.480","Text":"On the x-axis we have our average, that\u0027s 30."},{"Start":"00:25.580 ","End":"00:28.130","Text":"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":"00:33.865","Text":"that\u0027s the area under the graph,"},{"Start":"00:33.865 ","End":"00:36.650","Text":"where the time taken by the medication to work deviates"},{"Start":"00:36.650 ","End":"00:39.140","Text":"from the average by less than 3 minutes,"},{"Start":"00:39.140 ","End":"00:44.120","Text":"that means that we\u0027re looking for 30 plus minus 3 minutes."},{"Start":"00:44.120 ","End":"00:46.390","Text":"That\u0027s 27 in this side,"},{"Start":"00:46.390 ","End":"00:48.870","Text":"and 33 in 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":"01:11.485","Text":"that\u0027s the proportion that we\u0027re looking for."},{"Start":"01:11.485 ","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":"01:21.590","Text":"All we have to do is we have to calculate the area from minus infinity to"},{"Start":"01:21.590 ","End":"01:32.170","Text":"33 and subtract from that the area from minus infinity to 27."},{"Start":"01:32.170 ","End":"01:36.395","Text":"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":"01:50.360","Text":"These are the steps that we have to take to standardize the values of x."},{"Start":"01:50.360 ","End":"01:54.650","Text":"We have x, 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":"02:02.940","Text":"Now here, x, we have 2 values of x."},{"Start":"02:02.940 ","End":"02:05.250","Text":"We have 27 and 33."},{"Start":"02:05.250 ","End":"02:09.980","Text":"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":"02:25.975","Text":"Now, that means that we have the probability here of x minus Mu divided by Sigma."},{"Start":"02:25.975 ","End":"02:30.900","Text":"That has to be between 33 minus Mu,"},{"Start":"02:30.900 ","End":"02:36.375","Text":"Mu is 30 divided by Sigma, where Sigma is 3."},{"Start":"02:36.375 ","End":"02:41.660","Text":"On the other side that will be 27 minus 30 divided by"},{"Start":"02:41.660 ","End":"02:47.030","Text":"3 and that equals to the probability of,"},{"Start":"02:47.030 ","End":"02:50.095","Text":"x minus Mu divided by Sigma, that\u0027s z."},{"Start":"02:50.095 ","End":"02:54.900","Text":"Z then has to be between 33 minus 30."},{"Start":"02:54.900 ","End":"02:57.345","Text":"Well, that\u0027s 3 divided by 3, that\u0027s 1,"},{"Start":"02:57.345 ","End":"03:02.815","Text":"and 27 minus 30 divided by 3, that\u0027s minus 1."},{"Start":"03:02.815 ","End":"03:09.485","Text":"These values are comparable to 27 and 33 on the z scale."},{"Start":"03:09.485 ","End":"03:12.890","Text":"For 27, we have minus 1 and for 33,"},{"Start":"03:12.890 ","End":"03:15.030","Text":"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":"03:29.450","Text":"we have to subtract the areas from minus infinity to 1 on the z scale,"},{"Start":"03:29.450 ","End":"03:31.250","Text":"and we have to subtract from that,"},{"Start":"03:31.250 ","End":"03:34.475","Text":"the area from minus infinity to minus 1."},{"Start":"03:34.475 ","End":"03:42.940","Text":"That means that we\u0027re looking at Phi of 1 minus Phi of minus 1."},{"Start":"03:42.950 ","End":"03:48.180","Text":"Now we\u0027re ready to look at our standard table."},{"Start":"03:48.180 ","End":"03:53.820","Text":"Here\u0027s our standard table and we\u0027re interested in our standard score"},{"Start":"03:53.820 ","End":"03:58.770","Text":"where z equals 1 and we\u0027re looking for Phi of 1."},{"Start":"03:58.770 ","End":"04:05.420","Text":"Well, let\u0027s look at z 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":"04:10.130","Text":"This is a row and it\u0027s 1.0,"},{"Start":"04:10.130 ","End":"04:12.710","Text":"so 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":"04:21.210","Text":"That means that Phi of 1 equals to 0.8413."},{"Start":"04:22.790 ","End":"04:25.070","Text":"Now that we have that,"},{"Start":"04:25.070 ","End":"04:28.730","Text":"let\u0027s go back to our question. Here we go."},{"Start":"04:28.730 ","End":"04:37.095","Text":"Phi of 1 is we said that equals to 0.8413 minus,"},{"Start":"04:37.095 ","End":"04:39.430","Text":"now, what\u0027s Phi of minus 1?"},{"Start":"04:39.430 ","End":"04:45.065","Text":"Well, we learn previously that 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":"04:57.140","Text":"Now here, minus a is equal to minus 1, a equals 1."},{"Start":"04:57.140 ","End":"05:00.650","Text":"So that means that Phi of 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":"05:36.730","Text":"that\u0027ll be equal to 68.26 percent."},{"Start":"05:36.730 ","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":13137},{"Watched":false,"Name":"Exercise 3 - Part a","Duration":"6m 17s","ChapterTopicVideoID":12659,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12659.jpeg","UploadDate":"2018-07-23T16:16:49.1570000","DurationForVideoObject":"PT6M17S","Description":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":"00:15.030","Text":"We\u0027re asked what\u0027s the proportion of people who weigh less than 55 kilograms?"},{"Start":"00:15.030 ","End":"00:18.990","Text":"The first thing that we need to know is that we\u0027re dealing with"},{"Start":"00:18.990 ","End":"00:23.475","Text":"a normal probability distribution and that\u0027s right here."},{"Start":"00:23.475 ","End":"00:26.430","Text":"That\u0027s given to us."},{"Start":"00:26.430 ","End":"00:32.055","Text":"The next thing that we need to do is to define our random variable."},{"Start":"00:32.055 ","End":"00:38.825","Text":"Well, we\u0027ll call that x and that will be the weight of the people in kilograms."},{"Start":"00:38.825 ","End":"00:43.370","Text":"Let\u0027s just write that down in kilograms."},{"Start":"00:43.370 ","End":"00:45.365","Text":"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":"00:56.450","Text":"that\u0027s right here and a standard deviation of 8 kilograms."},{"Start":"00:56.450 ","End":"00:59.090","Text":"That means that Sigma squared,"},{"Start":"00:59.090 ","End":"01:02.185","Text":"that equals to 8 squared."},{"Start":"01:02.185 ","End":"01:05.030","Text":"What are we looking for?"},{"Start":"01:05.030 ","End":"01:08.900","Text":"The proportion of people who weigh less than 55 kilograms."},{"Start":"01:08.900 ","End":"01:16.120","Text":"We\u0027re looking for the probability that X is less than 55."},{"Start":"01:16.120 ","End":"01:19.295","Text":"Now we have all this data."},{"Start":"01:19.295 ","End":"01:26.065","Text":"Let\u0027s just draw the density function and put all this data on the graph."},{"Start":"01:26.065 ","End":"01:28.890","Text":"Here\u0027s our density function."},{"Start":"01:28.890 ","End":"01:33.380","Text":"We have both the x axis and the z axis."},{"Start":"01:33.380 ","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":"01:38.690","Text":"Well, that\u0027s this line right here,"},{"Start":"01:38.690 ","End":"01:41.275","Text":"so we\u0027ll put 60 here,"},{"Start":"01:41.275 ","End":"01:43.670","Text":"and what are we looking for?"},{"Start":"01:43.670 ","End":"01:48.425","Text":"We\u0027re looking for the probability of x being less than 55."},{"Start":"01:48.425 ","End":"01:57.190","Text":"Let\u0027s just put 55 here again on the x-axis and we\u0027re looking for this area right here."},{"Start":"01:57.190 ","End":"01:59.490","Text":"Now, as we can see,"},{"Start":"01:59.490 ","End":"02:03.680","Text":"this area right here is less than 50 percent and why is that?"},{"Start":"02:03.680 ","End":"02:06.575","Text":"Because from minus infinity to 60,"},{"Start":"02:06.575 ","End":"02:08.380","Text":"well, that\u0027s 50 percent."},{"Start":"02:08.380 ","End":"02:12.560","Text":"That\u0027s half of the area under the density function."},{"Start":"02:12.560 ","End":"02:17.375","Text":"If we\u0027re looking at the area from minus infinity to 55,"},{"Start":"02:17.375 ","End":"02:20.320","Text":"well, that\u0027s less than 50 percent."},{"Start":"02:20.320 ","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":"02:32.090","Text":"We have x, we have to standardize it into"},{"Start":"02:32.090 ","End":"02:36.680","Text":"z or standard score and then we have to go to the table,"},{"Start":"02:36.680 ","End":"02:39.530","Text":"the standard table, to find the proportion."},{"Start":"02:39.530 ","End":"02:44.975","Text":"In our case, x is 55 and z,"},{"Start":"02:44.975 ","End":"02:47.570","Text":"or the transformation of x,"},{"Start":"02:47.570 ","End":"02:51.455","Text":"that\u0027s x minus Mu divided by Sigma."},{"Start":"02:51.455 ","End":"02:57.020","Text":"Having said that, let\u0027s see what the proportion is."},{"Start":"02:57.020 ","End":"03:00.110","Text":"What\u0027s the probability of x being less than 55 is."},{"Start":"03:00.110 ","End":"03:06.655","Text":"Well, that equals to the probability of x minus Mu divided by Sigma,"},{"Start":"03:06.655 ","End":"03:09.030","Text":"that being less than,"},{"Start":"03:09.030 ","End":"03:11.895","Text":"now, 55 minus Mu."},{"Start":"03:11.895 ","End":"03:17.000","Text":"Now, Mu here is 60 divided by Sigma,"},{"Start":"03:17.000 ","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":"03:26.228","Text":"Now, x minus Mu divided by Sigma, well, that\u0027s z."},{"Start":"03:26.228 ","End":"03:32.555","Text":"That z has to be less than 55 minus 60 divided by 8,"},{"Start":"03:32.555 ","End":"03:34.835","Text":"that\u0027s minus 5 divided by 8,"},{"Start":"03:34.835 ","End":"03:38.660","Text":"that equals to 0.625."},{"Start":"03:38.660 ","End":"03:44.270","Text":"Let\u0027s just round that up to 0.63 and that\u0027s minus."},{"Start":"03:44.270 ","End":"03:55.220","Text":"We\u0027re looking for Phi of minus 0.63."},{"Start":"03:55.220 ","End":"03:57.745","Text":"Now, if we recall,"},{"Start":"03:57.745 ","End":"04:01.710","Text":"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":"04:12.715","Text":"Why is that? Again, because the normal density function"},{"Start":"04:12.715 ","End":"04:17.905","Text":"is symmetrical around the average."},{"Start":"04:17.905 ","End":"04:24.265","Text":"This area right here is equal to this area right here,"},{"Start":"04:24.265 ","End":"04:32.830","Text":"where here it\u0027s 65 and what are the comparable values on the z axis?"},{"Start":"04:32.830 ","End":"04:37.695","Text":"Well, at 55, it was minus 0.63."},{"Start":"04:37.695 ","End":"04:42.925","Text":"Here it would be plus 0.63."},{"Start":"04:42.925 ","End":"04:50.225","Text":"Now we\u0027re ready to go to our standard table."},{"Start":"04:50.225 ","End":"05:00.435","Text":"That\u0027ll be 1 minus Phi of 0.63 and let\u0027s see what Phi of 0.63 is."},{"Start":"05:00.435 ","End":"05:08.615","Text":"Here\u0027s 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":"05:16.750","Text":"that\u0027s this row right here and here\u0027s that 3."},{"Start":"05:16.750 ","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":"05:30.190","Text":"That equals to 0.7357."},{"Start":"05:30.190 ","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":"05:47.760","Text":"0.7357 and"},{"Start":"05:47.760 ","End":"05:54.595","Text":"that equals to 0.2643."},{"Start":"05:54.595 ","End":"05:59.840","Text":"If we\u0027re looking to show this in percentages, well,"},{"Start":"05:59.840 ","End":"06:05.050","Text":"that equals to 26.43 percent."},{"Start":"06:05.050 ","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":"06:14.630","Text":"this is less than 50 percent,"},{"Start":"06:14.630 ","End":"06:17.190","Text":"as we said previously here."}],"ID":13138},{"Watched":false,"Name":"Exercise 3 - Parts b-c","Duration":"7m 10s","ChapterTopicVideoID":12660,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12660.jpeg","UploadDate":"2018-07-23T16:17:10.3970000","DurationForVideoObject":"PT7M10S","Description":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 here, 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":"00:31.850","Text":"Well, the processes that we have our x,"},{"Start":"00:31.850 ","End":"00:36.710","Text":"and we have to standardize it into a standard score that\u0027s z."},{"Start":"00:36.710 ","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":"00:52.560","Text":"Now, the standardization transformation is x minus mu divided by sigma."},{"Start":"00:52.560 ","End":"00:55.965","Text":"Let\u0027s do this standardization right now."},{"Start":"00:55.965 ","End":"01:01.685","Text":"That\u0027s the probability of x minus mu divided by sigma,"},{"Start":"01:01.685 ","End":"01:04.670","Text":"there has to be greater than 50."},{"Start":"01:04.670 ","End":"01:10.740","Text":"Now, mu is 60 divided by sigma, that\u0027s 8."},{"Start":"01:10.740 ","End":"01:17.230","Text":"Now, that equals to the probability of z being"},{"Start":"01:17.230 ","End":"01:24.560","Text":"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":"01:41.560","Text":"Well, we have to remember one of the characteristics of"},{"Start":"01:41.560 ","End":"01:47.330","Text":"the normal distribution is that it\u0027s symmetrical around the average."},{"Start":"01:47.330 ","End":"01:49.385","Text":"What does that mean?"},{"Start":"01:49.385 ","End":"01:55.495","Text":"Well, let\u0027s take a look at these two identical distributions right here."},{"Start":"01:55.495 ","End":"02:01.345","Text":"Now, assuming that this right here is minus A, well,"},{"Start":"02:01.345 ","End":"02:05.620","Text":"the area from minus infinity to minus a,"},{"Start":"02:05.620 ","End":"02:12.345","Text":"well that equals to the area right from a here to plus infinity."},{"Start":"02:12.345 ","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":"02:28.665","Text":"If we have a to be equal to 1.25,"},{"Start":"02:28.665 ","End":"02:32.350","Text":"this thing right here will be minus 1.25,"},{"Start":"02:32.350 ","End":"02:36.850","Text":"and here that\u0027ll be 1.25."},{"Start":"02:36.990 ","End":"02:43.220","Text":"That\u0027ll be Phi of minus 1.25."},{"Start":"02:43.220 ","End":"02:49.460","Text":"This area right here is actually the same as this area right here."},{"Start":"02:49.460 ","End":"02:56.410","Text":"1 minus 1.25,"},{"Start":"02:56.410 ","End":"03:02.945","Text":"that equals to Phi of 1.25."},{"Start":"03:02.945 ","End":"03:05.680","Text":"This thing right here,"},{"Start":"03:05.680 ","End":"03:09.010","Text":"that equals 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":"03:39.400","Text":"to this row and this column right here,"},{"Start":"03:39.400 ","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":"04:05.405","Text":"Phi of 1.25, well that equals,"},{"Start":"04:05.405 ","End":"04:12.110","Text":"as we saw to 0.8944"},{"Start":"04:12.110 ","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":"04:58.759","Text":"Now, that\u0027s the probability of x being between 60 and 70 kilograms."},{"Start":"04:58.759 ","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":"05:26.210","Text":"Now, what does 70 compared to on the z scale?"},{"Start":"05:26.210 ","End":"05:35.525","Text":"Well, let\u0027s standardized 70. 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":"05:42.980 ","End":"05:52.385","Text":"1.25, we know what Phi of 1.25 is from the last section."},{"Start":"05:52.385 ","End":"05:57.380","Text":"That equals to 0.8944,"},{"Start":"05:57.380 ","End":"05:58.835","Text":"and a why do we need that?"},{"Start":"05:58.835 ","End":"06:04.610","Text":"Well, in order to calculate this area, that\u0027s 1.25 here."},{"Start":"06:04.610 ","End":"06:07.070","Text":"In order to calculate this area,"},{"Start":"06:07.070 ","End":"06:12.110","Text":"this is comparable to saying Phi of"},{"Start":"06:12.110 ","End":"06:18.755","Text":"1.25 minus Phi of 0."},{"Start":"06:18.755 ","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":"06:30.335","Text":"that\u0027s 0.8944 minus Phi of 0."},{"Start":"06:30.335 ","End":"06:32.945","Text":"Well, we don\u0027t have to go to the table for that."},{"Start":"06:32.945 ","End":"06:35.330","Text":"We know that this is 0.5,"},{"Start":"06:35.330 ","End":"06:36.785","Text":"and how do we know that?"},{"Start":"06:36.785 ","End":"06:45.005","Text":"Well, we know that half of the area lies between minus infinity and 0 on the z scale."},{"Start":"06:45.005 ","End":"06:47.390","Text":"So here,"},{"Start":"06:47.390 ","End":"06:55.800","Text":"that equals to 0.3944,"},{"Start":"06:55.800 ","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":13139},{"Watched":false,"Name":"Exercise 3 - Parts d-e","Duration":"5m 59s","ChapterTopicVideoID":12661,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12661.jpeg","UploadDate":"2018-07-23T16:17:31.3730000","DurationForVideoObject":"PT5M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"In this section, we\u0027re asked what\u0027s the proportion of the population"},{"Start":"00:03.780 ","End":"00:07.560","Text":"whose weight 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 a density function to 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 ranges 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":"01:07.535","Text":"Let\u0027s take, for example 64,"},{"Start":"01:07.535 ","End":"01:12.715","Text":"so z equals to x minus Mu divided by Sigma."},{"Start":"01:12.715 ","End":"01:18.490","Text":"That means that we\u0027re looking at 64 minus 60 divided by 8."},{"Start":"01:18.490 ","End":"01:20.210","Text":"That\u0027s a 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 1/2."},{"Start":"01:25.710 ","End":"01:29.025","Text":"Now, it\u0027s very easy to see."},{"Start":"01:29.025 ","End":"01:32.195","Text":"We could have calculated it in our heads"},{"Start":"01:32.195 ","End":"01:38.340","Text":"because 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 being"},{"Start":"01:44.960 ","End":"01:51.715","Text":"between 0.5 and minus 0.5."},{"Start":"01:51.715 ","End":"01:54.640","Text":"This right here is 0.5,"},{"Start":"01:54.640 ","End":"01:57.170","Text":"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":"02:16.150","Text":"Well, this is Phi of 0.5 minus,"},{"Start":"02:16.730 ","End":"02:24.390","Text":"that\u0027s this area right here from minus infinity to 0.5 from the z scale,"},{"Start":"02:24.390 ","End":"02:29.600","Text":"minus now Phi of minus 0.5."},{"Start":"02:29.600 ","End":"02:35.105","Text":"Well, that\u0027s minus infinity to minus 0.5 on the z scale."},{"Start":"02:35.105 ","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":"02:54.125","Text":"and 5 minus 0.5."},{"Start":"02:54.125 ","End":"03:00.350","Text":"Well, that equals to 1 minus Phi of 0.5,"},{"Start":"03:00.350 ","End":"03:10.265","Text":"and that equals to 0.6915 minus 1 minus"},{"Start":"03:10.265 ","End":"03:15.665","Text":"0.6915 and that comes out"},{"Start":"03:15.665 ","End":"03:22.835","Text":"to 0.383 or 38.3 percent."},{"Start":"03:22.835 ","End":"03:27.409","Text":"Now, that\u0027s the proportion right here of the population"},{"Start":"03:27.409 ","End":"03:32.555","Text":"whose weight deviates from the average by no more than 4 kilograms."},{"Start":"03:32.555 ","End":"03:35.510","Text":"In this section, we\u0027re asked what is the chances of"},{"Start":"03:35.510 ","End":"03:39.845","Text":"a randomly selected person weighing less than 140 kilograms?"},{"Start":"03:39.845 ","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":"03:58.955","Text":"We\u0027re looking to calculate the probability or the proportion"},{"Start":"03:58.955 ","End":"04:04.130","Text":"of the area in yellow here where x is below 149."},{"Start":"04:04.130 ","End":"04:08.575","Text":"In order to do that, let\u0027s standardize x at a 140."},{"Start":"04:08.575 ","End":"04:14.080","Text":"That means that z equals to x minus Mu divided by Sigma."},{"Start":"04:14.080 ","End":"04:20.450","Text":"That means that we\u0027re looking at 140 minus 60 divided by 8,"},{"Start":"04:20.450 ","End":"04:23.555","Text":"our standard deviation, 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 the probability then of z being less than 10."},{"Start":"04:40.145 ","End":"04:44.895","Text":"Now that equals to Phi at 10."},{"Start":"04:44.895 ","End":"04:52.925","Text":"Now let\u0027s go to our standard table to see what this value is."},{"Start":"04:52.925 ","End":"04:57.650","Text":"Here\u0027s this stable, and we\u0027re looking at the values of z."},{"Start":"04:57.650 ","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":"05:10.525","Text":"That means that we\u0027re looking at this value right here."},{"Start":"05:10.525 ","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":"05:31.865","Text":"For any value above 3.49,"},{"Start":"05:31.865 ","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":"well, that almost equals to 1."},{"Start":"05:43.920 ","End":"05:47.390","Text":"Now, 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":"05:56.120","Text":"then the chances of that person weighing less than 140 kilograms,"},{"Start":"05:56.120 ","End":"05:59.190","Text":"that\u0027s almost a certainty."}],"ID":13140},{"Watched":false,"Name":"Exercise 4 - Part a","Duration":"6m 20s","ChapterTopicVideoID":12662,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12662.jpeg","UploadDate":"2018-07-23T16:17:52.0570000","DurationForVideoObject":"PT6M20S","Description":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":"00:24.540","Text":"Secondly, let\u0027s define our random variable."},{"Start":"00:24.540 ","End":"00:32.010","Text":"Let\u0027s call that x, and 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":"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":"that equals to 400 grams."},{"Start":"00:50.290 ","End":"00:57.265","Text":"Let\u0027s just look at our density function and see how we can solve this problem."},{"Start":"00:57.265 ","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_0.9,"},{"Start":"01:10.460 ","End":"01:16.630","Text":"where 10 percent of the area under the curve lies above this value,"},{"Start":"01:16.630 ","End":"01:23.465","Text":"and 90 percent of the area under the curve falls below this value."},{"Start":"01:23.465 ","End":"01:27.350","Text":"That\u0027s the meaning of the upper 10th percentile."},{"Start":"01:27.350 ","End":"01:32.735","Text":"In previous questions, 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":"01:41.164","Text":"Then we had to go to our standard table to find the proportion."},{"Start":"01:41.164 ","End":"01:44.194","Text":"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":"01:50.275","Text":"we have to go and calculate our z,"},{"Start":"01:50.275 ","End":"01:52.230","Text":"our standard score,"},{"Start":"01:52.230 ","End":"01:56.855","Text":"and from there we have to calculate our x."},{"Start":"01:56.855 ","End":"01:59.630","Text":"This is what we need to do."},{"Start":"01:59.630 ","End":"02:00.830","Text":"Let\u0027s go ahead and do this."},{"Start":"02:00.830 ","End":"02:04.780","Text":"We\u0027re looking for then, some value,"},{"Start":"02:04.780 ","End":"02:09.775","Text":"let\u0027s say Phi of z_0.9,"},{"Start":"02:09.775 ","End":"02:13.265","Text":"and that has to equal to 0.9."},{"Start":"02:13.265 ","End":"02:18.350","Text":"Remember this z_0.9 is"},{"Start":"02:18.350 ","End":"02:25.895","Text":"the comparable value of what we\u0027re looking for on the x-axis, that\u0027s x_0.9."},{"Start":"02:25.895 ","End":"02:31.765","Text":"Let\u0027s go to our table and see what this value is."},{"Start":"02:31.765 ","End":"02:34.470","Text":"Here\u0027s our standard table,"},{"Start":"02:34.470 ","End":"02:37.850","Text":"and we\u0027re going to have to look in the table right here"},{"Start":"02:37.850 ","End":"02:42.085","Text":"to find the value that\u0027s closest to 0.9."},{"Start":"02:42.085 ","End":"02:47.055","Text":"Remember we want Phi of some value z_0.9,"},{"Start":"02:47.055 ","End":"02:51.750","Text":"we want that to be equal to 0.9."},{"Start":"02:51.750 ","End":"02:56.235","Text":"We\u0027re looking for this value right here, z_0.9."},{"Start":"02:56.235 ","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":"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":"03:22.860","Text":"That\u0027ll be 0.8997,"},{"Start":"03:22.860 ","End":"03:28.770","Text":"we\u0027re looking at z_0.9,"},{"Start":"03:28.770 ","End":"03:36.090","Text":"that equals to 1.28."},{"Start":"03:36.090 ","End":"03:42.840","Text":"In some cases, we can be just a little bit more exact. Why is that?"},{"Start":"03:42.840 ","End":"03:52.620","Text":"We can scroll down and 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_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":"04:25.580","Text":"the comparable value on the x-axis."},{"Start":"04:25.580 ","End":"04:28.885","Text":"Remember we were working backwards."},{"Start":"04:28.885 ","End":"04:32.140","Text":"How did we get from x to z?"},{"Start":"04:32.140 ","End":"04:33.920","Text":"How did we standardize?"},{"Start":"04:33.920 ","End":"04:38.795","Text":"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":"04:57.900","Text":"We know that z equals to 1.282,"},{"Start":"04:57.900 ","End":"05:08.940","Text":"then has to equal to x_0.9 minus 3,300 divided by 400."},{"Start":"05:08.940 ","End":"05:11.420","Text":"This is our 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":"05:32.330","Text":"Therefore x_0.9, that"},{"Start":"05:32.330 ","End":"05:40.340","Text":"equals to 3,812.8 grams."},{"Start":"05:40.340 ","End":"05:46.375","Text":"This value, that\u0027s right here,"},{"Start":"05:46.375 ","End":"05:53.630","Text":"3,812.8 that\u0027s the comparable value"},{"Start":"05:53.630 ","End":"06:01.050","Text":"on the x-axis for z_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":13141},{"Watched":false,"Name":"Exercise 4 - Parts b-c","Duration":"8m 18s","ChapterTopicVideoID":12663,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12663.jpeg","UploadDate":"2018-07-23T16:18:15.1500000","DurationForVideoObject":"PT8M18S","Description":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":"00:07.485","Text":"Let\u0027s draw our density function."},{"Start":"00:07.485 ","End":"00:12.150","Text":"Here it is, and we\u0027re asked to find the 95th percentile."},{"Start":"00:12.150 ","End":"00:15.780","Text":"What does that mean? 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":"00:25.740","Text":"where 95 percent of the area under the curve falls below this value,"},{"Start":"00:25.740 ","End":"00:33.430","Text":"and only 5 percent of the area under the curve falls above this value."},{"Start":"00:34.400 ","End":"00:36.840","Text":"How are we going to do that?"},{"Start":"00:36.840 ","End":"00:43.820","Text":"In previous questions, 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":"00:56.410","Text":"Now here, we\u0027re given the proportion."},{"Start":"00:56.410 ","End":"01:01.145","Text":"What we need to do is we need to go to the table and find z."},{"Start":"01:01.145 ","End":"01:03.095","Text":"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 right now,"},{"Start":"01:09.350 ","End":"01:13.760","Text":"our first step is to find a specific z."},{"Start":"01:13.760 ","End":"01:16.895","Text":"We\u0027ll call this z_0.95,"},{"Start":"01:16.895 ","End":"01:25.465","Text":"which is the comparable value of x_0.95 on the z scale."},{"Start":"01:25.465 ","End":"01:32.425","Text":"We\u0027re looking then for Phi of z_0.95,"},{"Start":"01:32.425 ","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":"This 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 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 where at Phi of z being equal to 95,"},{"Start":"02:16.215 ","End":"02:20.750","Text":"then z then would be equal to 1.645."},{"Start":"02:20.750 ","End":"02:24.930","Text":"We\u0027ll be using this value right here."},{"Start":"02:25.360 ","End":"02:31.175","Text":"We know then that z_ 0.95,"},{"Start":"02:31.175 ","End":"02:36.655","Text":"that equals to 1.645."},{"Start":"02:36.655 ","End":"02:40.055","Text":"We found z. Now we have to calculate x."},{"Start":"02:40.055 ","End":"02:42.640","Text":"Now, how did we get from x to z?"},{"Start":"02:42.640 ","End":"02:51.165","Text":"Well, we said that z was equal to x minus Mu divided by Sigma."},{"Start":"02:51.165 ","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":"03:03.590","Text":"We know that z equals to 1.645 and then"},{"Start":"03:03.590 ","End":"03:09.665","Text":"has to be equal to x_ 0.95 minus,"},{"Start":"03:09.665 ","End":"03:16.065","Text":"now Mu is 3,300 divided by 400."},{"Start":"03:16.065 ","End":"03:20.330","Text":"Again, we have an equation with only 1 variable,"},{"Start":"03:20.330 ","End":"03:22.520","Text":"so let\u0027s just solve that."},{"Start":"03:22.520 ","End":"03:27.675","Text":"Now, we\u0027ll just multiply both sides by 400,"},{"Start":"03:27.675 ","End":"03:30.840","Text":"and they\u0027ll be 658,"},{"Start":"03:30.840 ","End":"03:36.000","Text":"and that will be equal to x_0.95"},{"Start":"03:36.000 ","End":"03:42.885","Text":"minus 3,300. x_0.95,"},{"Start":"03:42.885 ","End":"03:50.325","Text":"that 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":"04:26.670","Text":"Here\u0027s our density function and we\u0027re looking for a specific x right here."},{"Start":"04:26.670 ","End":"04:29.010","Text":"We\u0027ll call this x_0.1,"},{"Start":"04:29.010 ","End":"04:31.339","Text":"for the bottom 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":"04:49.385","Text":"Now, because we know that this value is below the average,"},{"Start":"04:49.385 ","End":"04:54.040","Text":"then we can expect a negative z value."},{"Start":"04:54.040 ","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":"05:03.170","Text":"We\u0027ll call this the comparable z value."},{"Start":"05:03.170 ","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":"05:18.045","Text":"then we have to remember that this density function is symmetrical."},{"Start":"05:18.045 ","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":"05:36.750","Text":"This then on the x-axis would be x_0.9,"},{"Start":"05:36.750 ","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":"05:56.260","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":"06:08.120","Text":"calculated in section a the value of z_0.9."},{"Start":"06:08.120 ","End":"06:14.010","Text":"z_ 0.9, that equals 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, that\u0027s a minus value,"},{"Start":"06:25.654 ","End":"06:29.690","Text":"but it\u0027s the same value right here as this only with a minus sign."},{"Start":"06:29.690 ","End":"06:37.955","Text":"z _0.1 is equal to minus 1.282. Why is that?"},{"Start":"06:37.955 ","End":"06:40.870","Text":"Again, because of the symmetry."},{"Start":"06:40.870 ","End":"06:49.145","Text":"This distance right here from 0 under 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":"07:02.880","Text":"Now that we know the value of z_0.1,"},{"Start":"07:04.190 ","End":"07:07.935","Text":"then let\u0027s calculate x."},{"Start":"07:07.935 ","End":"07:10.095","Text":"We know that z,"},{"Start":"07:10.095 ","End":"07:13.935","Text":"that equals to x minus Mu divided by Sigma."},{"Start":"07:13.935 ","End":"07:15.915","Text":"Let\u0027s just plug in the numbers."},{"Start":"07:15.915 ","End":"07:20.130","Text":"That\u0027ll be minus 1.282,"},{"Start":"07:20.130 ","End":"07:23.290","Text":"and that has to be equal to x_0.1."},{"Start":"07:23.290 ","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":"07:38.255","Text":"We\u0027ll multiply both sides by 400."},{"Start":"07:38.255 ","End":"07:42.140","Text":"We have minus 512.8,"},{"Start":"07:42.140 ","End":"07:47.680","Text":"that equals to x_0.1 minus 3,300."},{"Start":"07:47.680 ","End":"07:51.110","Text":"That means that x_0.1,"},{"Start":"07:51.110 ","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":"08:12.905","Text":"and 90 percent of the babies weigh more than this value."},{"Start":"08:12.905 ","End":"08:17.610","Text":"That\u0027s the meaning of the bottom 10th percentile."}],"ID":13142},{"Watched":false,"Name":"Exercise 5 - Parts a-b","Duration":"10m 33s","ChapterTopicVideoID":12664,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12664.jpeg","UploadDate":"2018-07-23T16:18:51.9170000","DurationForVideoObject":"PT10M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"This question we\u0027re given that the marks in an intelligence tests have"},{"Start":"00:03.450 ","End":"00:08.820","Text":"a normal probability distribution with an average of 100 and a variance of 225,"},{"Start":"00:08.820 ","End":"00:14.610","Text":"and 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 a random variable."},{"Start":"00:28.200 ","End":"00:32.640","Text":"Now let\u0027s call that X and that\u0027ll be the marks."},{"Start":"00:32.640 ","End":"00:35.060","Text":"That\u0027s given on the intelligence test."},{"Start":"00:35.060 ","End":"00:41.300","Text":"Now we know that has a normal probability distribution where our average Mu,"},{"Start":"00:41.300 ","End":"00:45.420","Text":"well, that equals to 100 and that\u0027s given right here."},{"Start":"00:45.420 ","End":"00:49.565","Text":"The variance, not the standard deviation, the variance,"},{"Start":"00:49.565 ","End":"00:55.160","Text":"Sigma squared, that equals to 225."},{"Start":"00:55.160 ","End":"01:01.960","Text":"Now here, I\u0027d prefer it if you write this out like this."},{"Start":"01:01.960 ","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":"01:22.890","Text":"Here\u0027s our density function and on the x-axis,"},{"Start":"01:22.890 ","End":"01:24.650","Text":"well, 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":"01:39.230","Text":"That means that we\u0027re looking for X_0.9."},{"Start":"01:39.230 ","End":"01:44.600","Text":"That\u0027s the value where 10 percent of the area under"},{"Start":"01:44.600 ","End":"01:50.890","Text":"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 0.9,"},{"Start":"02:18.970 ","End":"02:24.130","Text":"we have to go to our standard table and see"},{"Start":"02:24.130 ","End":"02:29.785","Text":"what our Z is that\u0027s comparable to our probability and from Z,"},{"Start":"02:29.785 ","End":"02:33.350","Text":"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":"02:47.840","Text":"That means that we\u0027re looking now for Z_0.9."},{"Start":"02:49.220 ","End":"02:53.685","Text":"We\u0027re looking then for Phi of"},{"Start":"02:53.685 ","End":"03:00.945","Text":"Z_0.9 and we know that this is equal to 0.9."},{"Start":"03:00.945 ","End":"03:06.627","Text":"Let\u0027s go to our standard table and see what our Z value is."},{"Start":"03:06.627 ","End":"03:13.364","Text":"Again, we\u0027re looking for Phi of a specific Z."},{"Start":"03:13.364 ","End":"03:16.460","Text":"We\u0027ll call that Z_0.9."},{"Start":"03:16.460 ","End":"03:19.520","Text":"We want that to be equal to 0.9."},{"Start":"03:19.520 ","End":"03:23.840","Text":"We\u0027re looking for this value 0.9 within"},{"Start":"03:23.840 ","End":"03:29.445","Text":"the table and then we can go and see what our Z value is."},{"Start":"03:29.445 ","End":"03:37.355","Text":"0.9 what\u0027s the closest value that we have to 0.9 within the table right here?"},{"Start":"03:37.355 ","End":"03:42.790","Text":"Well, that looks like this value right here."},{"Start":"03:42.790 ","End":"03:52.930","Text":"Now this Phi value occurs in the intersection of 1.28,"},{"Start":"03:53.230 ","End":"03:56.030","Text":"so Z then,"},{"Start":"03:56.030 ","End":"04:01.400","Text":"Z_0.9, that equals to 1.28."},{"Start":"04:01.400 ","End":"04:05.690","Text":"Now let\u0027s see if we can get a more exact figure for Z."},{"Start":"04:05.690 ","End":"04:07.645","Text":"Let\u0027s scroll down,"},{"Start":"04:07.645 ","End":"04:12.570","Text":"and here for 5Z being equal to 0.9,"},{"Start":"04:12.570 ","End":"04:17.265","Text":"we get a Z value of 1.282."},{"Start":"04:17.265 ","End":"04:22.340","Text":"Let\u0027s just write that down here, that\u0027s 1.282."},{"Start":"04:22.340 ","End":"04:27.905","Text":"Now really doesn\u0027t matter if you write 1.28 or 1.282."},{"Start":"04:27.905 ","End":"04:33.130","Text":"The minute you understand this principle than all is well."},{"Start":"04:33.130 ","End":"04:39.855","Text":"Let\u0027s go back to our question and continue with our calculations."},{"Start":"04:39.855 ","End":"04:45.810","Text":"Now we know then that Z_0.9,"},{"Start":"04:45.810 ","End":"04:48.580","Text":"that equals to 1.282."},{"Start":"04:49.670 ","End":"04:52.005","Text":"Let\u0027s write that down."},{"Start":"04:52.005 ","End":"04:54.340","Text":"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 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":"05:19.520","Text":"that will be 1.282 and that has to be equal to X_0.9 minus Mu."},{"Start":"05:19.520 ","End":"05:25.690","Text":"Now I\u0027m using 100 divided by Sigma where Sigma is 15."},{"Start":"05:25.690 ","End":"05:28.540","Text":"The moment we have this then,"},{"Start":"05:28.540 ","End":"05:30.365","Text":"it\u0027s just basic math."},{"Start":"05:30.365 ","End":"05:34.864","Text":"Let\u0027s just continue here and let\u0027s just scroll this up."},{"Start":"05:34.864 ","End":"05:38.740","Text":"We\u0027ll multiply both sides by 15."},{"Start":"05:38.740 ","End":"05:42.210","Text":"We\u0027ll get 19.23,"},{"Start":"05:42.210 ","End":"05:46.560","Text":"that equals to X_0.9 minus 100,"},{"Start":"05:46.560 ","End":"05:52.870","Text":"so X_0.9, that equals to 119.23."},{"Start":"05:53.900 ","End":"06:00.020","Text":"This then is the upper 10th percentile."},{"Start":"06:00.020 ","End":"06:01.775","Text":"Let\u0027s just scroll this down here."},{"Start":"06:01.775 ","End":"06:06.800","Text":"This is the upper 10th percentile of the marks on the intelligence test."},{"Start":"06:06.800 ","End":"06:11.270","Text":"Where 10 percent of the grades are above"},{"Start":"06:11.270 ","End":"06:16.370","Text":"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":"06:31.010","Text":"What we\u0027re looking for is some value of X that\u0027s right here."},{"Start":"06:31.010 ","End":"06:33.935","Text":"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":"06:53.020","Text":"Now, this is the value that we\u0027re looking for."},{"Start":"06:53.020 ","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":"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, our standard score."},{"Start":"07:15.575 ","End":"07:19.280","Text":"From there we have to calculate our X."},{"Start":"07:19.280 ","End":"07:26.370","Text":"Now, this is very similar to what we did in section a."},{"Start":"07:26.370 ","End":"07:28.470","Text":"Where in section a,"},{"Start":"07:28.470 ","End":"07:33.510","Text":"we had to calculate the top 10th percentile."},{"Start":"07:33.510 ","End":"07:36.480","Text":"Let\u0027s just see what we did there."},{"Start":"07:36.480 ","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 does 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 the specific value."},{"Start":"08:14.725 ","End":"08:19.140","Text":"Let\u0027s see what our corresponding Z value is right here."},{"Start":"08:19.140 ","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":"08:39.645","Text":"We don\u0027t really need to go to the table,"},{"Start":"08:39.645 ","End":"08:41.450","Text":"and in any case,"},{"Start":"08:41.450 ","End":"08:46.145","Text":"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 we 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":"09:06.405","Text":"where in our case that\u0027s Z_0.1."},{"Start":"09:06.405 ","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":"In that term, Mu divided by Sigma,"},{"Start":"09:18.605 ","End":"09:20.210","Text":"where 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":"09:34.680","Text":"That will be minus 19.23 and that equals to X_0.1 minus 100,"},{"Start":"09:34.680 ","End":"09:45.135","Text":"or X_0.1, that equals to 80.77."},{"Start":"09:45.135 ","End":"09:51.110","Text":"This then is the bottom 10th percentile of the probability distribution."},{"Start":"09:51.110 ","End":"09:55.460","Text":"Now, we could have calculated this a little bit differently."},{"Start":"09:55.460 ","End":"10:00.990","Text":"We know from section A that the distance between"},{"Start":"10:00.990 ","End":"10:08.825","Text":"the average and the 90th percentile, or that\u0027s 19.23."},{"Start":"10:08.825 ","End":"10:17.990","Text":"That\u0027s this distance right here or the symmetrical distance right here."},{"Start":"10:17.990 ","End":"10:21.680","Text":"Well, that\u0027s also 19.23."},{"Start":"10:21.680 ","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":13143},{"Watched":false,"Name":"Exercise 5 - Parts c-e","Duration":"13m 19s","ChapterTopicVideoID":12665,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12665.jpeg","UploadDate":"2018-07-23T16:19:38.1430000","DurationForVideoObject":"PT13M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.380","Text":"In this section we\u0027re asked,"},{"Start":"00:01.380 ","End":"00:05.585","Text":"20 percent of those taking the test receive marks higher than what number?"},{"Start":"00:05.585 ","End":"00:08.565","Text":"Okay, so let\u0027s look at our density function."},{"Start":"00:08.565 ","End":"00:10.560","Text":"So this is our density function,"},{"Start":"00:10.560 ","End":"00:15.165","Text":"we\u0027re looking for some value of x and we\u0027ll call that x right here;"},{"Start":"00:15.165 ","End":"00:21.030","Text":"where 80 percent of"},{"Start":"00:21.030 ","End":"00:25.815","Text":"the area underneath the density function falls below this value,"},{"Start":"00:25.815 ","End":"00:32.490","Text":"and 20 percent of the area under the density function fall above this value."},{"Start":"00:32.490 ","End":"00:34.380","Text":"So that\u0027ll be 0.2 here,"},{"Start":"00:34.380 ","End":"00:36.388","Text":"that\u0027ll be 0.8 here,"},{"Start":"00:36.388 ","End":"00:40.850","Text":"and we\u0027ll call this value x sub 0.8 because,"},{"Start":"00:40.850 ","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 our standard score"},{"Start":"00:51.995 ","End":"00:55.280","Text":"and then go to our 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":"00:58.820","Text":"We\u0027re given the probability."},{"Start":"00:58.820 ","End":"01:05.130","Text":"We have to see what our z value is or the comparable z value is,"},{"Start":"01:05.130 ","End":"01:08.240","Text":"and then we have to calculate our x."},{"Start":"01:08.240 ","End":"01:14.120","Text":"So the first thing that we need to do is to see what our comparable z value is,"},{"Start":"01:14.120 ","End":"01:17.390","Text":"that\u0027ll be z sub 0.8,"},{"Start":"01:17.390 ","End":"01:18.970","Text":"that\u0027s what we\u0027ll call it."},{"Start":"01:18.970 ","End":"01:23.520","Text":"Okay. So what we\u0027re looking for"},{"Start":"01:23.520 ","End":"01:31.370","Text":"then is Phi of z sub 0.8."},{"Start":"01:31.370 ","End":"01:39.035","Text":"Okay? So we\u0027re looking for this z value which makes Phi of this value equals to 0.8."},{"Start":"01:39.035 ","End":"01:40.775","Text":"So let\u0027s go to the table."},{"Start":"01:40.775 ","End":"01:43.520","Text":"So this is our table, and as I said,"},{"Start":"01:43.520 ","End":"01:47.495","Text":"we\u0027re looking for our z value,"},{"Start":"01:47.495 ","End":"01:51.360","Text":"where Phi of the z values,"},{"Start":"01:51.360 ","End":"01:55.595","Text":"z sub 0.8, that equals to 0.8."},{"Start":"01:55.595 ","End":"02:01.430","Text":"So what\u0027s the closest value to 0.8 that we have within the table?"},{"Start":"02:01.430 ","End":"02:04.295","Text":"That\u0027s right here."},{"Start":"02:04.295 ","End":"02:11.990","Text":"So the comparable z value or the z value that we\u0027re looking for, z sub 0.8."},{"Start":"02:11.990 ","End":"02:17.135","Text":"Well, that equals to 0.84."},{"Start":"02:17.135 ","End":"02:23.015","Text":"All right, 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":"02:31.665","Text":"let\u0027s see if we have a comparable z value?"},{"Start":"02:31.665 ","End":"02:34.335","Text":"We don\u0027t, okay? We don\u0027t."},{"Start":"02:34.335 ","End":"02:41.900","Text":"This smaller table doesn\u0027t give us the z value for 5 z being equal to"},{"Start":"02:41.900 ","End":"02:51.890","Text":"0.8 but I can tell you that this value right here is 0.842."},{"Start":"02:51.890 ","End":"02:59.245","Text":"Let\u0027s now go back to our question and see how we can continue."},{"Start":"02:59.245 ","End":"03:03.510","Text":"So z sub 0.8,"},{"Start":"03:03.510 ","End":"03:10.250","Text":"well that equals to 0.842."},{"Start":"03:10.250 ","End":"03:13.730","Text":"Now, our standardization transformation,"},{"Start":"03:13.730 ","End":"03:17.510","Text":"that equals to x minus Mu divided by Sigma."},{"Start":"03:17.510 ","End":"03:19.670","Text":"So in our case,"},{"Start":"03:19.670 ","End":"03:24.665","Text":"we have 0.842 or that equals to"},{"Start":"03:24.665 ","End":"03:31.190","Text":"x sub 0.8 minus Mu that\u0027s 100, divided by Sigma."},{"Start":"03:31.190 ","End":"03:39.030","Text":"Well, that\u0027s 15. Let\u0027s multiply both sides by 15 and we get 12.6,"},{"Start":"03:39.030 ","End":"03:43.030","Text":"that equals to x sub 0.8 minus 100."},{"Start":"03:43.030 ","End":"03:46.880","Text":"That means that x sub 0.8"},{"Start":"03:46.880 ","End":"03:55.735","Text":"that equals to 112.6."},{"Start":"03:55.735 ","End":"03:58.370","Text":"So this then is the value,"},{"Start":"03:58.370 ","End":"04:02.690","Text":"where 80 percent of the marks fall below"},{"Start":"04:02.690 ","End":"04:08.460","Text":"this value and 20 percent of the marks are higher than this value."},{"Start":"04:08.860 ","End":"04:12.920","Text":"In this section, we\u0027re asked what\u0027s the 20th percentile?"},{"Start":"04:12.920 ","End":"04:16.250","Text":"Okay. So let\u0027s draw our density function."},{"Start":"04:16.250 ","End":"04:21.155","Text":"So here\u0027s our density function and we\u0027re looking for some value of x."},{"Start":"04:21.155 ","End":"04:24.050","Text":"We 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":"We\u0027ll call this x sub 0.2 because,"},{"Start":"04:43.100 ","End":"04:46.940","Text":"in essence, we\u0027re looking for the 20th percentile."},{"Start":"04:46.940 ","End":"04:51.335","Text":"So 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. Why is that?"},{"Start":"05:03.680 ","End":"05:06.320","Text":"Well, because we have our probability,"},{"Start":"05:06.320 ","End":"05:07.840","Text":"we\u0027re given a probability,"},{"Start":"05:07.840 ","End":"05:12.470","Text":"and we have to find out what our x is. Right here."},{"Start":"05:12.470 ","End":"05:18.605","Text":"So we have to go from here to here and then calculate our x."},{"Start":"05:18.605 ","End":"05:21.455","Text":"So this though,"},{"Start":"05:21.455 ","End":"05:25.250","Text":"looks vaguely familiar from Section C. So let\u0027s see what we did in"},{"Start":"05:25.250 ","End":"05:30.665","Text":"Section C and see how we can use that to help us solve this problem."},{"Start":"05:30.665 ","End":"05:34.280","Text":"Okay, so this is what we did in Section C. We were looking"},{"Start":"05:34.280 ","End":"05:37.610","Text":"for the 80th percentile, right?"},{"Start":"05:37.610 ","End":"05:41.055","Text":"x sub 0.8, where 20 percent"},{"Start":"05:41.055 ","End":"05:45.065","Text":"of the area underneath that density function was above this value."},{"Start":"05:45.065 ","End":"05:48.170","Text":"Now here in this section in our question right here,"},{"Start":"05:48.170 ","End":"05:50.330","Text":"we\u0027re looking for the 20th percentile."},{"Start":"05:50.330 ","End":"05:53.480","Text":"That means that we\u0027re looking for the same 20 percent of"},{"Start":"05:53.480 ","End":"05:57.155","Text":"the area under the density function but on the other side."},{"Start":"05:57.155 ","End":"06:01.070","Text":"The symmetrical side of the density function."},{"Start":"06:01.070 ","End":"06:06.305","Text":"Now, because we\u0027re dealing with asymmetrical distribution,"},{"Start":"06:06.305 ","End":"06:12.395","Text":"then the comparable z value here for z sub 0.2,"},{"Start":"06:12.395 ","End":"06:17.555","Text":"well, that equals to minus 0.842."},{"Start":"06:17.555 ","End":"06:23.345","Text":"That\u0027s the symmetrical value for z sub 0.8."},{"Start":"06:23.345 ","End":"06:26.525","Text":"So now that we have our z,"},{"Start":"06:26.525 ","End":"06:29.180","Text":"let\u0027s calculate our x."},{"Start":"06:29.180 ","End":"06:35.420","Text":"Now, we know that z equals to x minus Mu divided by Sigma,"},{"Start":"06:35.420 ","End":"06:37.325","Text":"so let\u0027s just plug in the numbers."},{"Start":"06:37.325 ","End":"06:40.655","Text":"That\u0027ll be minus 0.842,"},{"Start":"06:40.655 ","End":"06:44.510","Text":"and that equals to x sub 0.2 minus 100."},{"Start":"06:44.510 ","End":"06:48.515","Text":"Mu is 100 divided by 15; Sigma is 15."},{"Start":"06:48.515 ","End":"06:55.595","Text":"Let\u0027s multiply both sides by 15 and they\u0027ll be 12.6."},{"Start":"06:55.595 ","End":"06:57.964","Text":"Let\u0027s just write this up nicely."},{"Start":"06:57.964 ","End":"07:04.520","Text":"That\u0027s 12.6 and that equals to x sub 0.2 minus"},{"Start":"07:04.520 ","End":"07:13.790","Text":"100 or x sub 0.2 that equals to 87.4."},{"Start":"07:13.790 ","End":"07:18.485","Text":"Now, this then is the 20th percentile where"},{"Start":"07:18.485 ","End":"07:21.050","Text":"20 percent of the marks falls below"},{"Start":"07:21.050 ","End":"07:24.950","Text":"this value and 80 percent of the marks fall above this value."},{"Start":"07:24.950 ","End":"07:28.655","Text":"Now again, we could have solved this differently."},{"Start":"07:28.655 ","End":"07:33.008","Text":"How? Well, we know that this distance right here"},{"Start":"07:33.008 ","End":"07:40.850","Text":"between this average and the 80th percentile, that\u0027s 12.6."},{"Start":"07:40.850 ","End":"07:43.130","Text":"Okay, that\u0027s this distance right here."},{"Start":"07:43.130 ","End":"07:51.080","Text":"So symmetrically, all we would have had to do was to subtract 12.6 from"},{"Start":"07:51.080 ","End":"07:54.770","Text":"100 to get the symmetrical point of x sub 0.2 or"},{"Start":"07:54.770 ","End":"07:59.885","Text":"the 20th percentile but this is what we did here."},{"Start":"07:59.885 ","End":"08:04.650","Text":"This is exactly what we did here but in a more formal fashion."},{"Start":"08:05.120 ","End":"08:07.400","Text":"In this section we\u0027re asked,"},{"Start":"08:07.400 ","End":"08:12.800","Text":"5 percent of those taking the test receive marks lower than what number?"},{"Start":"08:12.800 ","End":"08:15.515","Text":"So let\u0027s draw our density function."},{"Start":"08:15.515 ","End":"08:17.705","Text":"So here\u0027s our density function,"},{"Start":"08:17.705 ","End":"08:21.885","Text":"and we\u0027re looking then for some value of x,"},{"Start":"08:21.885 ","End":"08:25.080","Text":"call this x sub 0.05."},{"Start":"08:25.080 ","End":"08:30.575","Text":"Where 5 percent of the marks fall below this value."},{"Start":"08:30.575 ","End":"08:38.735","Text":"That\u0027s 0.05, and 95 percent of the marks fall above this value."},{"Start":"08:38.735 ","End":"08:41.420","Text":"Now, how do we do this?"},{"Start":"08:41.420 ","End":"08:44.735","Text":"Well, we usually have our x."},{"Start":"08:44.735 ","End":"08:48.200","Text":"We standardize this into a standard score and then we go to"},{"Start":"08:48.200 ","End":"08:52.580","Text":"our standard table to find the proportion or the probability."},{"Start":"08:52.580 ","End":"08:54.890","Text":"Now here we have to work backwards."},{"Start":"08:54.890 ","End":"08:57.335","Text":"We\u0027re given a proportion,"},{"Start":"08:57.335 ","End":"09:00.050","Text":"we have to see what the standard score is,"},{"Start":"09:00.050 ","End":"09:06.470","Text":"then we have to calculate our x but we have another complication here."},{"Start":"09:06.470 ","End":"09:11.300","Text":"We know that the comparable z value for this is negative,"},{"Start":"09:11.300 ","End":"09:18.094","Text":"so that\u0027s negative minus z sub 0.05."},{"Start":"09:18.094 ","End":"09:23.330","Text":"So since the standard table doesn\u0027t support negative values of z,"},{"Start":"09:23.330 ","End":"09:25.970","Text":"we\u0027re going to have to go to the symmetrical point"},{"Start":"09:25.970 ","End":"09:28.384","Text":"on the other side of the distribution,"},{"Start":"09:28.384 ","End":"09:31.445","Text":"and that\u0027s right here."},{"Start":"09:31.445 ","End":"09:37.205","Text":"Now here, that\u0027ll be z sub 0.95,"},{"Start":"09:37.205 ","End":"09:43.880","Text":"and that\u0027s the comparable x value would be x sub 0.95."},{"Start":"09:43.880 ","End":"09:45.635","Text":"What\u0027s this value right here?"},{"Start":"09:45.635 ","End":"09:51.430","Text":"Well, that\u0027s the value where 0.05 percent of"},{"Start":"09:51.430 ","End":"09:57.650","Text":"the area under the density function fall above this value right here."},{"Start":"09:57.650 ","End":"10:06.354","Text":"So we\u0027re looking then for Phi of z sub 0.95,"},{"Start":"10:06.354 ","End":"10:15.020","Text":"and we want to look for z where the Phi of z equals to 0.95."},{"Start":"10:15.020 ","End":"10:19.470","Text":"So let\u0027s go to our table now."},{"Start":"10:19.470 ","End":"10:20.870","Text":"So this is our table and again,"},{"Start":"10:20.870 ","End":"10:26.165","Text":"we\u0027re looking for Phi of z sub 0.95,"},{"Start":"10:26.165 ","End":"10:29.675","Text":"and we want that to be equal to 0.95."},{"Start":"10:29.675 ","End":"10:38.210","Text":"So we\u0027re looking for a value that\u0027s closest to 0.95 within the table right here."},{"Start":"10:38.210 ","End":"10:40.160","Text":"Now, where would that be?"},{"Start":"10:40.160 ","End":"10:45.110","Text":"That would be here, somewhere between here."},{"Start":"10:45.110 ","End":"10:51.830","Text":"So the comparable z would be 1.64 or 1.65."},{"Start":"10:51.830 ","End":"10:58.285","Text":"Let\u0027s see if we can get much more exact value for z."},{"Start":"10:58.285 ","End":"11:00.210","Text":"So we\u0027ll scroll down,"},{"Start":"11:00.210 ","End":"11:04.610","Text":"and we see here that Phi of z being equal to 0.95."},{"Start":"11:04.610 ","End":"11:09.635","Text":"Well, the comparable z value is 1.645."},{"Start":"11:09.635 ","End":"11:12.410","Text":"So let\u0027s just write that down here."},{"Start":"11:12.410 ","End":"11:15.965","Text":"Z sub 0.95,"},{"Start":"11:15.965 ","End":"11:19.730","Text":"that equals to 1.645."},{"Start":"11:19.730 ","End":"11:22.720","Text":"Okay, so now that we have our z value,"},{"Start":"11:22.720 ","End":"11:24.695","Text":"let\u0027s get back to our question."},{"Start":"11:24.695 ","End":"11:29.965","Text":"So now we know that z sub 0.95,"},{"Start":"11:29.965 ","End":"11:38.320","Text":"that equals to 1.645."},{"Start":"11:38.320 ","End":"11:44.500","Text":"So the symmetrical point on the other side of the distribution,"},{"Start":"11:44.500 ","End":"11:50.650","Text":"which would be z sub 0.05 would be"},{"Start":"11:50.650 ","End":"11:57.930","Text":"equal to minus 1.645."},{"Start":"11:57.930 ","End":"12:03.375","Text":"So now that we have our z value,"},{"Start":"12:03.375 ","End":"12:08.240","Text":"let\u0027s calculate our x, x sub 0.5."},{"Start":"12:08.240 ","End":"12:09.650","Text":"Let\u0027s just plug in the numbers."},{"Start":"12:09.650 ","End":"12:13.900","Text":"We know that the standardization transformation,"},{"Start":"12:13.900 ","End":"12:16.760","Text":"that\u0027s x minus Mu divided by Sigma."},{"Start":"12:16.760 ","End":"12:22.385","Text":"So let\u0027s just plug in the numbers that will be minus 1.645,"},{"Start":"12:22.385 ","End":"12:27.590","Text":"and that has to be equal to x sub 0.05 minus Mu,"},{"Start":"12:27.590 ","End":"12:32.035","Text":"that\u0027s 100 divided by Sigma, that\u0027s 15."},{"Start":"12:32.035 ","End":"12:36.725","Text":"Okay, so that equals to,"},{"Start":"12:36.725 ","End":"12:41.600","Text":"let\u0027s just multiply both sides by 15 and add 100."},{"Start":"12:41.600 ","End":"12:45.075","Text":"So x sub 0.05,"},{"Start":"12:45.075 ","End":"12:48.940","Text":"that would be equal to 75.325."},{"Start":"12:50.900 ","End":"12:59.755","Text":"So this then is the 5th percentile. That\u0027s right here."},{"Start":"12:59.755 ","End":"13:03.200","Text":"That would be this value right here,"},{"Start":"13:03.200 ","End":"13:07.700","Text":"where 5 percent of the marks"},{"Start":"13:07.700 ","End":"13:13.430","Text":"fall below this value and 95 percent of the marks fall above this value."},{"Start":"13:13.430 ","End":"13:16.320","Text":"That\u0027s 75.325."}],"ID":13144},{"Watched":false,"Name":"Exercise 6 - Part c","Duration":"5m 34s","ChapterTopicVideoID":12666,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12666.jpeg","UploadDate":"2018-07-23T16:19:59.5500000","DurationForVideoObject":"PT5M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.640","Text":"In this section, we\u0027re given that 1 percent of the bottles with"},{"Start":"00:02.640 ","End":"00:05.700","Text":"the lowest volume are donated to charity and we\u0027re asked,"},{"Start":"00:05.700 ","End":"00:09.795","Text":"what\u0027s the maximum volume of bottles that are donated to charity?"},{"Start":"00:09.795 ","End":"00:13.095","Text":"Let\u0027s just draw our density function."},{"Start":"00:13.095 ","End":"00:20.135","Text":"Here\u0027s our density function and we\u0027re asked to find the specific value of X right here."},{"Start":"00:20.135 ","End":"00:24.750","Text":"We\u0027ll call this X_0.01 and why is that?"},{"Start":"00:24.750 ","End":"00:29.800","Text":"Well, this is the value of X where"},{"Start":"00:29.800 ","End":"00:35.714","Text":"the area under the density function from minus infinity till this value right here,"},{"Start":"00:35.714 ","End":"00:39.860","Text":"well, that equals 1 percent and that"},{"Start":"00:39.860 ","End":"00:44.720","Text":"means that from this value right here to plus infinity,"},{"Start":"00:44.720 ","End":"00:48.490","Text":"the area under the density function is 99 percent."},{"Start":"00:48.490 ","End":"00:53.220","Text":"From here, forwards that\u0027s 99 percent."},{"Start":"00:53.220 ","End":"00:58.170","Text":"Now how do we calculate this value of X?"},{"Start":"00:58.170 ","End":"01:00.790","Text":"Usually, we\u0027re given X,"},{"Start":"01:00.790 ","End":"01:03.875","Text":"we have to standardize it into a standard score,"},{"Start":"01:03.875 ","End":"01:07.760","Text":"and then we have to go to the standard table to find the probability."},{"Start":"01:07.760 ","End":"01:10.100","Text":"But here, we\u0027re working backwards."},{"Start":"01:10.100 ","End":"01:12.035","Text":"We\u0027re given the probability."},{"Start":"01:12.035 ","End":"01:18.840","Text":"We have to go see what Z is and from that, calculate our X."},{"Start":"01:19.780 ","End":"01:22.835","Text":"Let\u0027s see how we can do that."},{"Start":"01:22.835 ","End":"01:31.610","Text":"Well, the comparable Z value for X_0.01, well that\u0027s Z_0.01."},{"Start":"01:31.610 ","End":"01:36.320","Text":"Now, this is a negative value right here in the table, the standard table,"},{"Start":"01:36.320 ","End":"01:41.825","Text":"it doesn\u0027t support negative values for Z. What do we need to do?"},{"Start":"01:41.825 ","End":"01:49.880","Text":"We need to take the opposite symmetrical value of this number right here,"},{"Start":"01:49.880 ","End":"01:51.620","Text":"and that\u0027s right here."},{"Start":"01:51.620 ","End":"01:55.595","Text":"This is Z_0.99,"},{"Start":"01:55.595 ","End":"02:01.450","Text":"which is comparable to X_0.99."},{"Start":"02:01.450 ","End":"02:03.320","Text":"What\u0027s this value right here?"},{"Start":"02:03.320 ","End":"02:06.650","Text":"Well, this is a value where the area under"},{"Start":"02:06.650 ","End":"02:11.479","Text":"the density function from this value until plus infinity,"},{"Start":"02:11.479 ","End":"02:13.270","Text":"well that\u0027s 1 percent."},{"Start":"02:13.270 ","End":"02:19.160","Text":"That means that the area from minus infinity until this point,"},{"Start":"02:19.160 ","End":"02:22.070","Text":"well that\u0027s 99 percent."},{"Start":"02:22.070 ","End":"02:28.320","Text":"This then is the value that we\u0027re looking for right now."},{"Start":"02:28.330 ","End":"02:34.770","Text":"We\u0027re looking for Z_0.99 and we want to"},{"Start":"02:34.770 ","End":"02:41.051","Text":"know what that Z is that makes Phi of Z_0.99,"},{"Start":"02:41.051 ","End":"02:45.630","Text":"what makes that equal to 0.99?"},{"Start":"02:45.630 ","End":"02:48.520","Text":"Let\u0027s go to our table."},{"Start":"02:49.220 ","End":"02:55.890","Text":"Again, we\u0027re looking for Z_0.99,"},{"Start":"02:55.890 ","End":"03:00.765","Text":"where Phi of this number equals 0.99."},{"Start":"03:00.765 ","End":"03:07.670","Text":"Let\u0027s see what number within the table comes close to 0.99."},{"Start":"03:07.670 ","End":"03:10.980","Text":"Let\u0027s just scroll down here."},{"Start":"03:11.600 ","End":"03:18.525","Text":"We\u0027ve seen that this is the number right here."},{"Start":"03:18.525 ","End":"03:22.435","Text":"It\u0027s between this number right here and this number right here."},{"Start":"03:22.435 ","End":"03:29.340","Text":"That\u0027s where Z equals 2.3 to something."},{"Start":"03:29.340 ","End":"03:34.495","Text":"Now, let\u0027s see if we can get a more accurate value for Z."},{"Start":"03:34.495 ","End":"03:40.590","Text":"At Phi of Z, where Phi of Z equals 0.99,"},{"Start":"03:40.590 ","End":"03:45.060","Text":"well, Z then equals to 2.326."},{"Start":"03:45.060 ","End":"03:47.925","Text":"We\u0027ll use this value right here."},{"Start":"03:47.925 ","End":"03:51.520","Text":"Let\u0027s just scroll up."},{"Start":"03:52.070 ","End":"03:57.105","Text":"That means that Z_0.99,"},{"Start":"03:57.105 ","End":"04:01.285","Text":"well that equals to 2.326."},{"Start":"04:01.285 ","End":"04:03.710","Text":"Great, so let\u0027s use this value."},{"Start":"04:03.710 ","End":"04:08.760","Text":"Let\u0027s get back to our question and see how we can continue."},{"Start":"04:08.950 ","End":"04:12.950","Text":"We know that Z_0.99,"},{"Start":"04:12.950 ","End":"04:19.930","Text":"that equals to 2.326."},{"Start":"04:19.930 ","End":"04:26.270","Text":"The comparable or the symmetrical opposite value of this number,"},{"Start":"04:26.270 ","End":"04:32.045","Text":"which is this number, that would be minus 2.326."},{"Start":"04:32.045 ","End":"04:36.120","Text":"Great, so having said that,"},{"Start":"04:36.120 ","End":"04:45.510","Text":"we know that the standardization transformation Z equals to X minus Mu divided by Sigma."},{"Start":"04:45.510 ","End":"04:46.995","Text":"Let\u0027s just plug in the numbers."},{"Start":"04:46.995 ","End":"04:54.300","Text":"Z here would be minus 2.326 and that equals to X, in our case,"},{"Start":"04:54.300 ","End":"04:58.075","Text":"sub 0.01 minus 500,"},{"Start":"04:58.075 ","End":"04:59.540","Text":"that\u0027s our Mu,"},{"Start":"04:59.540 ","End":"05:02.720","Text":"divided by Sigma, that\u0027s 20."},{"Start":"05:02.720 ","End":"05:06.155","Text":"Let\u0027s multiply both sides by 20."},{"Start":"05:06.155 ","End":"05:10.565","Text":"We get minus 46 dot 52,"},{"Start":"05:10.565 ","End":"05:19.730","Text":"and that equals to X_0.01 minus 500 and that means that X_0.01,"},{"Start":"05:19.730 ","End":"05:26.215","Text":"that equals to 453.48."},{"Start":"05:26.215 ","End":"05:34.710","Text":"This then is the maximum volume in the bottles that are donated to charity."}],"ID":13145},{"Watched":false,"Name":"Exercise 6 - Parts a-b","Duration":"9m 24s","ChapterTopicVideoID":12667,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12667.jpeg","UploadDate":"2018-07-23T16:20:34.3000000","DurationForVideoObject":"PT9M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.840","Text":"In this question, we\u0027re given that the volume of a bottled beverage"},{"Start":"00:03.840 ","End":"00:07.635","Text":"has a normal distribution with a standard deviation of 20 milliliters."},{"Start":"00:07.635 ","End":"00:13.920","Text":"We assume that 33 percent of the bottles have a volume of over 508.8 milliliters."},{"Start":"00:13.920 ","End":"00:18.480","Text":"We\u0027re asked, what\u0027s the average volume of the bottled beverage?"},{"Start":"00:18.480 ","End":"00:23.460","Text":"First, we have to determine that we\u0027re dealing with a normal distribution. We are."},{"Start":"00:23.460 ","End":"00:25.785","Text":"That\u0027s given to us right here."},{"Start":"00:25.785 ","End":"00:28.935","Text":"The next step is to define a random variable."},{"Start":"00:28.935 ","End":"00:36.495","Text":"Call that X, and X would be the volume in milliliters."},{"Start":"00:36.495 ","End":"00:41.570","Text":"Now that has a normal distribution where we don\u0027t"},{"Start":"00:41.570 ","End":"00:46.535","Text":"know what our Mu is and our standard deviation,"},{"Start":"00:46.535 ","End":"00:50.400","Text":"that equals to 20 milliliters."},{"Start":"00:50.750 ","End":"00:53.585","Text":"Now that we have this information,"},{"Start":"00:53.585 ","End":"00:56.785","Text":"let\u0027s just draw our density function."},{"Start":"00:56.785 ","End":"00:59.240","Text":"Here\u0027s our density function."},{"Start":"00:59.240 ","End":"01:03.215","Text":"Now, this is our average right here, this is our Mu."},{"Start":"01:03.215 ","End":"01:04.310","Text":"We don\u0027t know what that is,"},{"Start":"01:04.310 ","End":"01:06.140","Text":"that\u0027s what we\u0027re looking for."},{"Start":"01:06.140 ","End":"01:07.790","Text":"But what do we know?"},{"Start":"01:07.790 ","End":"01:15.845","Text":"Well, we assume that 33 percent of the bottles have a volume of over 508.8 milliliters."},{"Start":"01:15.845 ","End":"01:19.355","Text":"That means we have this point right here."},{"Start":"01:19.355 ","End":"01:21.265","Text":"We\u0027ll call that x,"},{"Start":"01:21.265 ","End":"01:22.400","Text":"right on the x axis,"},{"Start":"01:22.400 ","End":"01:26.555","Text":"we\u0027ll call that the 508.8 milliliters."},{"Start":"01:26.555 ","End":"01:31.880","Text":"We know that 33 percent"},{"Start":"01:31.880 ","End":"01:37.980","Text":"of the area under the density function falls above this value right here."},{"Start":"01:38.030 ","End":"01:42.680","Text":"That means that 67 percent of"},{"Start":"01:42.680 ","End":"01:47.340","Text":"the area under the density function from below this area right here."},{"Start":"01:48.590 ","End":"01:56.510","Text":"How do we use this information to calculate our average?"},{"Start":"01:56.510 ","End":"02:01.990","Text":"Well, we know that the standardization transformation is this,"},{"Start":"02:01.990 ","End":"02:07.855","Text":"z equals to x minus Mu divided by Sigma."},{"Start":"02:07.855 ","End":"02:09.895","Text":"Now we have Sigma,"},{"Start":"02:09.895 ","End":"02:15.090","Text":"that\u0027s 20 and x, that\u0027s 508.8."},{"Start":"02:15.090 ","End":"02:17.410","Text":"What we don\u0027t have is Mu,"},{"Start":"02:17.410 ","End":"02:18.805","Text":"the thing that we\u0027re looking for,"},{"Start":"02:18.805 ","End":"02:23.800","Text":"but also we don\u0027t have the standard score for 508.8,"},{"Start":"02:23.800 ","End":"02:27.640","Text":"the comparable value on the z scale."},{"Start":"02:27.640 ","End":"02:30.520","Text":"Now, how do we get that?"},{"Start":"02:30.520 ","End":"02:33.196","Text":"Well, we have the proportion,"},{"Start":"02:33.196 ","End":"02:34.660","Text":"it\u0027s given to us."},{"Start":"02:34.660 ","End":"02:38.875","Text":"We know that Phi of z,"},{"Start":"02:38.875 ","End":"02:41.110","Text":"this z right here,"},{"Start":"02:41.110 ","End":"02:45.935","Text":"that has to equal to 0.67."},{"Start":"02:45.935 ","End":"02:49.665","Text":"We\u0027re looking for Phi of z,"},{"Start":"02:49.665 ","End":"02:54.600","Text":"that gives us 67 percent."},{"Start":"02:54.600 ","End":"02:59.919","Text":"What z will give us 67 percent."},{"Start":"02:59.919 ","End":"03:04.565","Text":"Now for that, we need to go to the standard table."},{"Start":"03:04.565 ","End":"03:08.040","Text":"Again, we\u0027re looking for z,"},{"Start":"03:08.040 ","End":"03:13.875","Text":"where Phi of z will equal 0.67."},{"Start":"03:13.875 ","End":"03:19.990","Text":"We\u0027re looking for the value here within the table that\u0027s closest to 0.67,"},{"Start":"03:19.990 ","End":"03:23.080","Text":"and that\u0027s here, right here."},{"Start":"03:23.080 ","End":"03:32.520","Text":"It\u0027s exactly 0.67 so we can say that the comparable z value right here,"},{"Start":"03:32.520 ","End":"03:37.515","Text":"that would be equal to 0.44."},{"Start":"03:37.515 ","End":"03:42.150","Text":"Phi of 0.44 will give us 0.67."},{"Start":"03:42.150 ","End":"03:46.420","Text":"Great. Let\u0027s now go back to our question."},{"Start":"03:47.570 ","End":"03:49.670","Text":"Z as we said,"},{"Start":"03:49.670 ","End":"03:54.365","Text":"0.44, so we\u0027ll just plug that in right here."},{"Start":"03:54.365 ","End":"03:57.935","Text":"Z, that\u0027s 0.44,"},{"Start":"03:57.935 ","End":"03:59.480","Text":"that equals to x."},{"Start":"03:59.480 ","End":"04:02.520","Text":"Now x is 508.8"},{"Start":"04:07.550 ","End":"04:09.860","Text":"minus Mu,"},{"Start":"04:09.860 ","End":"04:13.820","Text":"minus what we\u0027re looking for, divided by Sigma."},{"Start":"04:13.820 ","End":"04:19.200","Text":"We\u0027ve figured out that z right here 0.44."},{"Start":"04:19.460 ","End":"04:22.170","Text":"Now Sigma here."},{"Start":"04:22.170 ","End":"04:26.820","Text":"We said that that equals to 20."},{"Start":"04:26.820 ","End":"04:33.135","Text":"In essence we have an equation with 1 variable so let\u0027s just solve that."},{"Start":"04:33.135 ","End":"04:36.725","Text":"Let\u0027s multiply both sides by 20."},{"Start":"04:36.725 ","End":"04:41.330","Text":"That gives us 8.8"},{"Start":"04:41.830 ","End":"04:47.990","Text":"and that equals to 508.8 minus Mu."},{"Start":"04:47.990 ","End":"04:49.915","Text":"Let\u0027s isolate Mu."},{"Start":"04:49.915 ","End":"04:53.730","Text":"Mu then would be equal to 500."},{"Start":"04:53.730 ","End":"05:01.560","Text":"This then is the average volume of a bottled beverage right here, 500."},{"Start":"05:02.170 ","End":"05:05.900","Text":"In this section, we\u0027re given that 5 percent of the bottles that are"},{"Start":"05:05.900 ","End":"05:09.120","Text":"produced with the largest volume are sent for testing,"},{"Start":"05:09.120 ","End":"05:13.060","Text":"and we\u0027re asked starting from what volume are bottles sent for testing."},{"Start":"05:13.060 ","End":"05:16.160","Text":"Let\u0027s draw our density function."},{"Start":"05:16.160 ","End":"05:19.896","Text":"Here\u0027s our density function with an average of 500."},{"Start":"05:19.896 ","End":"05:22.715","Text":"We\u0027ve calculated that in Section A."},{"Start":"05:22.715 ","End":"05:29.380","Text":"What are we given? We want to find the value of x,"},{"Start":"05:29.380 ","End":"05:34.130","Text":"where 5 percent of the area under"},{"Start":"05:34.130 ","End":"05:39.760","Text":"the density function falls above this value is 5 percent."},{"Start":"05:39.760 ","End":"05:43.490","Text":"That means that 95 percent of the area under"},{"Start":"05:43.490 ","End":"05:48.650","Text":"the density function should fall below this value right here."},{"Start":"05:48.650 ","End":"05:53.615","Text":"This x would be called x_0.95,"},{"Start":"05:53.615 ","End":"05:56.370","Text":"that\u0027s the 95th percentile."},{"Start":"05:57.260 ","End":"05:59.780","Text":"How do we calculate this?"},{"Start":"05:59.780 ","End":"06:02.120","Text":"Well, usually we\u0027re given x."},{"Start":"06:02.120 ","End":"06:04.940","Text":"We have to standardize this into a standard score,"},{"Start":"06:04.940 ","End":"06:08.285","Text":"and then we have to go through the standard table to find a proportion."},{"Start":"06:08.285 ","End":"06:10.280","Text":"Here we have to work backwards."},{"Start":"06:10.280 ","End":"06:13.010","Text":"We\u0027re given the proportion that\u0027s 95 percent."},{"Start":"06:13.010 ","End":"06:17.855","Text":"We have to go to the table and see what our z score is."},{"Start":"06:17.855 ","End":"06:21.770","Text":"From that, we can calculate our x."},{"Start":"06:21.770 ","End":"06:27.055","Text":"We\u0027re looking for z_0.95."},{"Start":"06:27.055 ","End":"06:35.535","Text":"That\u0027s our special z that will give us Phi of z being equal to 95 percent."},{"Start":"06:35.535 ","End":"06:39.570","Text":"Again, we\u0027re looking for z_0.95,"},{"Start":"06:39.570 ","End":"06:45.720","Text":"where Phi of this z equals to 0.95."},{"Start":"06:45.720 ","End":"06:48.210","Text":"Let\u0027s go to our table."},{"Start":"06:48.210 ","End":"06:50.605","Text":"This is a table and, again,"},{"Start":"06:50.605 ","End":"06:58.695","Text":"we\u0027re looking for z sub 0.95 where Phi of this z,"},{"Start":"06:58.695 ","End":"07:02.685","Text":"that would be equal to 0.95."},{"Start":"07:02.685 ","End":"07:11.045","Text":"Let\u0027s see what value we have here within the table that\u0027s closest to 0.95."},{"Start":"07:11.045 ","End":"07:14.690","Text":"That\u0027s right in this row right here,"},{"Start":"07:14.690 ","End":"07:22.560","Text":"that\u0027s between 1.64 and 1.65."},{"Start":"07:22.560 ","End":"07:26.100","Text":"Between this value and this value."},{"Start":"07:26.100 ","End":"07:36.100","Text":"Our z_0.95 is between 1.64 and 1.65."},{"Start":"07:36.100 ","End":"07:38.930","Text":"Let\u0027s see if we can get a more exact value."},{"Start":"07:38.930 ","End":"07:41.270","Text":"Let\u0027s just scroll down here,"},{"Start":"07:41.270 ","End":"07:42.890","Text":"and here we go."},{"Start":"07:42.890 ","End":"07:53.890","Text":"Phi of z being 0.95 for the z value is 1.645 so we\u0027ll use that value right here."},{"Start":"07:55.790 ","End":"08:01.365","Text":"Z_0.95, that equals to 1.645."},{"Start":"08:01.365 ","End":"08:03.900","Text":"Let\u0027s use this value as we said,"},{"Start":"08:03.900 ","End":"08:07.050","Text":"and let\u0027s get back to our question now."},{"Start":"08:07.050 ","End":"08:11.445","Text":"Here we go. Z_0.95,"},{"Start":"08:11.445 ","End":"08:15.900","Text":"that equals to 1.645."},{"Start":"08:15.900 ","End":"08:22.075","Text":"What\u0027s the standardization transformation from x to z?"},{"Start":"08:22.075 ","End":"08:28.310","Text":"Well, we know that z equals to x minus Mu divided by Sigma."},{"Start":"08:28.310 ","End":"08:30.665","Text":"Let\u0027s just plug in the numbers here."},{"Start":"08:30.665 ","End":"08:36.540","Text":"Z is 1.645, so that\u0027ll be 1.645."},{"Start":"08:36.540 ","End":"08:41.055","Text":"That equals to x_0.95,"},{"Start":"08:41.055 ","End":"08:43.380","Text":"that\u0027s what we\u0027re looking for minus Mu,"},{"Start":"08:43.380 ","End":"08:45.190","Text":"where Mu is 500,"},{"Start":"08:45.190 ","End":"08:47.480","Text":"divided by Sigma,"},{"Start":"08:47.480 ","End":"08:49.375","Text":"where Sigma is 20."},{"Start":"08:49.375 ","End":"08:57.560","Text":"Let\u0027s multiply both sides by 20 and we get 32.9 and that equals"},{"Start":"08:57.560 ","End":"09:05.960","Text":"to x_0.95 minus 500 so x_0.95,"},{"Start":"09:05.960 ","End":"09:11.790","Text":"that equals to 532.9."},{"Start":"09:14.000 ","End":"09:24.060","Text":"This is the volume of the bottles above which the bottles are sent for testing."}],"ID":13146},{"Watched":false,"Name":"Exercise 7 - Part a-b","Duration":"6m 4s","ChapterTopicVideoID":12668,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12668.jpeg","UploadDate":"2018-07-23T16:20:56.5170000","DurationForVideoObject":"PT6M4S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.160","Text":"In this question, we\u0027re given that the lifespan of"},{"Start":"00:02.160 ","End":"00:05.070","Text":"a device has a normal probability distribution."},{"Start":"00:05.070 ","End":"00:11.430","Text":"Now it\u0027s known that half of the devices last less than 500 hours and that 67 percent of"},{"Start":"00:11.430 ","End":"00:14.730","Text":"the devices last less than 544 hours"},{"Start":"00:14.730 ","End":"00:18.720","Text":"and we\u0027re asked what\u0027s the average lifespan of a device?"},{"Start":"00:18.720 ","End":"00:21.750","Text":"Well, the first thing that we need to do is to make sure we\u0027re dealing with"},{"Start":"00:21.750 ","End":"00:24.840","Text":"a normal probability distribution and we are,"},{"Start":"00:24.840 ","End":"00:26.640","Text":"that\u0027s given right here."},{"Start":"00:26.640 ","End":"00:30.300","Text":"The next step is to define a random variable."},{"Start":"00:30.300 ","End":"00:40.140","Text":"Let\u0027s call that X and that\u0027ll be defined as the lifespan of a device in hours."},{"Start":"00:40.140 ","End":"00:44.570","Text":"As we said, that has a normal probability distribution,"},{"Start":"00:44.570 ","End":"00:52.385","Text":"but we don\u0027t know what the average is and we don\u0027t know what the standard deviation is."},{"Start":"00:52.385 ","End":"00:58.460","Text":"What do we know? Well, it\u0027s known that half of the devices last less than 500 hours."},{"Start":"00:58.460 ","End":"01:05.645","Text":"That means that we\u0027re looking at the probability of X being less than 500."},{"Start":"01:05.645 ","End":"01:11.015","Text":"Well, that equals 0.5 or 1/2."},{"Start":"01:11.015 ","End":"01:15.109","Text":"Well, that means that we\u0027re dealing with the median."},{"Start":"01:15.109 ","End":"01:22.654","Text":"Now, in a symmetrical distribution where the symmetry is around the average,"},{"Start":"01:22.654 ","End":"01:29.045","Text":"then this also means that we\u0027re dealing with the average."},{"Start":"01:29.045 ","End":"01:36.320","Text":"In another words, if we\u0027re given that half of the devices last less than 500 hours."},{"Start":"01:36.320 ","End":"01:41.655","Text":"Then we\u0027re dealing with here with 50 percent."},{"Start":"01:41.655 ","End":"01:47.975","Text":"Now, since this is a symmetrical distribution around the average right here,"},{"Start":"01:47.975 ","End":"01:51.715","Text":"then the average must equal 500."},{"Start":"01:51.715 ","End":"02:01.880","Text":"Now we know that X has a normal distribution where Mu equals 500."},{"Start":"02:01.880 ","End":"02:07.050","Text":"Still don\u0027t know what the standard deviation is."},{"Start":"02:07.750 ","End":"02:13.025","Text":"In this section, we\u0027re asked what\u0027s the standard deviation of the lifespan of the device."},{"Start":"02:13.025 ","End":"02:18.815","Text":"We know that X is distributed with a normal distribution."},{"Start":"02:18.815 ","End":"02:22.490","Text":"Now our Mu equals 500."},{"Start":"02:22.490 ","End":"02:25.910","Text":"We\u0027ve calculated then the last section and we still"},{"Start":"02:25.910 ","End":"02:29.180","Text":"don\u0027t know what the standard deviation is."},{"Start":"02:29.180 ","End":"02:33.710","Text":"Now, what piece of information do we still need to use?"},{"Start":"02:33.710 ","End":"02:36.940","Text":"Well, let\u0027s go take a look at the question."},{"Start":"02:36.940 ","End":"02:45.560","Text":"We\u0027re told that 67 percent of the devices last less than 544 hours."},{"Start":"02:45.560 ","End":"02:50.304","Text":"This is the extra piece of information that we need to use."},{"Start":"02:50.304 ","End":"03:00.470","Text":"This is our density function and we\u0027re told that when X is 544,"},{"Start":"03:00.470 ","End":"03:06.520","Text":"well, 33 percent of the devices are above this number."},{"Start":"03:06.520 ","End":"03:12.395","Text":"That means that 67 percent of the devices are below this number."},{"Start":"03:12.395 ","End":"03:19.235","Text":"Now, how do we extract the standard deviation?"},{"Start":"03:19.235 ","End":"03:24.605","Text":"Well, we know that the standardization transformation goes like this."},{"Start":"03:24.605 ","End":"03:33.050","Text":"It\u0027s Z that equals to X minus Mu divided by Sigma. There\u0027s our Sigma."},{"Start":"03:33.050 ","End":"03:35.959","Text":"Now, if we just plug in the numbers,"},{"Start":"03:35.959 ","End":"03:37.790","Text":"what are we going to get here?"},{"Start":"03:37.790 ","End":"03:40.562","Text":"Well, it\u0027ll be X,"},{"Start":"03:40.562 ","End":"03:50.145","Text":"544 minus 500 divided by Sigma."},{"Start":"03:50.145 ","End":"03:53.900","Text":"Now, that equals to what?"},{"Start":"03:53.900 ","End":"04:00.210","Text":"Well, we need to know what this Z value is right here."},{"Start":"04:00.440 ","End":"04:06.165","Text":"We know that Phi of Z,"},{"Start":"04:06.165 ","End":"04:10.934","Text":"well that equals to 0.67 or 67 percent"},{"Start":"04:10.934 ","End":"04:16.535","Text":"because from minus infinity till 544 on the x axis,"},{"Start":"04:16.535 ","End":"04:22.175","Text":"or from minus infinity till this specific Z on the z axis,"},{"Start":"04:22.175 ","End":"04:25.310","Text":"well, that equals to 67 percent."},{"Start":"04:25.310 ","End":"04:30.300","Text":"Let\u0027s go to our standard table to see what Phi of Z."},{"Start":"04:31.790 ","End":"04:42.665","Text":"Again, we\u0027re looking for a specific Z value where phi of this value equals 0.67."},{"Start":"04:42.665 ","End":"04:47.810","Text":"Now, let\u0027s see which value within"},{"Start":"04:47.810 ","End":"04:53.480","Text":"the table comes the closest to 0.67 and that\u0027s it right here."},{"Start":"04:53.480 ","End":"04:56.060","Text":"It comes exactly to 0.67,"},{"Start":"04:56.060 ","End":"05:00.770","Text":"so the Z value is 0.44."},{"Start":"05:00.770 ","End":"05:06.540","Text":"The Z we\u0027re looking for equals to 0.44."},{"Start":"05:06.540 ","End":"05:10.240","Text":"Great. Let\u0027s now go back to our question."},{"Start":"05:14.740 ","End":"05:17.195","Text":"We know then that Z,"},{"Start":"05:17.195 ","End":"05:20.735","Text":"well that equals to 0.44."},{"Start":"05:20.735 ","End":"05:23.800","Text":"Let\u0027s just plug that in right here."},{"Start":"05:23.800 ","End":"05:25.470","Text":"That\u0027ll be X,"},{"Start":"05:25.470 ","End":"05:35.490","Text":"544 minus 500 divided by Sigma are variable and that equals to 0.44."},{"Start":"05:35.490 ","End":"05:38.490","Text":"That\u0027s our Z value right here."},{"Start":"05:38.490 ","End":"05:41.955","Text":"Let\u0027s just do some basic math."},{"Start":"05:41.955 ","End":"05:45.360","Text":"That\u0027s 0.44 Sigma,"},{"Start":"05:45.360 ","End":"05:50.080","Text":"that equals to 544 minus 500."},{"Start":"05:50.080 ","End":"06:00.060","Text":"Now, that means that Sigma then equals to 44 divided by 0.44 and that equals to 100."},{"Start":"06:00.060 ","End":"06:05.319","Text":"This then is our standard deviation."}],"ID":13147},{"Watched":false,"Name":"Exercise 7 - Parts c-d","Duration":"9m 6s","ChapterTopicVideoID":12670,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12670.jpeg","UploadDate":"2018-07-23T16:21:44.3900000","DurationForVideoObject":"PT9M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.290","Text":"In this section we\u0027re asked,"},{"Start":"00:01.290 ","End":"00:07.350","Text":"what are the chances that a randomly selected device will last less than 460 hours?"},{"Start":"00:07.350 ","End":"00:11.400","Text":"First of all, let\u0027s remind ourselves X is a random variable,"},{"Start":"00:11.400 ","End":"00:12.990","Text":"that\u0027s the lifespan in hours,"},{"Start":"00:12.990 ","End":"00:19.333","Text":"and that has a normal distribution where Mu now equals to 500,"},{"Start":"00:19.333 ","End":"00:21.440","Text":"and the standard deviation,"},{"Start":"00:21.440 ","End":"00:24.765","Text":"that equals to 100 hours."},{"Start":"00:24.765 ","End":"00:28.140","Text":"Let\u0027s look at our density function."},{"Start":"00:28.140 ","End":"00:30.155","Text":"This is our density functions,"},{"Start":"00:30.155 ","End":"00:34.700","Text":"and we\u0027re interested in this value right here of X,"},{"Start":"00:34.700 ","End":"00:38.765","Text":"where X equals 460 hours."},{"Start":"00:38.765 ","End":"00:42.979","Text":"Why is that? Because we want to calculate"},{"Start":"00:42.979 ","End":"00:50.235","Text":"this probability right here from minus infinity till 460 hours."},{"Start":"00:50.235 ","End":"00:59.080","Text":"We\u0027re looking at the probability of X being less than 460 hours."},{"Start":"00:59.080 ","End":"01:02.130","Text":"Let\u0027s just write this out nicely."},{"Start":"01:02.130 ","End":"01:04.320","Text":"That\u0027s 460 hours."},{"Start":"01:04.320 ","End":"01:06.725","Text":"Now, whenever we have X,"},{"Start":"01:06.725 ","End":"01:09.995","Text":"the first thing that we want to do is we want to standardize it"},{"Start":"01:09.995 ","End":"01:14.405","Text":"and then go to the standard table to see what the probability is."},{"Start":"01:14.405 ","End":"01:20.930","Text":"We\u0027re looking for the comparable value of X on the Z scale."},{"Start":"01:20.930 ","End":"01:22.460","Text":"How do we do that?"},{"Start":"01:22.460 ","End":"01:24.835","Text":"Well, we have to standardize."},{"Start":"01:24.835 ","End":"01:30.970","Text":"How do we standardize? Well, that\u0027s Z equals to X minus Mu divided by Sigma."},{"Start":"01:30.970 ","End":"01:34.070","Text":"In our case, well, that\u0027s the probability."},{"Start":"01:34.070 ","End":"01:38.135","Text":"Now, it\u0027s X minus Mu divided by Sigma,"},{"Start":"01:38.135 ","End":"01:45.480","Text":"that has to be less than 460, minus Mu."},{"Start":"01:45.480 ","End":"01:47.265","Text":"Now, Mu is 500,"},{"Start":"01:47.265 ","End":"01:50.845","Text":"divided by 100, that\u0027s Sigma."},{"Start":"01:50.845 ","End":"01:54.600","Text":"Now, we know that this expression right here is Z,"},{"Start":"01:54.600 ","End":"01:59.830","Text":"so that\u0027s the probability of Z being less than."},{"Start":"01:59.830 ","End":"02:03.830","Text":"Now, that\u0027ll be 460 minus 500,"},{"Start":"02:03.830 ","End":"02:08.760","Text":"that\u0027s minus 40 divided by 100."},{"Start":"02:08.800 ","End":"02:20.100","Text":"That equals to Phi of Z at minus 0.4."},{"Start":"02:20.110 ","End":"02:26.220","Text":"We know that the standard table doesn\u0027t support negative values of Z,"},{"Start":"02:26.220 ","End":"02:28.055","Text":"so what do we need to do?"},{"Start":"02:28.055 ","End":"02:32.510","Text":"Well, we need to take the opposite symmetrical point."},{"Start":"02:32.510 ","End":"02:35.510","Text":"We know that this Z right here,"},{"Start":"02:35.510 ","End":"02:38.135","Text":"that equals to minus 0.4,"},{"Start":"02:38.135 ","End":"02:43.585","Text":"and that\u0027s comparable to 460 on the x-axis."},{"Start":"02:43.585 ","End":"02:49.985","Text":"But now what we need to do is we need to find this area right here,"},{"Start":"02:49.985 ","End":"02:55.985","Text":"which has the same value as this area right here."},{"Start":"02:55.985 ","End":"03:01.015","Text":"Now, because the density function is symmetrical,"},{"Start":"03:01.015 ","End":"03:07.550","Text":"then we\u0027re looking at Z being equal to 0.4 right here."},{"Start":"03:08.750 ","End":"03:13.780","Text":"When we say Phi of 0.4,"},{"Start":"03:13.780 ","End":"03:22.780","Text":"then we\u0027re actually looking at this probability right here from minus infinity till 0.4."},{"Start":"03:22.780 ","End":"03:24.150","Text":"But we\u0027re not interested in that."},{"Start":"03:24.150 ","End":"03:26.845","Text":"We\u0027re interested in this area right here."},{"Start":"03:26.845 ","End":"03:36.180","Text":"That means that this area right here is 1 minus Z at 0.4."},{"Start":"03:36.180 ","End":"03:41.985","Text":"That equals to 1 minus Phi at 0.4."},{"Start":"03:41.985 ","End":"03:49.990","Text":"Now we\u0027re ready to go to our table to see what Phi of 0.4 is."},{"Start":"03:50.180 ","End":"03:55.995","Text":"Again, we\u0027re looking at Phi of 0.4."},{"Start":"03:55.995 ","End":"04:03.390","Text":"Now, let\u0027s look at 0.4 as a Z value right here."},{"Start":"04:03.390 ","End":"04:06.880","Text":"This is the row right here, 0.40."},{"Start":"04:07.340 ","End":"04:18.465","Text":"This then is the value of Phi at 0.4 that equals to 0.6554."},{"Start":"04:18.465 ","End":"04:20.990","Text":"Great. Now that we have that,"},{"Start":"04:20.990 ","End":"04:23.430","Text":"let\u0027s get back to our question."},{"Start":"04:24.370 ","End":"04:30.230","Text":"That then equals to 1 minus Phi at 0.4,"},{"Start":"04:30.230 ","End":"04:36.650","Text":"we said that was 0.6554,"},{"Start":"04:36.650 ","End":"04:44.485","Text":"and that equals to 0.3446."},{"Start":"04:44.485 ","End":"04:48.470","Text":"The chances that a randomly selected device will last"},{"Start":"04:48.470 ","End":"04:54.365","Text":"less than 460 hours, that\u0027s 0.3446."},{"Start":"04:54.365 ","End":"05:00.110","Text":"In this section, we\u0027re asked what\u0027s the upper 1 percentile of a device\u0027s lifespan?"},{"Start":"05:00.110 ","End":"05:04.165","Text":"First of all, let\u0027s just draw the density function."},{"Start":"05:04.165 ","End":"05:06.525","Text":"This is our density function,"},{"Start":"05:06.525 ","End":"05:10.070","Text":"and we\u0027re looking for that special value of X."},{"Start":"05:10.070 ","End":"05:14.555","Text":"We\u0027ll call this X_0.99. Why is that?"},{"Start":"05:14.555 ","End":"05:18.230","Text":"Well, we\u0027re looking for the value where the area"},{"Start":"05:18.230 ","End":"05:21.790","Text":"under the density function for Xs above this value,"},{"Start":"05:21.790 ","End":"05:24.470","Text":"that\u0027s this shaded area right here,"},{"Start":"05:24.470 ","End":"05:27.425","Text":"that has to be equal to 1 percent."},{"Start":"05:27.425 ","End":"05:33.650","Text":"That means that the area under the density function from minus infinity to this value,"},{"Start":"05:33.650 ","End":"05:37.440","Text":"that has to be equal to 99 percent."},{"Start":"05:37.730 ","End":"05:41.255","Text":"How do we calculate this value right here?"},{"Start":"05:41.255 ","End":"05:43.625","Text":"Usually we\u0027re given X,"},{"Start":"05:43.625 ","End":"05:46.170","Text":"and then we have to standardize it,"},{"Start":"05:46.170 ","End":"05:50.300","Text":"and then we have to go to the standard table to find the probability."},{"Start":"05:50.300 ","End":"05:53.360","Text":"Well, here we have to work backwards because we\u0027re"},{"Start":"05:53.360 ","End":"05:56.705","Text":"given the probability or the proportion,"},{"Start":"05:56.705 ","End":"05:59.875","Text":"and we\u0027ll have to go back to calculate X."},{"Start":"05:59.875 ","End":"06:03.005","Text":"We\u0027re going to have to go to the table,"},{"Start":"06:03.005 ","End":"06:05.150","Text":"see what Z is,"},{"Start":"06:05.150 ","End":"06:07.865","Text":"and then calculate our X."},{"Start":"06:07.865 ","End":"06:10.560","Text":"Now, when we\u0027re talking about this Z,"},{"Start":"06:10.560 ","End":"06:13.155","Text":"well, this is a special Z."},{"Start":"06:13.155 ","End":"06:16.883","Text":"We\u0027ll call this Z_0.99,"},{"Start":"06:16.883 ","End":"06:24.825","Text":"and that\u0027ll be the value of Z that\u0027s comparable to X_0.99 on the x-axis."},{"Start":"06:24.825 ","End":"06:29.670","Text":"Again, we\u0027re looking for Z_0.99,"},{"Start":"06:29.670 ","End":"06:32.436","Text":"where Phi of Z,"},{"Start":"06:32.436 ","End":"06:36.390","Text":"this special Z, that equals to 0.99."},{"Start":"06:36.390 ","End":"06:44.094","Text":"The area under the density function from minus infinity to Z_0.99,"},{"Start":"06:44.094 ","End":"06:46.285","Text":"that\u0027s under Z scale."},{"Start":"06:46.285 ","End":"06:49.500","Text":"Let\u0027s go to the table."},{"Start":"06:49.500 ","End":"06:51.950","Text":"This is our table, and again,"},{"Start":"06:51.950 ","End":"06:55.475","Text":"we\u0027re looking for Z_0.99,"},{"Start":"06:55.475 ","End":"06:57.880","Text":"where Phi of this Z,"},{"Start":"06:57.880 ","End":"07:00.930","Text":"that has to be equal to 0.99."},{"Start":"07:00.930 ","End":"07:09.215","Text":"We\u0027re looking within the table right here for a value that\u0027s closest to 0.99."},{"Start":"07:09.215 ","End":"07:12.570","Text":"Let\u0027s just scroll down here."},{"Start":"07:13.610 ","End":"07:19.260","Text":"Right in this row, we see these 2 values."},{"Start":"07:19.260 ","End":"07:25.375","Text":"Z then has to be equal to 2.32 something."},{"Start":"07:25.375 ","End":"07:29.990","Text":"Now, let\u0027s scroll down and see if we can get a more exact value for Z."},{"Start":"07:29.990 ","End":"07:36.210","Text":"For Phi being equal to 0.99,"},{"Start":"07:36.210 ","End":"07:40.860","Text":"the exact value of Z is 2.326."},{"Start":"07:40.860 ","End":"07:44.410","Text":"We\u0027ll be using this value for Z."},{"Start":"07:44.720 ","End":"07:50.385","Text":"We know then that Z_0.99,"},{"Start":"07:50.385 ","End":"07:57.420","Text":"that equals to 2.326."},{"Start":"07:57.420 ","End":"07:58.955","Text":"Now how do we use that?"},{"Start":"07:58.955 ","End":"08:04.640","Text":"Well, we know that the transformation for standardization is this."},{"Start":"08:04.640 ","End":"08:08.555","Text":"That\u0027s Z, that\u0027s equal to X minus Mu divided by Sigma."},{"Start":"08:08.555 ","End":"08:12.660","Text":"In our case, Z would be 2.326."},{"Start":"08:12.660 ","End":"08:15.990","Text":"Let\u0027s just plug in the numbers, 2.326,"},{"Start":"08:15.990 ","End":"08:20.490","Text":"that equals to X_0.99,"},{"Start":"08:20.490 ","End":"08:23.475","Text":"that\u0027s what we want to find out, minus Mu."},{"Start":"08:23.475 ","End":"08:27.300","Text":"Now, Mu is 500, divided by Sigma."},{"Start":"08:27.300 ","End":"08:29.385","Text":"That\u0027s 100."},{"Start":"08:29.385 ","End":"08:32.345","Text":"Let\u0027s just do a little bit of quick math."},{"Start":"08:32.345 ","End":"08:35.240","Text":"We\u0027ll multiply both sides by 100,"},{"Start":"08:35.240 ","End":"08:39.645","Text":"so that\u0027ll be 232.6,"},{"Start":"08:39.645 ","End":"08:45.620","Text":"and that equals to X_0.99 minus 500,"},{"Start":"08:45.620 ","End":"08:52.720","Text":"or X_0.99, and that equals to 732.6."},{"Start":"08:53.270 ","End":"08:56.660","Text":"This value for X right here,"},{"Start":"08:56.660 ","End":"09:01.310","Text":"that\u0027s the upper 1 percentile of a device\u0027s lifespan,"},{"Start":"09:01.310 ","End":"09:05.130","Text":"or the 99th percentile."}],"ID":13148},{"Watched":false,"Name":"Exercise 7 - Part e","Duration":"4m 3s","ChapterTopicVideoID":12669,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12669.jpeg","UploadDate":"2018-07-23T16:21:17.2370000","DurationForVideoObject":"PT4M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.819","Text":"In this section, we\u0027re given the 1 percent of the devices"},{"Start":"00:02.819 ","End":"00:05.549","Text":"with the shortest lifespan are sent to the laboratory"},{"Start":"00:05.549 ","End":"00:07.980","Text":"for a thorough check and we\u0027re asked what\u0027s"},{"Start":"00:07.980 ","End":"00:12.090","Text":"the maximum lifespan of a device sent to the laboratory?"},{"Start":"00:12.090 ","End":"00:16.440","Text":"Let\u0027s just look at our density function here."},{"Start":"00:16.440 ","End":"00:23.820","Text":"Here\u0027s our density function and we\u0027re looking for that special value of X right here."},{"Start":"00:23.820 ","End":"00:26.880","Text":"We\u0027ll call this X_0.01,"},{"Start":"00:26.880 ","End":"00:33.090","Text":"where 1 percent of the area under the density function"},{"Start":"00:33.090 ","End":"00:40.420","Text":"falls between minus infinity and X_0.01."},{"Start":"00:40.420 ","End":"00:43.575","Text":"That means that this shaded area right here,"},{"Start":"00:43.575 ","End":"00:48.920","Text":"that\u0027s 1 percent and that means that the rest of the area under the density functional,"},{"Start":"00:48.920 ","End":"00:54.545","Text":"that\u0027s 99 percent from X_0.01 till plus infinity."},{"Start":"00:54.545 ","End":"00:56.180","Text":"Now, how do we do that?"},{"Start":"00:56.180 ","End":"00:58.700","Text":"How do we calculate this X?"},{"Start":"00:58.700 ","End":"01:04.160","Text":"Usually, we\u0027re given our X and then we have to"},{"Start":"01:04.160 ","End":"01:06.890","Text":"standardize it into our z score and then"},{"Start":"01:06.890 ","End":"01:09.890","Text":"go to the standard table to calculate the probability."},{"Start":"01:09.890 ","End":"01:11.960","Text":"Well, here we have to work backwards."},{"Start":"01:11.960 ","End":"01:14.550","Text":"We\u0027re given the probability."},{"Start":"01:14.550 ","End":"01:18.750","Text":"We have to go to our table to see what the z value is,"},{"Start":"01:18.750 ","End":"01:22.600","Text":"and then calculate our X."},{"Start":"01:23.000 ","End":"01:32.630","Text":"We\u0027re looking here for this value which is comparable to the X value on the x-axis."},{"Start":"01:32.630 ","End":"01:35.940","Text":"We\u0027ll call this z_0.01."},{"Start":"01:37.010 ","End":"01:40.970","Text":"Now, if we take a look at this problem,"},{"Start":"01:40.970 ","End":"01:45.320","Text":"we see that it\u0027s the exact mirror image of what we did in section"},{"Start":"01:45.320 ","End":"01:48.470","Text":"D. Let\u0027s just recall what we did there"},{"Start":"01:48.470 ","End":"01:52.650","Text":"and see how this can help us in solving this problem."},{"Start":"01:53.060 ","End":"01:55.670","Text":"Here\u0027s what we did in section D,"},{"Start":"01:55.670 ","End":"02:03.900","Text":"where we\u0027re looking for the 99th percentile or the top 1 percentile of this distribution."},{"Start":"02:03.900 ","End":"02:08.535","Text":"We calculated that to be 732.6 and"},{"Start":"02:08.535 ","End":"02:15.395","Text":"the comparable z value was 2.326."},{"Start":"02:15.395 ","End":"02:17.380","Text":"Now, as we can see,"},{"Start":"02:17.380 ","End":"02:19.795","Text":"this is the exact mirror image."},{"Start":"02:19.795 ","End":"02:24.160","Text":"Instead of looking for the 1 percentile on the right,"},{"Start":"02:24.160 ","End":"02:26.485","Text":"we\u0027re looking for the 1 percentile on the left."},{"Start":"02:26.485 ","End":"02:31.824","Text":"Now because the distribution is symmetrical around the average,"},{"Start":"02:31.824 ","End":"02:35.545","Text":"then that means that z_0.01,"},{"Start":"02:35.545 ","End":"02:42.195","Text":"well that equals to minus 2.326."},{"Start":"02:42.195 ","End":"02:46.155","Text":"It\u0027s the same value only minus."},{"Start":"02:46.155 ","End":"02:50.205","Text":"We\u0027ve figured out then what our z value is."},{"Start":"02:50.205 ","End":"02:53.125","Text":"How do we calculate our x value?"},{"Start":"02:53.125 ","End":"03:00.335","Text":"Well, we know that z equals to x minus Mu divided by Sigma."},{"Start":"03:00.335 ","End":"03:02.765","Text":"Let\u0027s just plug in the numbers."},{"Start":"03:02.765 ","End":"03:07.820","Text":"We get minus 2.326."},{"Start":"03:07.820 ","End":"03:09.470","Text":"That\u0027s this value right here."},{"Start":"03:09.470 ","End":"03:13.670","Text":"Well, that equals to x_0.01."},{"Start":"03:13.670 ","End":"03:16.730","Text":"That\u0027s what we\u0027re trying to find out, minus Mu,"},{"Start":"03:16.730 ","End":"03:22.475","Text":"where Mu is 500, divided by 100."},{"Start":"03:22.475 ","End":"03:26.065","Text":"That\u0027s our standard deviation."},{"Start":"03:26.065 ","End":"03:31.070","Text":"Let\u0027s just multiply both sides by 100 and we get minus"},{"Start":"03:31.070 ","End":"03:38.345","Text":"232.6 and that equals to x sub 0.01 minus 500."},{"Start":"03:38.345 ","End":"03:41.532","Text":"That means that x_0.0,"},{"Start":"03:41.532 ","End":"03:44.420","Text":"that\u0027s the value that we\u0027re looking for,"},{"Start":"03:44.420 ","End":"03:50.630","Text":"well that equals to 267.4."},{"Start":"03:50.630 ","End":"03:54.380","Text":"The units here are in hours."},{"Start":"03:54.380 ","End":"04:02.760","Text":"This then is the maximum lifespan of the device sent to the laboratory."}],"ID":13149},{"Watched":false,"Name":"Exercise 8","Duration":"3m 31s","ChapterTopicVideoID":12671,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12671.jpeg","UploadDate":"2018-07-23T16:21:54.1830000","DurationForVideoObject":"PT3M31S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.070","Text":"In this question, we\u0027re given that the following are"},{"Start":"00:02.070 ","End":"00:04.680","Text":"3 normal probability distributions of"},{"Start":"00:04.680 ","End":"00:08.685","Text":"3 different groups sketched on a system of coordinate axes."},{"Start":"00:08.685 ","End":"00:12.960","Text":"Now, that means that we have 3 different distributions"},{"Start":"00:12.960 ","End":"00:18.045","Text":"here that are sketched on the same x axis,"},{"Start":"00:18.045 ","End":"00:20.760","Text":"where x could be a random variable."},{"Start":"00:20.760 ","End":"00:25.455","Text":"If we look at the direction of the arrow here,"},{"Start":"00:25.455 ","End":"00:26.970","Text":"we see that here we have"},{"Start":"00:26.970 ","End":"00:30.990","Text":"the smaller numbers of x and here we have the higher number of x."},{"Start":"00:30.990 ","End":"00:36.037","Text":"Also, we see that the average of distributions 1 and 2 are the same,"},{"Start":"00:36.037 ","End":"00:40.535","Text":"so that means that Mu_1 equals to Mu_2."},{"Start":"00:40.535 ","End":"00:44.320","Text":"We also see that Mu_3,"},{"Start":"00:44.320 ","End":"00:47.105","Text":"that\u0027s the average of distribution number 3,"},{"Start":"00:47.105 ","End":"00:50.060","Text":"is to the right of these guys right here,"},{"Start":"00:50.060 ","End":"00:53.270","Text":"that means that it\u0027s greater than these guys."},{"Start":"00:53.270 ","End":"00:58.955","Text":"So that means that Mu_3 then is greater than Mu_1,"},{"Start":"00:58.955 ","End":"01:00.905","Text":"which equals to Mu_2."},{"Start":"01:00.905 ","End":"01:04.970","Text":"So the answer to Section A right here is"},{"Start":"01:04.970 ","End":"01:09.745","Text":"that distribution number 3 has the higher average."},{"Start":"01:09.745 ","End":"01:11.690","Text":"In Section B, we\u0027re asked,"},{"Start":"01:11.690 ","End":"01:16.100","Text":"in which of the following measures are distributions 1 and 2 the same?"},{"Start":"01:16.100 ","End":"01:17.989","Text":"In their upper 10th percentile,"},{"Start":"01:17.989 ","End":"01:20.890","Text":"in their average, or in their variance?"},{"Start":"01:20.890 ","End":"01:23.689","Text":"Let\u0027s first take a look at the variance."},{"Start":"01:23.689 ","End":"01:26.710","Text":"The variance of distribution number 1,"},{"Start":"01:26.710 ","End":"01:28.205","Text":"as we can see,"},{"Start":"01:28.205 ","End":"01:33.635","Text":"is much smaller than that of variance of distribution 2."},{"Start":"01:33.635 ","End":"01:39.170","Text":"The probability of falling around the average in distribution"},{"Start":"01:39.170 ","End":"01:46.410","Text":"1 is much higher than the probability of falling around the average in distribution 2."},{"Start":"01:46.410 ","End":"01:51.560","Text":"So their averages are not the same."},{"Start":"01:51.560 ","End":"01:54.515","Text":"Now, what about their upper 10th percentile?"},{"Start":"01:54.515 ","End":"01:57.260","Text":"Well, the upper 10th percentile of distribution 2,"},{"Start":"01:57.260 ","End":"01:58.640","Text":"that\u0027s somewhere around here."},{"Start":"01:58.640 ","End":"02:00.590","Text":"We call that x_2."},{"Start":"02:00.590 ","End":"02:04.555","Text":"Of x_1, well, that\u0027s right here."},{"Start":"02:04.555 ","End":"02:09.420","Text":"We can see that x_1 does not equal to x_2,"},{"Start":"02:09.420 ","End":"02:16.400","Text":"so the upper 10th percentile is not the same in the 2 distributions."},{"Start":"02:16.400 ","End":"02:17.480","Text":"Now what about their average?"},{"Start":"02:17.480 ","End":"02:23.735","Text":"Well, we saw that in Section A Mu_1 does equal to Mu_2."},{"Start":"02:23.735 ","End":"02:29.975","Text":"The averages are the same right there so that means that Mu_1 then"},{"Start":"02:29.975 ","End":"02:37.265","Text":"equals to Mu_2 so they\u0027re the same in their averages."},{"Start":"02:37.265 ","End":"02:39.200","Text":"In this section, we\u0027re asked,"},{"Start":"02:39.200 ","End":"02:42.002","Text":"which distribution has the smallest standard deviation?"},{"Start":"02:42.002 ","End":"02:44.155","Text":"Distribution 1, 2, 3,"},{"Start":"02:44.155 ","End":"02:46.720","Text":"or no opinion whatsoever?"},{"Start":"02:46.720 ","End":"02:49.960","Text":"Let\u0027s just scroll up and look at our distributions."},{"Start":"02:49.960 ","End":"02:56.185","Text":"As we can see, let\u0027s compare distribution 2 and distribution 3 right here."},{"Start":"02:56.185 ","End":"02:59.720","Text":"We see that they\u0027re pretty much the same with respect to"},{"Start":"02:59.720 ","End":"03:03.515","Text":"how scattered they are around the average."},{"Start":"03:03.515 ","End":"03:05.120","Text":"They\u0027re pretty wide."},{"Start":"03:05.120 ","End":"03:08.915","Text":"They\u0027re not as concentrated around their average."},{"Start":"03:08.915 ","End":"03:13.305","Text":"On the other hand, distribution 1 is not as scattered;"},{"Start":"03:13.305 ","End":"03:15.270","Text":"it\u0027s thinner and taller."},{"Start":"03:15.270 ","End":"03:19.654","Text":"That means that it\u0027s much more compact around the average."},{"Start":"03:19.654 ","End":"03:24.020","Text":"That means that our answer then would be"},{"Start":"03:24.020 ","End":"03:30.780","Text":"that distribution 1 that has the smallest standard deviation."}],"ID":13150},{"Watched":false,"Name":"Exercise 9 - Parts a-b","Duration":"8m 18s","ChapterTopicVideoID":12672,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12672.jpeg","UploadDate":"2018-07-23T16:22:30.4770000","DurationForVideoObject":"PT8M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.630","Text":"In this question we\u0027re given that the time it takes a person to get work has"},{"Start":"00:03.630 ","End":"00:06.060","Text":"a normal probability distribution with an average of"},{"Start":"00:06.060 ","End":"00:09.360","Text":"40 minutes and a standard deviation of 5 minutes."},{"Start":"00:09.360 ","End":"00:12.060","Text":"We\u0027re asked, what\u0027s the probability that it takes"},{"Start":"00:12.060 ","End":"00:15.750","Text":"at least 45 minutes for a person to get to work?"},{"Start":"00:15.750 ","End":"00:18.710","Text":"The first thing that we need to know is that we\u0027re dealing with"},{"Start":"00:18.710 ","End":"00:21.220","Text":"a normal probability distribution,"},{"Start":"00:21.220 ","End":"00:23.150","Text":"that\u0027s given right here."},{"Start":"00:23.150 ","End":"00:26.810","Text":"The next thing that we need to do is define a random variable."},{"Start":"00:26.810 ","End":"00:28.475","Text":"Let\u0027s call it X."},{"Start":"00:28.475 ","End":"00:35.520","Text":"That\u0027ll be defined as the time we get to work,"},{"Start":"00:35.520 ","End":"00:38.920","Text":"and that\u0027ll be in minutes."},{"Start":"00:38.920 ","End":"00:44.625","Text":"That has a normal distribution where Mu, the average,"},{"Start":"00:44.625 ","End":"00:47.670","Text":"equals 40, that\u0027s given right here,"},{"Start":"00:47.670 ","End":"00:49.830","Text":"and the standard deviation,"},{"Start":"00:49.830 ","End":"00:52.810","Text":"well that\u0027s 5 minutes."},{"Start":"00:53.420 ","End":"00:58.160","Text":"We\u0027re asked about the probability"},{"Start":"00:58.160 ","End":"01:01.920","Text":"that it takes at least 45 minutes for a person to get work."},{"Start":"01:01.920 ","End":"01:08.755","Text":"We\u0027re looking for the probability of X being greater than 45."},{"Start":"01:08.755 ","End":"01:12.480","Text":"Let\u0027s just draw a density function."},{"Start":"01:12.480 ","End":"01:14.520","Text":"Here\u0027s our density function."},{"Start":"01:14.520 ","End":"01:16.665","Text":"Let\u0027s just put it on the data."},{"Start":"01:16.665 ","End":"01:19.065","Text":"The average is 40."},{"Start":"01:19.065 ","End":"01:22.340","Text":"That\u0027ll be right here, this is the average,"},{"Start":"01:22.340 ","End":"01:25.495","Text":"that\u0027s on the x-axis."},{"Start":"01:25.495 ","End":"01:30.350","Text":"We\u0027re looking for the probability of x being greater than 45,"},{"Start":"01:30.350 ","End":"01:34.480","Text":"so we\u0027ll just define here the point 45."},{"Start":"01:34.480 ","End":"01:39.140","Text":"We\u0027re interested in this area right here,"},{"Start":"01:39.140 ","End":"01:44.440","Text":"we want to know the probability where x is greater than 45."},{"Start":"01:44.440 ","End":"01:51.170","Text":"What we need to do is we need to standardize this number right here, 45."},{"Start":"01:51.170 ","End":"01:53.555","Text":"How do we do that?"},{"Start":"01:53.555 ","End":"01:58.040","Text":"Well, we know that the standardization process,"},{"Start":"01:58.040 ","End":"02:04.580","Text":"that\u0027s z, and that equals to x minus Mu divided by Sigma."},{"Start":"02:04.580 ","End":"02:08.190","Text":"We have our x, that\u0027s 45."},{"Start":"02:08.190 ","End":"02:11.100","Text":"We have to standardize that,"},{"Start":"02:11.100 ","End":"02:16.745","Text":"and then we have to go to our standard table to see what the probability is."},{"Start":"02:16.745 ","End":"02:21.045","Text":"Let\u0027s write now standardize x."},{"Start":"02:21.045 ","End":"02:28.190","Text":"We\u0027re looking at the probability of X minus Mu divided by Sigma,"},{"Start":"02:28.190 ","End":"02:31.970","Text":"that has to be greater than 45,"},{"Start":"02:31.970 ","End":"02:34.015","Text":"that\u0027s our x minus Mu."},{"Start":"02:34.015 ","End":"02:37.110","Text":"Mu is 40, so that\u0027ll be 40,"},{"Start":"02:37.110 ","End":"02:41.140","Text":"divided by Sigma where Sigma is 5."},{"Start":"02:41.300 ","End":"02:44.625","Text":"We\u0027re looking at the probability."},{"Start":"02:44.625 ","End":"02:47.760","Text":"X minus Mu divided by Sigma, well, that\u0027s z."},{"Start":"02:47.760 ","End":"02:54.260","Text":"That\u0027ll be the probability of z being greater than 45 minus 40,"},{"Start":"02:54.260 ","End":"02:57.020","Text":"that\u0027s 5 divided by 5, that\u0027s 1."},{"Start":"02:57.020 ","End":"03:02.225","Text":"Here, the comparable value of x equaling 45,"},{"Start":"03:02.225 ","End":"03:04.865","Text":"that\u0027s z equaling 1."},{"Start":"03:04.865 ","End":"03:14.040","Text":"Since the table gives us the probability of z being less than a value,"},{"Start":"03:14.040 ","End":"03:21.850","Text":"then this equals to 1 minus the probability of z being less than 1."},{"Start":"03:21.850 ","End":"03:25.010","Text":"The table gives us this area right here,"},{"Start":"03:25.010 ","End":"03:28.370","Text":"not this area but this area right here."},{"Start":"03:28.370 ","End":"03:30.580","Text":"Well, that\u0027s 1 minus this area."},{"Start":"03:30.580 ","End":"03:32.255","Text":"This is what we\u0027ve written down here."},{"Start":"03:32.255 ","End":"03:37.040","Text":"That equals to 1 minus Phi of 1."},{"Start":"03:37.040 ","End":"03:40.970","Text":"You can believe me or not,"},{"Start":"03:40.970 ","End":"03:48.060","Text":"but Phi of 1 equals to 0.8413,"},{"Start":"03:48.060 ","End":"03:55.676","Text":"and I invite you to look that up and check me."},{"Start":"03:55.676 ","End":"04:00.720","Text":"That means that 1 minus Phi of 1 is 1 minus"},{"Start":"04:00.720 ","End":"04:09.360","Text":"0.8413 and that equals to 0.1587."},{"Start":"04:09.360 ","End":"04:16.100","Text":"This is the probability that it takes at least 45 minutes for a person to get to work."},{"Start":"04:16.100 ","End":"04:22.655","Text":"In section b, we\u0027re given that a person leaves home at 8:10 to get to work by 9 o\u0027clock."},{"Start":"04:22.655 ","End":"04:27.070","Text":"We\u0027re asked, what are the chances that that person will be late?"},{"Start":"04:27.070 ","End":"04:32.100","Text":"That means then that it took the person"},{"Start":"04:32.100 ","End":"04:38.595","Text":"50 minutes to get to work from 8:10-9:00,"},{"Start":"04:38.595 ","End":"04:39.750","Text":"that\u0027s 50 minutes,"},{"Start":"04:39.750 ","End":"04:44.090","Text":"and we\u0027re asked what are the chances that the person will be late?"},{"Start":"04:44.090 ","End":"04:48.365","Text":"That means that we\u0027re looking for the probability of X,"},{"Start":"04:48.365 ","End":"04:51.830","Text":"a random variable that says that\u0027s the time it gets to work,"},{"Start":"04:51.830 ","End":"04:54.660","Text":"has to be greater than"},{"Start":"04:54.660 ","End":"05:00.090","Text":"15 minutes because if he gets to work by 9 o\u0027clock then he won\u0027t be late."},{"Start":"05:00.090 ","End":"05:02.840","Text":"What are the chances that the person would be late?"},{"Start":"05:02.840 ","End":"05:05.715","Text":"That means that he\u0027ll arrive after 9:00."},{"Start":"05:05.715 ","End":"05:13.450","Text":"That means that we\u0027re looking for the probability of X being greater than 50 minutes."},{"Start":"05:13.450 ","End":"05:17.985","Text":"Let\u0027s just draw our density function."},{"Start":"05:17.985 ","End":"05:20.060","Text":"Here\u0027s our density function,"},{"Start":"05:20.060 ","End":"05:23.855","Text":"and we\u0027re looking for the probability of X being greater than 50."},{"Start":"05:23.855 ","End":"05:27.050","Text":"Let\u0027s just draw 50 right here."},{"Start":"05:27.050 ","End":"05:28.730","Text":"This is x equals 50,"},{"Start":"05:28.730 ","End":"05:29.930","Text":"that\u0027s on the x-axis,"},{"Start":"05:29.930 ","End":"05:33.410","Text":"and we\u0027re interested in the probability of X being greater than 50,"},{"Start":"05:33.410 ","End":"05:37.535","Text":"that means that we\u0027re looking at calculating the area"},{"Start":"05:37.535 ","End":"05:43.320","Text":"under the density function from 50 to plus infinity."},{"Start":"05:43.490 ","End":"05:47.075","Text":"How do we do that? Again, we have our x,"},{"Start":"05:47.075 ","End":"05:49.115","Text":"we need to standardize it,"},{"Start":"05:49.115 ","End":"05:56.555","Text":"and then go look in the standard table of x here, that\u0027s 50."},{"Start":"05:56.555 ","End":"05:58.730","Text":"How do we standardize?"},{"Start":"05:58.730 ","End":"06:03.515","Text":"Well, that\u0027s x minus Mu divided by Sigma."},{"Start":"06:03.515 ","End":"06:05.645","Text":"Let\u0027s see what we have here."},{"Start":"06:05.645 ","End":"06:10.640","Text":"This probability, well, that equals to the probability of X"},{"Start":"06:10.640 ","End":"06:16.190","Text":"minus Mu divided by Sigma that has to be greater than 50 minus 40,"},{"Start":"06:16.190 ","End":"06:19.235","Text":"Mu is 40, divided by Sigma."},{"Start":"06:19.235 ","End":"06:22.200","Text":"Well, Sigma is 5."},{"Start":"06:22.200 ","End":"06:26.965","Text":"That means that this is the probability of z,"},{"Start":"06:26.965 ","End":"06:28.900","Text":"X minus Mu divided by Sigma,"},{"Start":"06:28.900 ","End":"06:34.885","Text":"that\u0027s z, has to be greater than 50 minus 40 divided by 5,"},{"Start":"06:34.885 ","End":"06:37.820","Text":"that\u0027s 10 divided by 5, that\u0027s 2."},{"Start":"06:37.820 ","End":"06:43.180","Text":"Again, since it\u0027s a standard table"},{"Start":"06:43.180 ","End":"06:48.700","Text":"gives us the probability of z being less than a specific value,"},{"Start":"06:48.700 ","End":"06:52.000","Text":"that means that it gives us this probability right here"},{"Start":"06:52.000 ","End":"06:56.005","Text":"from minus infinity to a specific value."},{"Start":"06:56.005 ","End":"07:05.155","Text":"Then this then equals to 1 minus the probability of z being less than 2."},{"Start":"07:05.155 ","End":"07:10.205","Text":"That means that\u0027s 1 minus Phi of 2."},{"Start":"07:10.205 ","End":"07:14.550","Text":"Now we\u0027re ready to go to our standard table."},{"Start":"07:14.800 ","End":"07:16.970","Text":"Here\u0027s a standard table,"},{"Start":"07:16.970 ","End":"07:24.560","Text":"and we\u0027re looking for Phi where z equals 2 and we want to know what that is."},{"Start":"07:24.560 ","End":"07:28.620","Text":"Well, let\u0027s go down here, let\u0027s just scroll down."},{"Start":"07:29.240 ","End":"07:32.760","Text":"Well, that\u0027s z equaling 2."},{"Start":"07:32.760 ","End":"07:38.925","Text":"We\u0027re looking at this value right here, 0.9772."},{"Start":"07:38.925 ","End":"07:43.830","Text":"That equals to 0.9772,"},{"Start":"07:43.830 ","End":"07:48.720","Text":"and this is the value that we\u0027re interested in."},{"Start":"07:48.980 ","End":"07:51.465","Text":"Now, we know what Phi is."},{"Start":"07:51.465 ","End":"07:57.855","Text":"This then becomes 1 minus 0.9772,"},{"Start":"07:57.855 ","End":"08:02.170","Text":"and that equals to 0.0228."},{"Start":"08:05.000 ","End":"08:09.020","Text":"This then is the probability,"},{"Start":"08:09.020 ","End":"08:12.680","Text":"or this is the chance that the person will be late if he"},{"Start":"08:12.680 ","End":"08:18.240","Text":"leaves work at 8:10 to get to work by 9 o\u0027clock."}],"ID":13151},{"Watched":false,"Name":"Exercise 9 - Parts c-d","Duration":"7m 39s","ChapterTopicVideoID":12673,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12673.jpeg","UploadDate":"2018-07-23T16:22:50.1230000","DurationForVideoObject":"PT7M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"In Section C, we\u0027re asked if it\u0027s known"},{"Start":"00:02.910 ","End":"00:05.970","Text":"that it takes a person longer than 45 minutes to get to work."},{"Start":"00:05.970 ","End":"00:10.785","Text":"What\u0027s the probability that the total time to get to work is less than 50 minutes."},{"Start":"00:10.785 ","End":"00:13.170","Text":"Well, the first thing that we need to do is recognize that"},{"Start":"00:13.170 ","End":"00:15.705","Text":"we\u0027re dealing with a conditional probability."},{"Start":"00:15.705 ","End":"00:18.510","Text":"That means that we\u0027re looking at the probability."},{"Start":"00:18.510 ","End":"00:20.130","Text":"Now, what\u0027s known?"},{"Start":"00:20.130 ","End":"00:24.825","Text":"It\u0027s known that it takes a person longer than 45 minutes and that means X,"},{"Start":"00:24.825 ","End":"00:26.600","Text":"the time to get to work."},{"Start":"00:26.600 ","End":"00:28.670","Text":"That\u0027s longer than 45 minutes,"},{"Start":"00:28.670 ","End":"00:30.455","Text":"it\u0027s greater than 45."},{"Start":"00:30.455 ","End":"00:33.455","Text":"What are we asked? We\u0027re asked about the probability"},{"Start":"00:33.455 ","End":"00:36.560","Text":"that the total time to get work is less than 50 minutes."},{"Start":"00:36.560 ","End":"00:40.775","Text":"That means that X here is less than 50."},{"Start":"00:40.775 ","End":"00:44.060","Text":"This is our conditional probability."},{"Start":"00:44.060 ","End":"00:47.285","Text":"Now, how do we solve that?"},{"Start":"00:47.285 ","End":"00:48.710","Text":"Well, in the numerator,"},{"Start":"00:48.710 ","End":"00:53.660","Text":"we have the intersection of these 2 events in the denominator,"},{"Start":"00:53.660 ","End":"00:57.295","Text":"we have the probability of what\u0027s known."},{"Start":"00:57.295 ","End":"00:59.285","Text":"Let\u0027s just write that."},{"Start":"00:59.285 ","End":"01:01.100","Text":"That\u0027ll be the probability."},{"Start":"01:01.100 ","End":"01:05.660","Text":"Now x here is greater than 45 and less than 50."},{"Start":"01:05.660 ","End":"01:10.810","Text":"X is between 45 and 50."},{"Start":"01:10.810 ","End":"01:12.685","Text":"That\u0027s the numerator."},{"Start":"01:12.685 ","End":"01:14.620","Text":"In the denominator, well,"},{"Start":"01:14.620 ","End":"01:20.510","Text":"that\u0027s the probability of x being greater than 45."},{"Start":"01:21.820 ","End":"01:26.950","Text":"Let\u0027s now just draw our density function."},{"Start":"01:26.950 ","End":"01:29.800","Text":"This is our density function."},{"Start":"01:29.800 ","End":"01:34.960","Text":"Excellent. Let\u0027s just continue to solve this."},{"Start":"01:34.960 ","End":"01:37.750","Text":"Well, let\u0027s take a look at the denominator,"},{"Start":"01:37.750 ","End":"01:42.400","Text":"the probability of X being greater than 45,"},{"Start":"01:42.400 ","End":"01:44.830","Text":"where we\u0027ll solve that in section a,"},{"Start":"01:44.830 ","End":"01:49.955","Text":"that equal to 0.1587."},{"Start":"01:49.955 ","End":"01:53.180","Text":"Let\u0027s take a look now at the numerator."},{"Start":"01:53.180 ","End":"01:54.739","Text":"Well, in the numerator,"},{"Start":"01:54.739 ","End":"02:00.770","Text":"we have this X equaling 45 and x"},{"Start":"02:00.770 ","End":"02:06.830","Text":"equaling 50 and we\u0027re interested in the probability of x being here between 45 and 50."},{"Start":"02:06.830 ","End":"02:11.120","Text":"That means that we want to calculate the area under the curve,"},{"Start":"02:11.120 ","End":"02:14.630","Text":"under the density function between 45 and 50."},{"Start":"02:14.630 ","End":"02:16.340","Text":"Now in section a,"},{"Start":"02:16.340 ","End":"02:21.875","Text":"we\u0027ve solved the comparable values of Z for 45,"},{"Start":"02:21.875 ","End":"02:24.255","Text":"and that\u0027s Z equaling 1."},{"Start":"02:24.255 ","End":"02:28.220","Text":"In section B, the comparable value for Z,"},{"Start":"02:28.220 ","End":"02:29.885","Text":"where x equals 50."},{"Start":"02:29.885 ","End":"02:31.765","Text":"Well, that\u0027s 2."},{"Start":"02:31.765 ","End":"02:36.980","Text":"How do we isolate this area right here?"},{"Start":"02:36.980 ","End":"02:41.375","Text":"Well, we take the area from minus infinity"},{"Start":"02:41.375 ","End":"02:45.890","Text":"till 2 on the z-axis and we subtract from that,"},{"Start":"02:45.890 ","End":"02:51.035","Text":"the area from minus infinity to 1 and the z-axis."},{"Start":"02:51.035 ","End":"02:54.924","Text":"Let\u0027s do that."},{"Start":"02:54.924 ","End":"02:58.010","Text":"All we need to do now is say this,"},{"Start":"02:58.010 ","End":"03:04.385","Text":"the probability that X is between 45 and 50."},{"Start":"03:04.385 ","End":"03:14.145","Text":"Well, that equals to the probability of z being between 1 and 2."},{"Start":"03:14.145 ","End":"03:21.285","Text":"Now, that equals to phi of 2 minus phi of 1."},{"Start":"03:21.285 ","End":"03:25.480","Text":"That\u0027s how we isolated this area right here."},{"Start":"03:25.480 ","End":"03:30.605","Text":"Now, we\u0027ve calculated these values, 5, 2,"},{"Start":"03:30.605 ","End":"03:32.785","Text":"we\u0027ve calculated that in section B,"},{"Start":"03:32.785 ","End":"03:39.220","Text":"that\u0027s 0.9772 minus 51."},{"Start":"03:39.220 ","End":"03:46.030","Text":"That\u0027s what\u0027s calculated in section a, that\u0027s 0.8413."},{"Start":"03:46.030 ","End":"03:52.840","Text":"The difference of these 2 probabilities is 0.1359."},{"Start":"03:53.870 ","End":"04:02.150","Text":"This is the probability that the total time to get to work is less than 15 minutes,"},{"Start":"04:02.150 ","End":"04:08.430","Text":"given that a person takes longer than 45 minutes to get to work."},{"Start":"04:08.530 ","End":"04:12.740","Text":"In this section, we\u0027re asked what are the chances it will take a person"},{"Start":"04:12.740 ","End":"04:18.035","Text":"at least 45 minutes to get to work exactly once during a 5-day workweek?"},{"Start":"04:18.035 ","End":"04:20.900","Text":"Well, here we have to recognize that we\u0027re"},{"Start":"04:20.900 ","End":"04:23.870","Text":"dealing with a binomial distribution. Why is that?"},{"Start":"04:23.870 ","End":"04:29.680","Text":"Let\u0027s take a look at the criteria for a binomial distribution."},{"Start":"04:29.680 ","End":"04:32.285","Text":"Here we have the criteria."},{"Start":"04:32.285 ","End":"04:37.310","Text":"The first one says that we have the same Bernoulli trial that\u0027s repeated independently."},{"Start":"04:37.310 ","End":"04:41.340","Text":"Well, n here equals 5."},{"Start":"04:41.340 ","End":"04:45.630","Text":"We have 5 workdays that we have to deal with,"},{"Start":"04:45.630 ","End":"04:48.890","Text":"and each day\u0027s independent of each other."},{"Start":"04:48.890 ","End":"04:51.565","Text":"We have Bernoulli trial,"},{"Start":"04:51.565 ","End":"04:57.725","Text":"so that means that we have a success and failure, so what success?"},{"Start":"04:57.725 ","End":"05:06.540","Text":"Success is getting to work in more than 45 minutes."},{"Start":"05:06.540 ","End":"05:10.060","Text":"The trial is repeated n times well for"},{"Start":"05:10.060 ","End":"05:14.860","Text":"5 days and X is defined as the total number of successes obtained."},{"Start":"05:14.860 ","End":"05:20.343","Text":"Well, here instead of X, we\u0027ll use y because X is already taken."},{"Start":"05:20.343 ","End":"05:26.460","Text":"Our random variable y would be that number of successes."},{"Start":"05:26.460 ","End":"05:31.050","Text":"Now, that means that y"},{"Start":"05:31.050 ","End":"05:37.000","Text":"then is distributed with a binomial distribution where n equals to 5."},{"Start":"05:37.000 ","End":"05:39.610","Text":"What about our probability?"},{"Start":"05:39.610 ","End":"05:45.645","Text":"Well, we\u0027re looking at the probability of X being greater than 45."},{"Start":"05:45.645 ","End":"05:48.415","Text":"That\u0027s the probability of success."},{"Start":"05:48.415 ","End":"05:56.750","Text":"We calculated that in section a that was equal to 0.1587."},{"Start":"05:56.750 ","End":"06:03.540","Text":"P here is equal to 0.1587."},{"Start":"06:03.540 ","End":"06:08.795","Text":"This is our binomial distribution."},{"Start":"06:08.795 ","End":"06:14.600","Text":"Now, the probability of y being equal to some k,"},{"Start":"06:14.600 ","End":"06:19.795","Text":"well, that equals to n over k,"},{"Start":"06:19.795 ","End":"06:28.715","Text":"P to the power of k times 1 minus P to the power of n minus k. Now,"},{"Start":"06:28.715 ","End":"06:33.840","Text":"in our case, we\u0027re looking at k being equal to 1."},{"Start":"06:33.840 ","End":"06:40.805","Text":"We\u0027re looking for the time it gets to work exactly once during a 5-day workweek."},{"Start":"06:40.805 ","End":"06:44.120","Text":"k here, that equals to 1."},{"Start":"06:44.120 ","End":"06:49.402","Text":"I think we have all the information that we need to calculate this probability."},{"Start":"06:49.402 ","End":"06:55.080","Text":"We\u0027re looking then at the probability of Y being equal to 1."},{"Start":"06:55.080 ","End":"06:57.390","Text":"Well, that equals to n over k. Well,"},{"Start":"06:57.390 ","End":"06:59.400","Text":"that\u0027s 5 over 1."},{"Start":"06:59.400 ","End":"07:07.110","Text":"Now P is 0.1587 to the power of 1 times 1"},{"Start":"07:07.110 ","End":"07:15.930","Text":"minus 0.1587 to the power of 4,"},{"Start":"07:15.930 ","End":"07:19.145","Text":"n minus k that\u0027s zero-point."},{"Start":"07:19.145 ","End":"07:27.215","Text":"That equals to 0.3975."},{"Start":"07:27.215 ","End":"07:32.540","Text":"This then is the probability that it will take a person"},{"Start":"07:32.540 ","End":"07:39.600","Text":"at least 45 minutes to get to work exactly once during a 5-day workweek."}],"ID":13152},{"Watched":false,"Name":"Exercise 10 - Part a","Duration":"11m 28s","ChapterTopicVideoID":12674,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12674.jpeg","UploadDate":"2018-07-23T16:23:56.4270000","DurationForVideoObject":"PT11M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this question, we\u0027re given that the monthly household spending in Chicago"},{"Start":"00:03.510 ","End":"00:07.590","Text":"has a normal probability distribution with an average of $2000,"},{"Start":"00:07.590 ","End":"00:10.680","Text":"and a standard deviation of $300."},{"Start":"00:10.680 ","End":"00:13.800","Text":"5 households are randomly selected."},{"Start":"00:13.800 ","End":"00:18.090","Text":"The probability that at least 1 of the household"},{"Start":"00:18.090 ","End":"00:23.275","Text":"spends more than T dollars per month is 0.98976,"},{"Start":"00:23.275 ","End":"00:26.630","Text":"and we\u0027re asked, what\u0027s the value of T?"},{"Start":"00:26.630 ","End":"00:28.820","Text":"This isn\u0027t a trivial question,"},{"Start":"00:28.820 ","End":"00:32.195","Text":"so let\u0027s first write down what we know,"},{"Start":"00:32.195 ","End":"00:38.390","Text":"and then we\u0027ll see how we calculate the value of T. First of all,"},{"Start":"00:38.390 ","End":"00:41.180","Text":"we have to make sure that we\u0027re dealing with a normal probability,"},{"Start":"00:41.180 ","End":"00:43.430","Text":"and we are that\u0027s given to us right here."},{"Start":"00:43.430 ","End":"00:47.180","Text":"Secondly, let\u0027s define a random variable."},{"Start":"00:47.180 ","End":"00:56.248","Text":"Let\u0027s call that X and X will be defined as the monthly spending,"},{"Start":"00:56.248 ","End":"01:02.390","Text":"and that has a normal distribution where Mu equals to 2,000,"},{"Start":"01:02.390 ","End":"01:04.520","Text":"that\u0027s given to us right here,"},{"Start":"01:04.520 ","End":"01:10.415","Text":"and our standard deviation with that 300 that\u0027s given to us right here."},{"Start":"01:10.415 ","End":"01:12.680","Text":"What else are we given?"},{"Start":"01:12.680 ","End":"01:17.090","Text":"We\u0027re given that we\u0027re sampling 5 households."},{"Start":"01:17.090 ","End":"01:20.085","Text":"So n here equals to 5,"},{"Start":"01:20.085 ","End":"01:25.346","Text":"and what are we checking when we\u0027re sampling these households?"},{"Start":"01:25.346 ","End":"01:31.130","Text":"We\u0027re checking on a monthly basis whether they spend more than T dollars,"},{"Start":"01:31.130 ","End":"01:35.260","Text":"so here we have a Bernoulli trial,"},{"Start":"01:35.260 ","End":"01:42.335","Text":"where we define success in the trial as"},{"Start":"01:42.335 ","End":"01:50.930","Text":"the household spending more than T dollars."},{"Start":"01:50.930 ","End":"01:54.635","Text":"What else?"},{"Start":"01:54.635 ","End":"02:01.040","Text":"Let\u0027s define P as the probability of"},{"Start":"02:01.040 ","End":"02:07.325","Text":"success and we\u0027ll define also"},{"Start":"02:07.325 ","End":"02:15.360","Text":"a new random variable called y and that will be the number of successes."},{"Start":"02:16.540 ","End":"02:20.600","Text":"When we have this,"},{"Start":"02:20.600 ","End":"02:28.595","Text":"then we can say that y is distributed with a binomial distribution where n equals to 5,"},{"Start":"02:28.595 ","End":"02:34.318","Text":"and we don\u0027t know what the probability of successes is,"},{"Start":"02:34.318 ","End":"02:36.470","Text":"and we need to calculate them."},{"Start":"02:36.470 ","End":"02:39.960","Text":"Let\u0027s do that."},{"Start":"02:40.130 ","End":"02:46.020","Text":"We said that y has a binomial distribution with n equals to 5 and P,"},{"Start":"02:46.020 ","End":"02:50.120","Text":"and we need to calculate this value right here."},{"Start":"02:50.120 ","End":"02:56.495","Text":"We can say that the generic formula for a binomial distribution as well,"},{"Start":"02:56.495 ","End":"02:59.765","Text":"the probability of y equals k,"},{"Start":"02:59.765 ","End":"03:03.020","Text":"well that equals to n over k,"},{"Start":"03:03.020 ","End":"03:08.390","Text":"P to the power of k times 1 minus P to the power of n minus"},{"Start":"03:08.390 ","End":"03:13.378","Text":"k. We\u0027ll put that aside for now,"},{"Start":"03:13.378 ","End":"03:14.660","Text":"and what are we trying to do?"},{"Start":"03:14.660 ","End":"03:21.595","Text":"Well, we said that the probability of y,"},{"Start":"03:21.595 ","End":"03:23.629","Text":"the number of successes,"},{"Start":"03:23.629 ","End":"03:29.276","Text":"or the number of households that spend more than T dollars,"},{"Start":"03:29.276 ","End":"03:32.180","Text":"we want that to be greater or equal to 1."},{"Start":"03:32.180 ","End":"03:39.340","Text":"We said that that equals to 0.98976."},{"Start":"03:39.340 ","End":"03:45.939","Text":"Again, this probability means that there\u0027s at least 1 household,"},{"Start":"03:45.939 ","End":"03:53.141","Text":"that\u0027s the probability of at least 1 household spending more than T dollars,"},{"Start":"03:53.141 ","End":"03:54.915","Text":"and that\u0027s the probability,"},{"Start":"03:54.915 ","End":"03:57.110","Text":"that\u0027s what we\u0027re given right here."},{"Start":"03:57.110 ","End":"04:05.435","Text":"You see the probability that at least 1 household spends more than T dollars is 0.98976."},{"Start":"04:05.435 ","End":"04:15.725","Text":"That means that 1 minus the probability of y equaling 0,"},{"Start":"04:15.725 ","End":"04:17.345","Text":"that\u0027s the same probability,"},{"Start":"04:17.345 ","End":"04:25.115","Text":"0.98976 where this probability"},{"Start":"04:25.115 ","End":"04:27.980","Text":"was that of the households,"},{"Start":"04:27.980 ","End":"04:31.640","Text":"at least 1 household spending more than T dollars."},{"Start":"04:31.640 ","End":"04:37.255","Text":"This probability is that no household spends more than T dollars."},{"Start":"04:37.255 ","End":"04:40.875","Text":"Since these are complimentary sets,"},{"Start":"04:40.875 ","End":"04:47.600","Text":"then 1 minus this complimentary set is the same probability."},{"Start":"04:47.600 ","End":"04:51.454","Text":"If that\u0027s the case,"},{"Start":"04:51.454 ","End":"04:56.134","Text":"then the probability of Y being equal to 0,"},{"Start":"04:56.134 ","End":"05:01.830","Text":"that equals to 0.01024."},{"Start":"05:01.910 ","End":"05:08.720","Text":"We actually have the 2 sides"},{"Start":"05:08.720 ","End":"05:14.005","Text":"of an equation in order for us to solve P. On 1 hand,"},{"Start":"05:14.005 ","End":"05:18.070","Text":"the probability of y equaling 0."},{"Start":"05:18.070 ","End":"05:22.435","Text":"Well, let\u0027s look at the binomial equation right here."},{"Start":"05:22.435 ","End":"05:25.085","Text":"That\u0027s n over k in our case,"},{"Start":"05:25.085 ","End":"05:28.990","Text":"n is 5 and k here, that\u0027s 0."},{"Start":"05:28.990 ","End":"05:33.260","Text":"K equals 0, so that\u0027s 5/0."},{"Start":"05:33.260 ","End":"05:41.710","Text":"P^0, P^k equals to 0 times 1 minus P to the power of n minus k,"},{"Start":"05:41.710 ","End":"05:42.820","Text":"that\u0027s 5 minus 0,"},{"Start":"05:42.820 ","End":"05:45.800","Text":"that\u0027s 5 and on the other hand,"},{"Start":"05:45.800 ","End":"05:53.285","Text":"we\u0027ve calculated this probability that\u0027s right here, that\u0027s 0.01024."},{"Start":"05:53.285 ","End":"05:56.600","Text":"From these 2 sides of the equation,"},{"Start":"05:56.600 ","End":"06:00.580","Text":"we can extract P. So let\u0027s do that."},{"Start":"06:00.580 ","End":"06:05.045","Text":"Well, 5/0, that equals to 1,"},{"Start":"06:05.045 ","End":"06:08.000","Text":"P^0, that equals to 1."},{"Start":"06:08.000 ","End":"06:11.420","Text":"We\u0027re left with 1 minus P^5,"},{"Start":"06:11.420 ","End":"06:14.555","Text":"and that equals to 0.01024."},{"Start":"06:14.555 ","End":"06:20.790","Text":"So we can say that 1 minus P,"},{"Start":"06:20.790 ","End":"06:23.468","Text":"let\u0027s take the fifth root of both sides,"},{"Start":"06:23.468 ","End":"06:31.710","Text":"that equals to the fifth root of 0.01024,"},{"Start":"06:31.710 ","End":"06:35.060","Text":"and that equals to 0.4."},{"Start":"06:35.060 ","End":"06:40.275","Text":"That means that P then equals to 0.6."},{"Start":"06:40.275 ","End":"06:42.405","Text":"We\u0027re not done yet."},{"Start":"06:42.405 ","End":"06:52.959","Text":"All we did was to calculate the probability of a household spending more than T dollars."},{"Start":"06:52.959 ","End":"06:57.300","Text":"Excellent. Now we can go to the next stage."},{"Start":"06:57.500 ","End":"07:02.525","Text":"The next stage brings us back to the normal distribution."},{"Start":"07:02.525 ","End":"07:05.240","Text":"Let\u0027s just take a look at"},{"Start":"07:05.240 ","End":"07:10.880","Text":"the density function and put all the information that we know on it."},{"Start":"07:10.880 ","End":"07:13.813","Text":"This is a density function,"},{"Start":"07:13.813 ","End":"07:15.440","Text":"and this is our average right here,"},{"Start":"07:15.440 ","End":"07:18.425","Text":"2,000 on the x-axis right here."},{"Start":"07:18.425 ","End":"07:26.810","Text":"We know that the probability of a household spending more than T dollars,"},{"Start":"07:26.810 ","End":"07:34.835","Text":"the probability of X being greater than T that equals to 0.6."},{"Start":"07:34.835 ","End":"07:36.260","Text":"If that\u0027s the case,"},{"Start":"07:36.260 ","End":"07:39.965","Text":"then T has to be here on this side of the average,"},{"Start":"07:39.965 ","End":"07:42.785","Text":"X equals to T. Why is that?"},{"Start":"07:42.785 ","End":"07:46.670","Text":"Well, from the average onwards, it\u0027s 50 percent."},{"Start":"07:46.670 ","End":"07:51.920","Text":"So there\u0027s no way that T can be here in order to meet this condition right here."},{"Start":"07:51.920 ","End":"07:55.385","Text":"So T must be on this side of the average."},{"Start":"07:55.385 ","End":"08:01.540","Text":"We\u0027ll have an area here of 60 percent."},{"Start":"08:01.540 ","End":"08:08.405","Text":"That means that this area from minus infinity to T, well that\u0027s 0.4."},{"Start":"08:08.405 ","End":"08:11.885","Text":"That means that T is the 40th percentile."},{"Start":"08:11.885 ","End":"08:20.330","Text":"What is the Z value that correlates to our T value right here?"},{"Start":"08:20.330 ","End":"08:25.345","Text":"Well, first of all, we know that this Z has to be a negative value."},{"Start":"08:25.345 ","End":"08:31.160","Text":"We know then that if we go to the standard table,"},{"Start":"08:31.160 ","End":"08:33.803","Text":"the standard table doesn\u0027t support negative values,"},{"Start":"08:33.803 ","End":"08:37.775","Text":"so what we\u0027re going to have to do is take the mirror image of"},{"Start":"08:37.775 ","End":"08:42.955","Text":"this and find our Z value on this side."},{"Start":"08:42.955 ","End":"08:45.930","Text":"What would that z value be?"},{"Start":"08:45.930 ","End":"08:49.115","Text":"That\u0027ll be Phi of Z,"},{"Start":"08:49.115 ","End":"08:53.225","Text":"and that will have to be equal to 0.6."},{"Start":"08:53.225 ","End":"08:58.205","Text":"Let\u0027s just go to the standard table and see"},{"Start":"08:58.205 ","End":"09:04.160","Text":"what our Z value is that makes 5 that equals to 0.6."},{"Start":"09:04.160 ","End":"09:12.930","Text":"This is our standard table and we have Phi of Z being equal to 0.6."},{"Start":"09:12.930 ","End":"09:16.970","Text":"We\u0027re looking for some value here within the table"},{"Start":"09:16.970 ","End":"09:22.070","Text":"that comes closest to 0.6 in order to find our Z."},{"Start":"09:22.070 ","End":"09:25.580","Text":"Well, that is right here."},{"Start":"09:25.580 ","End":"09:27.290","Text":"This is the value right here,"},{"Start":"09:27.290 ","End":"09:32.468","Text":"0.5987 that\u0027s the closest that we can come,"},{"Start":"09:32.468 ","End":"09:37.520","Text":"and that\u0027s where Z is at 0.25."},{"Start":"09:37.520 ","End":"09:40.640","Text":"So Z here we\u0027ll use that value."},{"Start":"09:40.640 ","End":"09:44.580","Text":"That value would be 0.25."},{"Start":"09:45.680 ","End":"09:50.615","Text":"Let\u0027s now get back to our question. Here we go."},{"Start":"09:50.615 ","End":"09:52.730","Text":"If Phi of Z equals 0.6,"},{"Start":"09:52.730 ","End":"09:58.715","Text":"then Z has to be equal to 0.25 right here."},{"Start":"09:58.715 ","End":"10:01.205","Text":"But, we\u0027re looking at the mirror image."},{"Start":"10:01.205 ","End":"10:02.915","Text":"That\u0027s what interests us."},{"Start":"10:02.915 ","End":"10:08.645","Text":"So this right here on this side has to be equal to minus 0.25."},{"Start":"10:08.645 ","End":"10:13.020","Text":"This then is the relevant Z for us."},{"Start":"10:13.020 ","End":"10:15.465","Text":"Again, what needs to be done?"},{"Start":"10:15.465 ","End":"10:20.885","Text":"We had our probability,"},{"Start":"10:20.885 ","End":"10:22.845","Text":"we calculated our Z,"},{"Start":"10:22.845 ","End":"10:25.280","Text":"now we have to calculate our X."},{"Start":"10:25.280 ","End":"10:29.900","Text":"What\u0027s the transformation that brings us from Z to X?"},{"Start":"10:29.900 ","End":"10:34.685","Text":"Well, we know that Z equals X minus Mu divided by Sigma."},{"Start":"10:34.685 ","End":"10:41.780","Text":"In a nutcase, X equals T. So minus 0.25,"},{"Start":"10:41.780 ","End":"10:43.805","Text":"that\u0027s our Z of interest,"},{"Start":"10:43.805 ","End":"10:46.580","Text":"that equals to X where X equals T,"},{"Start":"10:46.580 ","End":"10:48.815","Text":"so it\u0027ll be T minus Mu."},{"Start":"10:48.815 ","End":"10:53.240","Text":"Mu is 2,000 divided by Sigma,"},{"Start":"10:53.240 ","End":"10:54.980","Text":"our Sigma is 300."},{"Start":"10:54.980 ","End":"10:58.835","Text":"Let\u0027s just do our basic math right here."},{"Start":"10:58.835 ","End":"11:03.290","Text":"We\u0027ll multiply both sides by 300."},{"Start":"11:03.290 ","End":"11:06.155","Text":"That\u0027ll be minus 75."},{"Start":"11:06.155 ","End":"11:10.528","Text":"That equals to T minus 2,000,"},{"Start":"11:10.528 ","End":"11:18.730","Text":"and that means that T then equals 1,925."},{"Start":"11:18.730 ","End":"11:29.470","Text":"So at last, we\u0027ve arrived at the value for T and it\u0027s $1,925."}],"ID":13153},{"Watched":false,"Name":"Exercise 10 - Parts b-c","Duration":"9m 51s","ChapterTopicVideoID":12675,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12675.jpeg","UploadDate":"2018-07-23T16:24:39.4870000","DurationForVideoObject":"PT9M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this section, we\u0027re asked what are the chances that a household in Chicago"},{"Start":"00:03.720 ","End":"00:07.740","Text":"spends at least 1 standard deviation more than T?"},{"Start":"00:07.740 ","End":"00:12.555","Text":"We\u0027re looking at T plus 1 standard deviation."},{"Start":"00:12.555 ","End":"00:14.475","Text":"From the previous section,"},{"Start":"00:14.475 ","End":"00:18.915","Text":"we calculated T to be 1,925."},{"Start":"00:18.915 ","End":"00:23.160","Text":"We were given that a standard deviation is 300."},{"Start":"00:23.160 ","End":"00:27.015","Text":"We\u0027re looking at 2,225."},{"Start":"00:27.015 ","End":"00:31.410","Text":"We\u0027re looking for the probability that x,"},{"Start":"00:31.410 ","End":"00:32.670","Text":"the household spending,"},{"Start":"00:32.670 ","End":"00:37.635","Text":"is greater than 2,225."},{"Start":"00:37.635 ","End":"00:41.325","Text":"Let\u0027s just draw our density function."},{"Start":"00:41.325 ","End":"00:43.760","Text":"Here\u0027s our density function."},{"Start":"00:43.760 ","End":"00:49.100","Text":"We\u0027re looking for the probability of x being greater than this number right here."},{"Start":"00:49.100 ","End":"00:52.280","Text":"Let\u0027s just put that on the x-axis."},{"Start":"00:52.280 ","End":"00:55.970","Text":"We\u0027re looking for 2,225 and we\u0027re"},{"Start":"00:55.970 ","End":"01:00.305","Text":"looking for the probability that x is greater than this number."},{"Start":"01:00.305 ","End":"01:11.035","Text":"We\u0027re looking to calculate the area under the density function from 2,225 onto infinity."},{"Start":"01:11.035 ","End":"01:14.220","Text":"What do we need to do to calculate this area?"},{"Start":"01:14.220 ","End":"01:17.405","Text":"We need to standardize x. We have x."},{"Start":"01:17.405 ","End":"01:21.500","Text":"We need to standardize it into a standard score and then we have to go to"},{"Start":"01:21.500 ","End":"01:26.995","Text":"the standard table to calculate the area."},{"Start":"01:26.995 ","End":"01:30.270","Text":"Now, x is 2,225."},{"Start":"01:30.270 ","End":"01:36.020","Text":"Our standardization transformation, that\u0027s x minus Mu divided by Sigma."},{"Start":"01:36.020 ","End":"01:39.025","Text":"Let\u0027s just standardize."},{"Start":"01:39.025 ","End":"01:44.640","Text":"z equals to 2,225 minus Mu,"},{"Start":"01:44.640 ","End":"01:49.400","Text":"that\u0027s 2000 divided by Sigma."},{"Start":"01:49.400 ","End":"01:51.580","Text":"That\u0027s 300."},{"Start":"01:51.580 ","End":"01:56.055","Text":"That equals to 0.75."},{"Start":"01:56.055 ","End":"02:05.320","Text":"The corresponding z value for x being 2,225, that\u0027s 0.75."},{"Start":"02:06.260 ","End":"02:16.130","Text":"If we go to the standard table and we\u0027ll try to calculate what Phi of 0.75 is."},{"Start":"02:16.130 ","End":"02:23.940","Text":"Then we\u0027ll get the area between minus infinity to 0.75 on the z-scale."},{"Start":"02:23.940 ","End":"02:25.490","Text":"We\u0027ll get the area in white,"},{"Start":"02:25.490 ","End":"02:26.930","Text":"but we don\u0027t want that."},{"Start":"02:26.930 ","End":"02:33.585","Text":"We want the shaded area from 0.75 onto plus infinity on the z-scale."},{"Start":"02:33.585 ","End":"02:40.065","Text":"That means that we\u0027re looking for 1 minus this Phi right here."},{"Start":"02:40.065 ","End":"02:45.330","Text":"That will give us the probability that we\u0027re looking for."},{"Start":"02:45.380 ","End":"02:48.630","Text":"That\u0027s equal to 1 minus,"},{"Start":"02:48.630 ","End":"02:54.250","Text":"let\u0027s go to the table to see what Phi of 0.75 is."},{"Start":"02:54.350 ","End":"02:56.825","Text":"Here\u0027s our standard table."},{"Start":"02:56.825 ","End":"03:00.320","Text":"We\u0027re looking for Phi of 0.75."},{"Start":"03:00.320 ","End":"03:02.270","Text":"We want to know what that is."},{"Start":"03:02.270 ","End":"03:06.990","Text":"Where z is 0.75, here\u0027s z,"},{"Start":"03:06.990 ","End":"03:09.660","Text":"that\u0027s 0.7 right here,"},{"Start":"03:09.660 ","End":"03:13.440","Text":"and 5, that\u0027s this column right here."},{"Start":"03:13.440 ","End":"03:18.240","Text":"This is the value of z that we\u0027re looking for, 0.75."},{"Start":"03:18.240 ","End":"03:23.230","Text":"This is Phi. That equals to 0.7734."},{"Start":"03:24.680 ","End":"03:28.365","Text":"Let\u0027s get back to our question now."},{"Start":"03:28.365 ","End":"03:32.370","Text":"1 minus Phi of 0.75,"},{"Start":"03:32.370 ","End":"03:40.305","Text":"that equals to 1 minus 0.7734,"},{"Start":"03:40.305 ","End":"03:45.040","Text":"and that equals to 0.2266."},{"Start":"03:46.310 ","End":"03:51.110","Text":"This is the chance that the household in Chicago spends"},{"Start":"03:51.110 ","End":"03:57.965","Text":"at least 1 standard deviation more than t. In this section,"},{"Start":"03:57.965 ","End":"04:01.279","Text":"we\u0027re given that a mistake was discovered in the data and 100 dollars"},{"Start":"04:01.279 ","End":"04:05.090","Text":"must be added to the monthly spending of all households in Chicago."},{"Start":"04:05.090 ","End":"04:08.270","Text":"Given this correction, what\u0027s the probability that"},{"Start":"04:08.270 ","End":"04:12.725","Text":"a household\u0027s monthly spending is less than 1,800 dollars?"},{"Start":"04:12.725 ","End":"04:15.365","Text":"The first thing that we need to do is"},{"Start":"04:15.365 ","End":"04:19.640","Text":"understand what\u0027s the implication of adding 100 dollars."},{"Start":"04:19.640 ","End":"04:22.100","Text":"We had our old random variable,"},{"Start":"04:22.100 ","End":"04:24.820","Text":"now we added 100 dollars."},{"Start":"04:24.820 ","End":"04:27.070","Text":"That\u0027s x plus 100."},{"Start":"04:27.070 ","End":"04:30.685","Text":"We\u0027re adding 100 dollars to all the values of x."},{"Start":"04:30.685 ","End":"04:34.425","Text":"What happens to the average of x?"},{"Start":"04:34.425 ","End":"04:38.180","Text":"Beforehand, Mu was 2,000 dollars,"},{"Start":"04:38.180 ","End":"04:40.250","Text":"and now what happens?"},{"Start":"04:40.250 ","End":"04:42.035","Text":"After adding 100 dollars,"},{"Start":"04:42.035 ","End":"04:49.350","Text":"the average moves by 100 dollars and it becomes 2,100 dollars. Why is that?"},{"Start":"04:49.350 ","End":"04:53.025","Text":"We know that the expectation of x plus a,"},{"Start":"04:53.025 ","End":"04:57.760","Text":"that equals to the expectation of x plus a."},{"Start":"04:57.760 ","End":"05:04.895","Text":"In our case, the expectation of x was 2,000 and we added 100 dollars."},{"Start":"05:04.895 ","End":"05:07.475","Text":"It now becomes 2,100 dollars."},{"Start":"05:07.475 ","End":"05:11.035","Text":"What happens to the standard deviation of x?"},{"Start":"05:11.035 ","End":"05:14.745","Text":"The old standard deviation was 300 dollars."},{"Start":"05:14.745 ","End":"05:17.700","Text":"What happens after adding 100 dollars?"},{"Start":"05:17.700 ","End":"05:27.305","Text":"We know that the variance of x plus a that equals to the variance of x. Why is that?"},{"Start":"05:27.305 ","End":"05:32.030","Text":"By adding a constant to the random variable,"},{"Start":"05:32.030 ","End":"05:36.115","Text":"we\u0027re not influencing the disbursement of the data."},{"Start":"05:36.115 ","End":"05:42.350","Text":"We\u0027re just influencing the placement of all the data along the x-axis."},{"Start":"05:42.350 ","End":"05:46.700","Text":"The variance stays the same and if the variance stays the same,"},{"Start":"05:46.700 ","End":"05:50.135","Text":"then the standard deviation obviously stays the same."},{"Start":"05:50.135 ","End":"05:56.550","Text":"The new Sigma stays the same at 300 dollars."},{"Start":"05:57.050 ","End":"06:02.000","Text":"The random variable after we add 100 dollars,"},{"Start":"06:02.000 ","End":"06:04.670","Text":"after we add the correction,"},{"Start":"06:04.670 ","End":"06:10.182","Text":"that\u0027s distributed with a normal distribution where the average is 2,100"},{"Start":"06:10.182 ","End":"06:17.035","Text":"dollars and the standard deviation, that\u0027s 300 dollars."},{"Start":"06:17.035 ","End":"06:23.870","Text":"Having said that, we\u0027re interested right now in the probability of the monthly spendings,"},{"Start":"06:23.870 ","End":"06:28.895","Text":"that\u0027s x, being less than 1,800 dollars."},{"Start":"06:28.895 ","End":"06:32.765","Text":"Let\u0027s just draw our density function."},{"Start":"06:32.765 ","End":"06:35.480","Text":"Here we go. Here\u0027s our density function."},{"Start":"06:35.480 ","End":"06:40.250","Text":"We\u0027re looking for the probability of x being less than 1,800 dollars."},{"Start":"06:40.250 ","End":"06:43.160","Text":"Let\u0027s just put that on the x-axis."},{"Start":"06:43.160 ","End":"06:45.930","Text":"That\u0027s right here, that\u0027s 1,800."},{"Start":"06:45.930 ","End":"06:48.140","Text":"We\u0027re interested for this probability."},{"Start":"06:48.140 ","End":"06:50.405","Text":"We\u0027re interested to calculate"},{"Start":"06:50.405 ","End":"06:57.130","Text":"the area under the density function from minus infinity to 1,800."},{"Start":"06:57.130 ","End":"07:00.020","Text":"How do we do that? We have our x."},{"Start":"07:00.020 ","End":"07:03.290","Text":"We need to standardize it to get our standard score."},{"Start":"07:03.290 ","End":"07:08.845","Text":"Then we have to go to the standard table to find the appropriate probability."},{"Start":"07:08.845 ","End":"07:12.530","Text":"What\u0027s our standardization transformation?"},{"Start":"07:12.530 ","End":"07:15.860","Text":"That\u0027s x minus Mu divided by Sigma."},{"Start":"07:15.860 ","End":"07:18.185","Text":"Let\u0027s just plug in the numbers here."},{"Start":"07:18.185 ","End":"07:24.735","Text":"We\u0027re looking for the probability of x minus Mu divided by Sigma."},{"Start":"07:24.735 ","End":"07:30.390","Text":"We want that to be less than 1,800 minus Mu."},{"Start":"07:30.390 ","End":"07:34.740","Text":"The average is 2,100 divided by Sigma."},{"Start":"07:34.740 ","End":"07:37.605","Text":"Sigma is 300 dollars right here."},{"Start":"07:37.605 ","End":"07:40.380","Text":"That equals to the probability."},{"Start":"07:40.380 ","End":"07:43.485","Text":"x minus Mu divided by Sigma, that\u0027s z."},{"Start":"07:43.485 ","End":"07:48.110","Text":"That has to be less than 1,800 minus 2,100."},{"Start":"07:48.110 ","End":"07:50.180","Text":"That\u0027s minus 300 divided by 300."},{"Start":"07:50.180 ","End":"07:52.145","Text":"That\u0027s minus 1."},{"Start":"07:52.145 ","End":"07:58.890","Text":"This is what we\u0027re looking for and that equals to Phi of minus 1."},{"Start":"07:58.890 ","End":"08:04.235","Text":"We know that the standard table doesn\u0027t support negative z values."},{"Start":"08:04.235 ","End":"08:08.360","Text":"What we need to do is we need to go here to"},{"Start":"08:08.360 ","End":"08:14.515","Text":"the distribution curve and take the mirror image of that."},{"Start":"08:14.515 ","End":"08:19.365","Text":"If 1,800 was 1 on the z-scale,"},{"Start":"08:19.365 ","End":"08:23.915","Text":"1 is the comparable value of 1,800 on the x-scale."},{"Start":"08:23.915 ","End":"08:27.840","Text":"Then we have to take the mirror image of that. That\u0027s minus 1."},{"Start":"08:27.840 ","End":"08:33.390","Text":"Excuse me. Then we have to look at z equaling 1 right here."},{"Start":"08:33.390 ","End":"08:35.535","Text":"At z equaling 1,"},{"Start":"08:35.535 ","End":"08:41.360","Text":"the area under the curve from 1 to plus infinity will be the"},{"Start":"08:41.360 ","End":"08:48.080","Text":"same as the area from minus infinity to minus 1 on the z-scale."},{"Start":"08:48.140 ","End":"08:52.725","Text":"What does that mean? We know that"},{"Start":"08:52.725 ","End":"09:00.710","Text":"Phi of 1 is all the area from minus infinity to 1 on the z-scale."},{"Start":"09:00.710 ","End":"09:02.180","Text":"But we\u0027re not interested in that."},{"Start":"09:02.180 ","End":"09:07.050","Text":"We\u0027re interested in the area from 1 to plus infinity."},{"Start":"09:07.050 ","End":"09:13.600","Text":"So we need to take 1 minus Phi of 1 to get this area right here."},{"Start":"09:13.670 ","End":"09:16.260","Text":"1 minus Phi of 1,"},{"Start":"09:16.260 ","End":"09:18.165","Text":"that equals to 1 minus."},{"Start":"09:18.165 ","End":"09:23.440","Text":"Phi of 1 equals to 0.8413."},{"Start":"09:23.840 ","End":"09:26.620","Text":"You can check me."},{"Start":"09:26.620 ","End":"09:30.590","Text":"You can check this on the standard table."},{"Start":"09:30.590 ","End":"09:32.660","Text":"I invite you to do that."},{"Start":"09:32.660 ","End":"09:40.520","Text":"That means that this probability equals to 0.1587."},{"Start":"09:40.520 ","End":"09:45.770","Text":"This then is the probability of a household spending less"},{"Start":"09:45.770 ","End":"09:51.550","Text":"than 1,800 dollars after the 100 dollars correction."}],"ID":13154},{"Watched":false,"Name":"Exercise 11 - Part a","Duration":"5m 5s","ChapterTopicVideoID":12676,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12676.jpeg","UploadDate":"2018-07-23T16:25:34.6000000","DurationForVideoObject":"PT5M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"In this question, we\u0027re given that the length of"},{"Start":"00:02.280 ","End":"00:05.070","Text":"a random song that is broadcasted on the radio has"},{"Start":"00:05.070 ","End":"00:07.860","Text":"a normal probability distribution with an expectation of"},{"Start":"00:07.860 ","End":"00:11.835","Text":"3.5 minutes and a standard deviation of 30 seconds."},{"Start":"00:11.835 ","End":"00:14.850","Text":"We\u0027re asked what\u0027s the probability that the length of"},{"Start":"00:14.850 ","End":"00:18.785","Text":"a random song played on the radio is between 2.5 and 3 minutes?"},{"Start":"00:18.785 ","End":"00:20.860","Text":"Well, the first thing that we need to do,"},{"Start":"00:20.860 ","End":"00:24.690","Text":"is to make sure that we\u0027re dealing with a normal probability distribution, and we\u0027re."},{"Start":"00:24.690 ","End":"00:26.610","Text":"It\u0027s given right here."},{"Start":"00:26.610 ","End":"00:30.060","Text":"Second thing that we need to do is define a random variable,"},{"Start":"00:30.060 ","End":"00:40.860","Text":"so let\u0027s define x as a random variable and that\u0027ll be the length of the song in minutes."},{"Start":"00:40.860 ","End":"00:44.805","Text":"Now, that has a normal distribution where"},{"Start":"00:44.805 ","End":"00:50.535","Text":"Mu that equals to 3.5 minutes, that\u0027s right here,"},{"Start":"00:50.535 ","End":"00:53.275","Text":"and our standard deviation,"},{"Start":"00:53.275 ","End":"00:55.430","Text":"well, we want to use the same units,"},{"Start":"00:55.430 ","End":"00:57.860","Text":"so we\u0027ll translate this into minutes,"},{"Start":"00:57.860 ","End":"01:02.310","Text":"now 30 seconds is 0.5 minutes."},{"Start":"01:02.310 ","End":"01:06.020","Text":"So we want to know what\u0027s"},{"Start":"01:06.020 ","End":"01:12.410","Text":"the probability that the length of a random song is between 2.5 and 3 minutes?"},{"Start":"01:12.410 ","End":"01:19.925","Text":"That means that x is between 2.5 and 3 minutes."},{"Start":"01:19.925 ","End":"01:22.550","Text":"This is the probability that we\u0027re looking for."},{"Start":"01:22.550 ","End":"01:25.795","Text":"Let\u0027s just draw our density function."},{"Start":"01:25.795 ","End":"01:29.420","Text":"This is our density function and we\u0027re looking"},{"Start":"01:29.420 ","End":"01:32.735","Text":"for the probability of x between 2.5 and 3."},{"Start":"01:32.735 ","End":"01:36.155","Text":"Let\u0027s just write that down here."},{"Start":"01:36.155 ","End":"01:42.000","Text":"Here that\u0027s 2.5 and here that\u0027ll be 3,"},{"Start":"01:42.000 ","End":"01:44.945","Text":"so we\u0027re looking to calculate"},{"Start":"01:44.945 ","End":"01:51.395","Text":"this shaded area right here between 2.5 and 3 on the x-axis."},{"Start":"01:51.395 ","End":"01:52.655","Text":"Now how do we do that?"},{"Start":"01:52.655 ","End":"01:54.845","Text":"Again, we have our x."},{"Start":"01:54.845 ","End":"01:58.970","Text":"We have to standardize it and get our standard score."},{"Start":"01:58.970 ","End":"02:03.440","Text":"Then we have to go to our standard table to get the probabilities."},{"Start":"02:03.440 ","End":"02:09.950","Text":"Now, our transformation, our standardization transformation,"},{"Start":"02:09.950 ","End":"02:12.850","Text":"that\u0027s x minus Mu divided by Sigma."},{"Start":"02:12.850 ","End":"02:17.720","Text":"Let\u0027s just transform these values of x."},{"Start":"02:17.720 ","End":"02:24.605","Text":"Well, and that\u0027ll be the probability of x minus Mu divided by Sigma."},{"Start":"02:24.605 ","End":"02:34.909","Text":"Well, that has to be between 2.5 minus 3.5 divided by 0.5, that\u0027s here."},{"Start":"02:34.909 ","End":"02:41.675","Text":"Here it\u0027ll be 3 minus 3.5 divided by 0.5."},{"Start":"02:41.675 ","End":"02:48.965","Text":"Now, that equals to the probability of z being between;"},{"Start":"02:48.965 ","End":"02:52.620","Text":"now that\u0027ll be minus 2,"},{"Start":"02:52.620 ","End":"02:56.580","Text":"and here that\u0027ll be minus 1,"},{"Start":"02:56.580 ","End":"02:59.040","Text":"so on the z scale,"},{"Start":"02:59.040 ","End":"03:02.160","Text":"the comparable value of 2.5,"},{"Start":"03:02.160 ","End":"03:06.480","Text":"well that\u0027ll be minus 2 and the comparable value for 3,"},{"Start":"03:06.480 ","End":"03:09.540","Text":"well that\u0027ll be minus 1."},{"Start":"03:09.540 ","End":"03:14.610","Text":"How do we isolate this area right here?"},{"Start":"03:14.610 ","End":"03:20.840","Text":"Well, we take the area from minus infinity to minus 1 and then we"},{"Start":"03:20.840 ","End":"03:27.725","Text":"subtract the area from minus infinity till minus 2 on the z scale."},{"Start":"03:27.725 ","End":"03:31.915","Text":"That will leave us with this area right here."},{"Start":"03:31.915 ","End":"03:38.355","Text":"We\u0027re looking for the Phi of minus 1,"},{"Start":"03:38.355 ","End":"03:42.850","Text":"minus Phi of minus 2."},{"Start":"03:42.850 ","End":"03:44.885","Text":"Now, as we said,"},{"Start":"03:44.885 ","End":"03:50.465","Text":"the standard table doesn\u0027t support negative values of z,"},{"Start":"03:50.465 ","End":"03:53.380","Text":"so we need to translate that,"},{"Start":"03:53.380 ","End":"03:57.845","Text":"and we know that Phi of minus something,"},{"Start":"03:57.845 ","End":"04:02.500","Text":"well that\u0027s 1 minus Phi of that something."},{"Start":"04:02.900 ","End":"04:05.595","Text":"Let\u0027s just write this out."},{"Start":"04:05.595 ","End":"04:08.835","Text":"Phi of minus 1,"},{"Start":"04:08.835 ","End":"04:13.740","Text":"well that equals to 1 minus Phi of 1,"},{"Start":"04:13.740 ","End":"04:16.800","Text":"minus, now Phi of minus 2,"},{"Start":"04:16.800 ","End":"04:20.995","Text":"well that\u0027s 1 minus Phi of 2."},{"Start":"04:20.995 ","End":"04:28.010","Text":"Now if we go to our standard table and check me on this,"},{"Start":"04:28.010 ","End":"04:32.015","Text":"the Phi of 1 is equal to"},{"Start":"04:32.015 ","End":"04:40.440","Text":"0.8413 minus 1 minus"},{"Start":"04:40.440 ","End":"04:47.230","Text":"the Phi of 2, that\u0027s 0.9772."},{"Start":"04:47.660 ","End":"04:53.230","Text":"All this comes out to 0.1359."},{"Start":"04:54.200 ","End":"05:01.235","Text":"This then is the probability of a song that\u0027s broadcasted on the radio,"},{"Start":"05:01.235 ","End":"05:05.910","Text":"that its length is between 2.5 and 3 minutes."}],"ID":13155},{"Watched":false,"Name":"Exercise 11 - Parts c-d","Duration":"7m 52s","ChapterTopicVideoID":12678,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12678.jpeg","UploadDate":"2018-07-23T16:35:03.6730000","DurationForVideoObject":"PT7M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.470","Text":"In this section we\u0027re given,"},{"Start":"00:01.470 ","End":"00:05.030","Text":"the 200 songs are played on the radio on a given day and we\u0027re asked,"},{"Start":"00:05.030 ","End":"00:10.740","Text":"how many songs shorter than 3.5 minutes can we expect to be played?"},{"Start":"00:10.740 ","End":"00:15.600","Text":"Here we have other conditions that are right for a binomial distribution."},{"Start":"00:15.600 ","End":"00:18.780","Text":"Why is that? Well, let\u0027s take a look."},{"Start":"00:18.780 ","End":"00:22.830","Text":"Here are the criteria for a binomial distribution."},{"Start":"00:22.830 ","End":"00:27.510","Text":"The first 1 says that we have the same Bernoulli trial that\u0027s repeated independently."},{"Start":"00:27.510 ","End":"00:29.370","Text":"Well, in a Bernoulli trial,"},{"Start":"00:29.370 ","End":"00:31.410","Text":"we have successes and failures."},{"Start":"00:31.410 ","End":"00:35.250","Text":"Now, what\u0027s the definition of our success?"},{"Start":"00:35.250 ","End":"00:41.430","Text":"Success means that the length of"},{"Start":"00:41.430 ","End":"00:47.575","Text":"time of a song has to be less than 3.5 minutes."},{"Start":"00:47.575 ","End":"00:50.230","Text":"Now the trial is repeated n times,"},{"Start":"00:50.230 ","End":"00:52.610","Text":"so n equals 200."},{"Start":"00:52.610 ","End":"00:55.060","Text":"What else?"},{"Start":"00:55.060 ","End":"00:57.130","Text":"Well, what\u0027s the probability of success?"},{"Start":"00:57.130 ","End":"01:01.510","Text":"Well, the probability is that the length is less than 3.5."},{"Start":"01:01.510 ","End":"01:05.470","Text":"Well, we know that the length of time is"},{"Start":"01:05.470 ","End":"01:09.775","Text":"distributed with a normal distribution with an average of 3.5."},{"Start":"01:09.775 ","End":"01:14.799","Text":"That means that the probability of the length being less than 3.5,"},{"Start":"01:14.799 ","End":"01:16.870","Text":"well, that equals to a half."},{"Start":"01:16.870 ","End":"01:20.890","Text":"Let\u0027s define now a new random variable,"},{"Start":"01:20.890 ","End":"01:25.960","Text":"we\u0027ll call it Y, and that will count our successes."},{"Start":"01:26.300 ","End":"01:29.570","Text":"Now, what are we looking for?"},{"Start":"01:29.570 ","End":"01:33.830","Text":"We\u0027re looking for the expectation of Y. Why is that?"},{"Start":"01:33.830 ","End":"01:36.900","Text":"Well, we\u0027re asked for the expectation of"},{"Start":"01:36.900 ","End":"01:41.585","Text":"how many songs shorter than 3.5 minutes can we expect to be played?"},{"Start":"01:41.585 ","End":"01:47.120","Text":"Well, we know that the expectation in a binomial distribution, well,"},{"Start":"01:47.120 ","End":"01:50.985","Text":"that equals to n times p. In our case,"},{"Start":"01:50.985 ","End":"01:56.625","Text":"n equals 200 and p equals to 0.5."},{"Start":"01:56.625 ","End":"02:06.085","Text":"That means that we can expect 100 songs to be shorter than 3.5 minutes."},{"Start":"02:06.085 ","End":"02:10.415","Text":"In this section, we\u0027re given that 8 songs were broadcasted at a given hour."},{"Start":"02:10.415 ","End":"02:12.410","Text":"We\u0027re asked, what\u0027s the probability that"},{"Start":"02:12.410 ","End":"02:17.365","Text":"exactly a quarter of them were longer than 4 minutes, and the rest weren\u0027t?"},{"Start":"02:17.365 ","End":"02:23.605","Text":"Here we have the right conditions for a binomial distribution. Why is that?"},{"Start":"02:23.605 ","End":"02:26.550","Text":"Here, the criteria for a binomial distribution,"},{"Start":"02:26.550 ","End":"02:29.930","Text":"we see that we have Bernoulli trials."},{"Start":"02:29.930 ","End":"02:31.130","Text":"Now in a Bernoulli trial,"},{"Start":"02:31.130 ","End":"02:33.545","Text":"we have successes and failures."},{"Start":"02:33.545 ","End":"02:36.365","Text":"How do we define our success?"},{"Start":"02:36.365 ","End":"02:42.320","Text":"Well, success for us right now is the length of"},{"Start":"02:42.320 ","End":"02:48.440","Text":"time that a song is played that is over 4 minutes."},{"Start":"02:48.440 ","End":"02:51.500","Text":"Now, what\u0027s the probability of our success?"},{"Start":"02:51.500 ","End":"02:53.570","Text":"Well, we don\u0027t know that yet."},{"Start":"02:53.570 ","End":"02:56.090","Text":"What do we have here? What\u0027s our n?"},{"Start":"02:56.090 ","End":"02:57.755","Text":"How many trials do we have?"},{"Start":"02:57.755 ","End":"02:59.850","Text":"Well, we have 8 songs,"},{"Start":"02:59.850 ","End":"03:02.060","Text":"and they\u0027re independently played from each other,"},{"Start":"03:02.060 ","End":"03:04.355","Text":"so that means n equals 8."},{"Start":"03:04.355 ","End":"03:06.440","Text":"What do we have to do right now?"},{"Start":"03:06.440 ","End":"03:08.770","Text":"Well, let\u0027s define a new random variable,"},{"Start":"03:08.770 ","End":"03:14.500","Text":"we\u0027ll call that w, and that\u0027ll count our successes."},{"Start":"03:14.500 ","End":"03:19.160","Text":"We know then that w is distributed with"},{"Start":"03:19.160 ","End":"03:25.385","Text":"a binomial distribution where n equals to 8 and we don\u0027t know what our probability is."},{"Start":"03:25.385 ","End":"03:27.365","Text":"Let\u0027s calculate that."},{"Start":"03:27.365 ","End":"03:34.520","Text":"Well, what\u0027s the probability that the length of a song is greater than 4 minutes?"},{"Start":"03:34.520 ","End":"03:36.695","Text":"Well, we know that this variable,"},{"Start":"03:36.695 ","End":"03:38.000","Text":"the length of a song,"},{"Start":"03:38.000 ","End":"03:41.590","Text":"is distributed with a normal distribution."},{"Start":"03:41.590 ","End":"03:43.730","Text":"Let\u0027s just calculate that."},{"Start":"03:43.730 ","End":"03:46.865","Text":"Let\u0027s first draw our density function."},{"Start":"03:46.865 ","End":"03:52.640","Text":"Here we have our density function for the x random variable, the length of time."},{"Start":"03:52.640 ","End":"03:58.610","Text":"We know that the average time of a song that\u0027s broadcast is 3.5 minutes."},{"Start":"03:58.610 ","End":"04:05.630","Text":"Now we\u0027re looking for the probability that the length of time is greater than 4 minutes,"},{"Start":"04:05.630 ","End":"04:06.850","Text":"that\u0027s right here,"},{"Start":"04:06.850 ","End":"04:09.950","Text":"x equals 4, and we\u0027re looking for this probability."},{"Start":"04:09.950 ","End":"04:16.190","Text":"This area under the density function from 4 on to plus infinity."},{"Start":"04:16.190 ","End":"04:21.920","Text":"We\u0027re looking for the probability that x is greater than 4."},{"Start":"04:21.920 ","End":"04:24.290","Text":"Now, how do we calculate this?"},{"Start":"04:24.290 ","End":"04:26.555","Text":"Well, we usually have our x,"},{"Start":"04:26.555 ","End":"04:27.950","Text":"we have to standardize it,"},{"Start":"04:27.950 ","End":"04:30.155","Text":"and we have to go to the probability,"},{"Start":"04:30.155 ","End":"04:35.255","Text":"the standard table to calculate our probability."},{"Start":"04:35.255 ","End":"04:42.620","Text":"Now, how do we connect our x to our z, our standard score?"},{"Start":"04:42.620 ","End":"04:46.945","Text":"Well, that equals to x minus Mu divided by Sigma."},{"Start":"04:46.945 ","End":"04:49.325","Text":"Let\u0027s just calculate this."},{"Start":"04:49.325 ","End":"04:55.115","Text":"This is the probability then of x minus Mu divided by Sigma."},{"Start":"04:55.115 ","End":"04:58.715","Text":"Well, that has to be greater than 4 minus 3.5."},{"Start":"04:58.715 ","End":"05:02.220","Text":"That\u0027s our Mu divided by 0.5."},{"Start":"05:02.220 ","End":"05:04.139","Text":"That\u0027s our standard deviation."},{"Start":"05:04.139 ","End":"05:08.560","Text":"That means that we\u0027re looking at the probability now of z,"},{"Start":"05:08.560 ","End":"05:10.850","Text":"x minus Mu divided by Sigma,"},{"Start":"05:10.850 ","End":"05:16.040","Text":"being greater than 0.5 divided by 0.5,"},{"Start":"05:16.040 ","End":"05:17.735","Text":"that equals to 1."},{"Start":"05:17.735 ","End":"05:23.370","Text":"The comparable value of x on the z scale,"},{"Start":"05:23.370 ","End":"05:26.925","Text":"well, that equals to 1, that\u0027s right here."},{"Start":"05:26.925 ","End":"05:32.970","Text":"We\u0027re looking then at Phi of 1,"},{"Start":"05:32.970 ","End":"05:35.990","Text":"but phi of 1 gives me the area from"},{"Start":"05:35.990 ","End":"05:40.370","Text":"minus infinity till 1 on the z scale and we don\u0027t want that."},{"Start":"05:40.370 ","End":"05:43.325","Text":"We want the area from 1 to plus infinity."},{"Start":"05:43.325 ","End":"05:48.285","Text":"That means that we\u0027re looking at 1 minus Phi of 1."},{"Start":"05:48.285 ","End":"05:53.170","Text":"Let\u0027s just calculate that or you can believe me,"},{"Start":"05:53.170 ","End":"05:54.860","Text":"and I invite you to check this,"},{"Start":"05:54.860 ","End":"06:02.360","Text":"that this equals to 1 minus 0.8413."},{"Start":"06:02.360 ","End":"06:06.750","Text":"That\u0027s 5, 1, 0.8413. That means that"},{"Start":"06:06.750 ","End":"06:15.730","Text":"the probability of x being greater than 4 minutes that equals to 0.1587."},{"Start":"06:17.240 ","End":"06:21.300","Text":"Excellent. Now we know what our probability,"},{"Start":"06:21.300 ","End":"06:25.125","Text":"is that equals to 0.1587."},{"Start":"06:25.125 ","End":"06:27.005","Text":"What are we looking for then?"},{"Start":"06:27.005 ","End":"06:34.700","Text":"We\u0027re looking for the probability of w being equal to 2. Why is that?"},{"Start":"06:34.700 ","End":"06:40.880","Text":"Well, we\u0027re asked about the probability that exactly a quarter of the 8 songs."},{"Start":"06:40.880 ","End":"06:42.785","Text":"Now a quarter of 8 songs is 2."},{"Start":"06:42.785 ","End":"06:48.905","Text":"Now, let\u0027s take a look at this probability right here."},{"Start":"06:48.905 ","End":"06:50.420","Text":"Let\u0027s plug in the numbers."},{"Start":"06:50.420 ","End":"06:54.830","Text":"Well, when we look at w,"},{"Start":"06:54.830 ","End":"06:58.140","Text":"the probability of w being equal to k, well,"},{"Start":"06:58.140 ","End":"07:03.140","Text":"that equals to n over k, p^k,"},{"Start":"07:03.140 ","End":"07:10.265","Text":"and 1 minus p to the power of n minus k. Let\u0027s just plug in the numbers here."},{"Start":"07:10.265 ","End":"07:13.595","Text":"k equals 2 and n equals 8."},{"Start":"07:13.595 ","End":"07:16.030","Text":"We have 8 over 2."},{"Start":"07:16.030 ","End":"07:18.750","Text":"Now, what\u0027s our probability?"},{"Start":"07:18.750 ","End":"07:27.310","Text":"Times 0.1587^2 times 1 minus this number,"},{"Start":"07:27.310 ","End":"07:33.200","Text":"which is here, which is 0.8413^6,"},{"Start":"07:33.200 ","End":"07:38.140","Text":"n minus k. That\u0027s 8 minus 2 is 6."}],"ID":13156},{"Watched":false,"Name":"Exercise 11 - Part b","Duration":"7m 37s","ChapterTopicVideoID":12677,"CourseChapterTopicPlaylistID":245051,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12677.jpeg","UploadDate":"2018-07-23T16:26:00.9170000","DurationForVideoObject":"PT7M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.650","Text":"In this section we\u0027re asked what\u0027s"},{"Start":"00:01.650 ","End":"00:06.720","Text":"the interquartile range of the length of the song broadcast on the radio?"},{"Start":"00:06.720 ","End":"00:11.835","Text":"In order to answer that we need to understand what the interquartile range is."},{"Start":"00:11.835 ","End":"00:15.135","Text":"Let\u0027s draw our density function."},{"Start":"00:15.135 ","End":"00:17.460","Text":"Here\u0027s our density function,"},{"Start":"00:17.460 ","End":"00:19.665","Text":"here\u0027s our average at 3.5."},{"Start":"00:19.665 ","End":"00:21.780","Text":"That\u0027s on the x axis."},{"Start":"00:21.780 ","End":"00:25.770","Text":"Now, let\u0027s take a look at the 75th percentile."},{"Start":"00:25.770 ","End":"00:30.075","Text":"That\u0027s right here. We\u0027ll call that x_0.75."},{"Start":"00:30.075 ","End":"00:35.130","Text":"The definition of this value right here is that the area under"},{"Start":"00:35.130 ","End":"00:40.050","Text":"the density function from this value onto plus infinity,"},{"Start":"00:40.050 ","End":"00:42.115","Text":"that\u0027s this shaded area right here."},{"Start":"00:42.115 ","End":"00:46.670","Text":"That should equal to 0.25 or 25 percent."},{"Start":"00:46.670 ","End":"00:53.615","Text":"That means that the area in white from minus infinity to this value,"},{"Start":"00:53.615 ","End":"00:59.455","Text":"x_0.75, that should equal to 0.75 or 75 percent."},{"Start":"00:59.455 ","End":"01:07.400","Text":"Now, let\u0027s take a look at the mirror image right here, at x_0.25."},{"Start":"01:07.400 ","End":"01:10.265","Text":"That\u0027s the 25th percentile."},{"Start":"01:10.265 ","End":"01:16.805","Text":"The area under the density function from minus infinity till this value right here,"},{"Start":"01:16.805 ","End":"01:20.795","Text":"that should also equal to 0.25."},{"Start":"01:20.795 ","End":"01:24.050","Text":"That\u0027s the definition of the 25th percentile."},{"Start":"01:24.050 ","End":"01:27.200","Text":"That means that the area under the density function"},{"Start":"01:27.200 ","End":"01:30.875","Text":"from this value right here to plus infinity,"},{"Start":"01:30.875 ","End":"01:34.055","Text":"that should also equal to 0.75."},{"Start":"01:34.055 ","End":"01:38.605","Text":"We see here that we have a mirror image between"},{"Start":"01:38.605 ","End":"01:45.610","Text":"the values of x_0.25 and x_0.75."},{"Start":"01:45.610 ","End":"01:50.965","Text":"Now, what\u0027s the interquartile range?"},{"Start":"01:50.965 ","End":"01:56.285","Text":"Well, that\u0027s this range right here from x_0.25"},{"Start":"01:56.285 ","End":"02:03.515","Text":"to x_0.75 from the 25th percentile to the 75th percentile."},{"Start":"02:03.515 ","End":"02:13.230","Text":"That means that we\u0027re looking at x_0.75 minus x_0.25."},{"Start":"02:13.230 ","End":"02:15.005","Text":"This range right here,"},{"Start":"02:15.005 ","End":"02:19.370","Text":"that\u0027s defined as the interquartile range."},{"Start":"02:19.370 ","End":"02:24.770","Text":"Now, in order to calculate the interquartile range,"},{"Start":"02:24.770 ","End":"02:29.675","Text":"we need to calculate the values of x_0.75 and x_0.25."},{"Start":"02:29.675 ","End":"02:32.780","Text":"But we can make our lives a little bit easier."},{"Start":"02:32.780 ","End":"02:39.470","Text":"We can calculate only 1 of the values,"},{"Start":"02:39.470 ","End":"02:42.145","Text":"let\u0027s say x_0.75,"},{"Start":"02:42.145 ","End":"02:47.120","Text":"and then calculate the distance between the average."},{"Start":"02:47.120 ","End":"02:49.145","Text":"Now we already know what the average is,"},{"Start":"02:49.145 ","End":"02:55.555","Text":"so we calculate this distance right here from the average to the 75th percentile,"},{"Start":"02:55.555 ","End":"02:58.195","Text":"and then we can multiply it by 2."},{"Start":"02:58.195 ","End":"03:01.400","Text":"Because this distance right here,"},{"Start":"03:01.400 ","End":"03:04.920","Text":"that equals to this distance right here."},{"Start":"03:05.690 ","End":"03:10.480","Text":"We can say that this equals to"},{"Start":"03:11.060 ","End":"03:19.050","Text":"x_0.75 minus the average, times 2."},{"Start":"03:19.050 ","End":"03:24.680","Text":"That then is also the definition of the interquartile range."},{"Start":"03:24.680 ","End":"03:26.090","Text":"If that\u0027s the case,"},{"Start":"03:26.090 ","End":"03:30.740","Text":"let\u0027s just go ahead and see how we can calculate this."},{"Start":"03:30.740 ","End":"03:37.010","Text":"Well, usually we can calculate z values and x values how."},{"Start":"03:37.010 ","End":"03:39.380","Text":"Well we have the standardization process,"},{"Start":"03:39.380 ","End":"03:42.020","Text":"we have x and we have to calculate our z,"},{"Start":"03:42.020 ","End":"03:46.310","Text":"and then we have to go to our standard table to calculate our probabilities."},{"Start":"03:46.310 ","End":"03:53.700","Text":"Well, here in order to calculate z_0.75,"},{"Start":"03:53.700 ","End":"03:59.720","Text":"which is the comparable value of x_0.75 on the z scale."},{"Start":"03:59.720 ","End":"04:03.310","Text":"We have this probability and we have to"},{"Start":"04:03.310 ","End":"04:09.010","Text":"go and see what our z value is from the standard table."},{"Start":"04:09.010 ","End":"04:10.509","Text":"Why do we have this probability?"},{"Start":"04:10.509 ","End":"04:14.515","Text":"Because at x_0.75,"},{"Start":"04:14.515 ","End":"04:17.440","Text":"we have the proportion,"},{"Start":"04:17.440 ","End":"04:20.190","Text":"the probability here that\u0027s 75 percent."},{"Start":"04:20.190 ","End":"04:26.730","Text":"We\u0027re looking then for Phi of some z value."},{"Start":"04:26.730 ","End":"04:29.625","Text":"We\u0027ll call this z_0.75,"},{"Start":"04:29.625 ","End":"04:32.895","Text":"and that should equal to 0.75."},{"Start":"04:32.895 ","End":"04:35.070","Text":"We\u0027re looking for this z value."},{"Start":"04:35.070 ","End":"04:39.170","Text":"Let\u0027s go to our table and see what this z value is."},{"Start":"04:39.170 ","End":"04:41.210","Text":"Here\u0027s our table."},{"Start":"04:41.210 ","End":"04:47.510","Text":"Now, we\u0027re looking then for Phi of some z value,"},{"Start":"04:47.510 ","End":"04:52.600","Text":"z_0.75, and that should equal to 0.75."},{"Start":"04:52.600 ","End":"04:57.170","Text":"We\u0027re looking for a value within this table right"},{"Start":"04:57.170 ","End":"05:04.005","Text":"here that\u0027s closest to 0.75 in order to identify what our z value is."},{"Start":"05:04.005 ","End":"05:08.025","Text":"Let\u0027s look at values here."},{"Start":"05:08.025 ","End":"05:17.635","Text":"We see here that these 2 values here are the closest to 0.75."},{"Start":"05:17.635 ","End":"05:23.425","Text":"That happens when z equals 0.67."},{"Start":"05:23.425 ","End":"05:25.750","Text":"It\u0027s between 0.67, 0."},{"Start":"05:25.750 ","End":"05:30.150","Text":"68, so z_0.75."},{"Start":"05:30.150 ","End":"05:33.935","Text":"Well, let\u0027s take the middle value right here,"},{"Start":"05:33.935 ","End":"05:38.280","Text":"and that\u0027ll be equal to 0.675."},{"Start":"05:38.280 ","End":"05:42.315","Text":"This would be the z value that we\u0027ll be using."},{"Start":"05:42.315 ","End":"05:48.645","Text":"We know then that z_0.75,"},{"Start":"05:48.645 ","End":"05:52.515","Text":"that equals to 0.675."},{"Start":"05:52.515 ","End":"05:56.625","Text":"Now how do we get from z to our x?"},{"Start":"05:56.625 ","End":"06:00.675","Text":"Z we said equal to 0.675."},{"Start":"06:00.675 ","End":"06:06.830","Text":"Well, we know that z equals to x minus Mu divided by Sigma."},{"Start":"06:06.830 ","End":"06:09.420","Text":"Let\u0027s just plug in the numbers."},{"Start":"06:10.700 ","End":"06:14.265","Text":"This z right here, 0.675,"},{"Start":"06:14.265 ","End":"06:19.410","Text":"that equals to x_0.75 minus Mu."},{"Start":"06:19.410 ","End":"06:25.995","Text":"Well, Mu is 3.5 divided by Sigma, that\u0027s 0.5."},{"Start":"06:25.995 ","End":"06:32.895","Text":"That means that when we multiply things out,"},{"Start":"06:32.895 ","End":"06:43.020","Text":"then x_0.75 that equals to 3.8375."},{"Start":"06:43.020 ","End":"06:46.835","Text":"All we did right now is calculate this value right here."},{"Start":"06:46.835 ","End":"06:50.695","Text":"That\u0027s 3.8375."},{"Start":"06:50.695 ","End":"06:54.395","Text":"Excellent. Now we\u0027re still not done yet."},{"Start":"06:54.395 ","End":"06:57.320","Text":"Now we have to calculate this distance right here"},{"Start":"06:57.320 ","End":"07:01.070","Text":"between the average and this value and multiply it by 2."},{"Start":"07:01.070 ","End":"07:05.400","Text":"The interquartile range,"},{"Start":"07:05.480 ","End":"07:12.150","Text":"that\u0027ll be equal to 2 times this value,"},{"Start":"07:12.150 ","End":"07:17.535","Text":"the 75th percentile, that\u0027s 3.8375."},{"Start":"07:17.535 ","End":"07:24.280","Text":"That\u0027s our x value minus our Mu, that\u0027s 3.5."},{"Start":"07:24.280 ","End":"07:31.790","Text":"Now that equals to 0.675."},{"Start":"07:31.790 ","End":"07:37.710","Text":"Our interquartile range is 0.675."}],"ID":13157}],"Thumbnail":null,"ID":245051},{"Name":"Transformation of a Continuous Random Variable","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"23s","ChapterTopicVideoID":12679,"CourseChapterTopicPlaylistID":245052,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12679.jpeg","UploadDate":"2019-01-31T23:38:17.7800000","DurationForVideoObject":"PT23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.370","Text":"In this chapter, we\u0027ll be talking about"},{"Start":"00:02.370 ","End":"00:05.325","Text":"the transformation of a continuous random variable."},{"Start":"00:05.325 ","End":"00:08.280","Text":"Now, a new random variable is created from"},{"Start":"00:08.280 ","End":"00:12.360","Text":"a continuous random variable with a known probability distribution."},{"Start":"00:12.360 ","End":"00:15.270","Text":"We create this new random variable through"},{"Start":"00:15.270 ","End":"00:19.515","Text":"a transformation function of the known random variable."},{"Start":"00:19.515 ","End":"00:22.840","Text":"Let\u0027s just take a look at an example."}],"ID":13166},{"Watched":false,"Name":"Example","Duration":"5m 27s","ChapterTopicVideoID":12680,"CourseChapterTopicPlaylistID":245052,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12680.jpeg","UploadDate":"2019-01-31T23:35:46.9800000","DurationForVideoObject":"PT5M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"In this example, we\u0027re given that x is"},{"Start":"00:02.910 ","End":"00:08.310","Text":"a continuous random variable with a uniform probability distribution between 0 and 1."},{"Start":"00:08.310 ","End":"00:12.080","Text":"We\u0027re asked to find the cumulative distribution function of Y,"},{"Start":"00:12.080 ","End":"00:14.960","Text":"where Y equals e^x."},{"Start":"00:14.960 ","End":"00:18.900","Text":"Again, x is unknown random variable and it has"},{"Start":"00:18.900 ","End":"00:24.750","Text":"a known distribution that\u0027s a uniform probability distribution between 0 and 1."},{"Start":"00:24.750 ","End":"00:28.230","Text":"We\u0027re asked to find the cumulative distribution function of Y,"},{"Start":"00:28.230 ","End":"00:31.050","Text":"that\u0027s our new random variable,"},{"Start":"00:31.050 ","End":"00:37.305","Text":"where the transformation function is this thing right here, Y equals e^x."},{"Start":"00:37.305 ","End":"00:41.400","Text":"Now, how do we go about solving this problem?"},{"Start":"00:41.400 ","End":"00:43.970","Text":"Well, the first thing that we\u0027d like to do is,"},{"Start":"00:43.970 ","End":"00:46.460","Text":"we\u0027d like to see what this function looks like."},{"Start":"00:46.460 ","End":"00:50.345","Text":"Y equaling e^x."},{"Start":"00:50.345 ","End":"00:52.825","Text":"Let\u0027s draw that."},{"Start":"00:52.825 ","End":"00:57.380","Text":"Here\u0027s our function, y equals e^x,"},{"Start":"00:57.380 ","End":"01:03.345","Text":"and x is defined between 0 and 1."},{"Start":"01:03.345 ","End":"01:06.905","Text":"We\u0027re given that. The definition of x,"},{"Start":"01:06.905 ","End":"01:09.215","Text":"the domain of this function,"},{"Start":"01:09.215 ","End":"01:10.670","Text":"that\u0027s the values of x,"},{"Start":"01:10.670 ","End":"01:14.050","Text":"that\u0027s between 0 and 1."},{"Start":"01:14.050 ","End":"01:18.290","Text":"Now, what about the range of this function?"},{"Start":"01:18.290 ","End":"01:20.825","Text":"That\u0027s the Y values."},{"Start":"01:20.825 ","End":"01:24.685","Text":"Well, when x equals 0,"},{"Start":"01:24.685 ","End":"01:28.755","Text":"then Y equals e^0,"},{"Start":"01:28.755 ","End":"01:30.285","Text":"that equals to 1."},{"Start":"01:30.285 ","End":"01:31.935","Text":"That\u0027s right here."},{"Start":"01:31.935 ","End":"01:34.440","Text":"What about when x equals 1?"},{"Start":"01:34.440 ","End":"01:39.440","Text":"Well, then Y would be equal to e^1."},{"Start":"01:39.440 ","End":"01:42.335","Text":"That\u0027s right here."},{"Start":"01:42.335 ","End":"01:50.555","Text":"That equals to e. The range of this function is between 1 and e,"},{"Start":"01:50.555 ","End":"01:56.070","Text":"and the domain of this function is between 0 and 1."},{"Start":"01:56.510 ","End":"01:59.340","Text":"Let\u0027s just write this out."},{"Start":"01:59.340 ","End":"02:03.930","Text":"X then is between 0 and 1,"},{"Start":"02:03.930 ","End":"02:12.600","Text":"and y is between 1 and e. Let\u0027s continue from here."},{"Start":"02:12.600 ","End":"02:21.075","Text":"We know, and we\u0027re given that x has a uniform distribution where a equals 0,"},{"Start":"02:21.075 ","End":"02:23.815","Text":"and b equals to 1."},{"Start":"02:23.815 ","End":"02:28.505","Text":"What\u0027s the cumulative distribution function of X?"},{"Start":"02:28.505 ","End":"02:35.840","Text":"Well, that\u0027s the probability of x being less than some value of x."},{"Start":"02:35.840 ","End":"02:44.360","Text":"Now, they\u0027ll be x minus a divided by b minus a."},{"Start":"02:44.360 ","End":"02:49.665","Text":"That equals to x minus 0 divided by 1 minus 0,"},{"Start":"02:49.665 ","End":"02:52.530","Text":"and that equals to X."},{"Start":"02:52.530 ","End":"02:59.735","Text":"We know then that the cumulative distribution function of x,"},{"Start":"02:59.735 ","End":"03:01.970","Text":"that equals to x."},{"Start":"03:01.970 ","End":"03:04.565","Text":"Now, what about y?"},{"Start":"03:04.565 ","End":"03:10.060","Text":"We\u0027re looking for the cumulative distribution of y."},{"Start":"03:10.060 ","End":"03:14.075","Text":"That equals to the probability of"},{"Start":"03:14.075 ","End":"03:20.060","Text":"random variable Y being less than or equal to because really doesn\u0027t matter,"},{"Start":"03:20.060 ","End":"03:22.790","Text":"we\u0027re dealing with continuous random variables."},{"Start":"03:22.790 ","End":"03:29.360","Text":"The probability of a random variable Y being less than or equal to some value y."},{"Start":"03:29.360 ","End":"03:32.615","Text":"Now, what is Y?"},{"Start":"03:32.615 ","End":"03:36.470","Text":"Well, Y is defined as e^x."},{"Start":"03:36.470 ","End":"03:45.070","Text":"That equals to the probability of e^x being less than some value of y."},{"Start":"03:45.200 ","End":"03:48.520","Text":"Let\u0027s continue from here."},{"Start":"03:48.770 ","End":"03:52.340","Text":"Now, how do we extract x?"},{"Start":"03:52.340 ","End":"03:55.610","Text":"Well, let\u0027s take ln of both sides."},{"Start":"03:55.610 ","End":"04:01.310","Text":"We have here, that\u0027s the probability of ln of e^x,"},{"Start":"04:01.310 ","End":"04:06.910","Text":"that\u0027s x being less than ln of y."},{"Start":"04:08.570 ","End":"04:13.570","Text":"We have the solution for this thing right here."},{"Start":"04:13.570 ","End":"04:19.340","Text":"We know that the probability of x being less than some value x,"},{"Start":"04:19.340 ","End":"04:23.315","Text":"well, that equals to that value itself, x."},{"Start":"04:23.315 ","End":"04:27.590","Text":"The probability of x being less than some value here,"},{"Start":"04:27.590 ","End":"04:33.100","Text":"in this case, it\u0027ll be ln y would that equals to ln of y."},{"Start":"04:33.100 ","End":"04:37.280","Text":"But we started out with F of y,"},{"Start":"04:37.280 ","End":"04:39.590","Text":"the cumulative distribution of y."},{"Start":"04:39.590 ","End":"04:45.005","Text":"That\u0027s the probability of Y random variable Y being less than y."},{"Start":"04:45.005 ","End":"04:50.970","Text":"We can see here again that F of y,"},{"Start":"04:50.970 ","End":"04:54.030","Text":"that equals to ln of y,"},{"Start":"04:54.030 ","End":"05:02.340","Text":"where y is between 1 and e. For example,"},{"Start":"05:02.340 ","End":"05:12.320","Text":"now, we want to know what the cumulative distribution function of y at 1.5."},{"Start":"05:12.320 ","End":"05:13.580","Text":"This is an example."},{"Start":"05:13.580 ","End":"05:17.090","Text":"Well, that\u0027s the probability of Y"},{"Start":"05:17.090 ","End":"05:21.275","Text":"being less than or equal to 1.5 where now we know what it is."},{"Start":"05:21.275 ","End":"05:27.960","Text":"It equals to ln of 1.5 right here."}],"ID":13167},{"Watched":false,"Name":"Exercise 1","Duration":"7m 46s","ChapterTopicVideoID":12681,"CourseChapterTopicPlaylistID":245052,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12681.jpeg","UploadDate":"2019-01-31T23:36:24.4970000","DurationForVideoObject":"PT7M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.520","Text":"Let W be a random variable with an exponential distribution with an expectation of 1."},{"Start":"00:05.520 ","End":"00:09.900","Text":"We define a new variable Y that equals to e to the power of"},{"Start":"00:09.900 ","End":"00:15.495","Text":"minus w. We\u0027re asked first to find the cumulative distribution function of Y,"},{"Start":"00:15.495 ","End":"00:18.570","Text":"and secondly, we\u0027re asked what\u0027s the name of"},{"Start":"00:18.570 ","End":"00:23.025","Text":"the cumulative distribution and what are its parameters."},{"Start":"00:23.025 ","End":"00:26.310","Text":"The first thing that we want to do is we want to"},{"Start":"00:26.310 ","End":"00:29.980","Text":"draw this function right here, let\u0027s do that."},{"Start":"00:30.230 ","End":"00:32.550","Text":"Here\u0027s your function,"},{"Start":"00:32.550 ","End":"00:35.535","Text":"and this is a W-axis right here."},{"Start":"00:35.535 ","End":"00:40.290","Text":"We know that W is a random variable with an exponential distribution."},{"Start":"00:40.290 ","End":"00:44.540","Text":"Here\u0027s our Y axis and Y is"},{"Start":"00:44.540 ","End":"00:49.790","Text":"the new random variable and it equals to e to the power of minus w. Here is"},{"Start":"00:49.790 ","End":"00:54.980","Text":"the function e to the power of minus w. First thing that we want to"},{"Start":"00:54.980 ","End":"01:02.000","Text":"do is we want to write out this distribution right here."},{"Start":"01:02.000 ","End":"01:05.750","Text":"Well, we know that W is distributed with"},{"Start":"01:05.750 ","End":"01:10.700","Text":"an exponential distribution with parameter Lambda."},{"Start":"01:10.700 ","End":"01:12.020","Text":"Now that equals what?"},{"Start":"01:12.020 ","End":"01:14.300","Text":"We\u0027ll get that to that in a minute."},{"Start":"01:14.300 ","End":"01:18.920","Text":"We\u0027re also given that the expectation of W is 1."},{"Start":"01:18.920 ","End":"01:23.430","Text":"Now, we know that the expectation of W,"},{"Start":"01:23.430 ","End":"01:26.930","Text":"if W is distributed with an exponential distribution with that equals to"},{"Start":"01:26.930 ","End":"01:30.410","Text":"1 divided by Lambda, we know,"},{"Start":"01:30.410 ","End":"01:33.875","Text":"or we\u0027re given that equals to 1,"},{"Start":"01:33.875 ","End":"01:37.595","Text":"that means that Lambda here equals to 1."},{"Start":"01:37.595 ","End":"01:44.190","Text":"W is distributed with an exponential distribution where Lambda equals 1."},{"Start":"01:44.390 ","End":"01:48.200","Text":"The next thing that we need to do is we need to find"},{"Start":"01:48.200 ","End":"01:52.149","Text":"the domain and range of this function."},{"Start":"01:52.149 ","End":"01:55.310","Text":"In other words, we need to know what are"},{"Start":"01:55.310 ","End":"02:01.850","Text":"the defined values for W and the defined values for Y."},{"Start":"02:01.850 ","End":"02:10.460","Text":"The domain or the defined values for W. Since W has an exponential distribution,"},{"Start":"02:10.460 ","End":"02:18.580","Text":"we know that W is defined from 0 to plus infinity."},{"Start":"02:18.580 ","End":"02:26.095","Text":"We say then that W is between 0 and plus infinity."},{"Start":"02:26.095 ","End":"02:29.160","Text":"What happens to y?"},{"Start":"02:30.470 ","End":"02:33.779","Text":"When W equals 0,"},{"Start":"02:33.779 ","End":"02:38.040","Text":"then Y equals to e to the power of minus 0,"},{"Start":"02:38.040 ","End":"02:39.920","Text":"e to the power of 0,"},{"Start":"02:39.920 ","End":"02:42.450","Text":"that equals to 1."},{"Start":"02:42.710 ","End":"02:47.060","Text":"What happens as W goes to infinity?"},{"Start":"02:47.060 ","End":"02:50.135","Text":"Well, as W goes to infinity,"},{"Start":"02:50.135 ","End":"02:54.135","Text":"this function goes to 0."},{"Start":"02:54.135 ","End":"02:59.190","Text":"Y then is between 0 and 1?"},{"Start":"02:59.190 ","End":"03:04.390","Text":"These are the defined values for Y."},{"Start":"03:05.530 ","End":"03:09.545","Text":"Great. Now that we know all this information,"},{"Start":"03:09.545 ","End":"03:13.895","Text":"let\u0027s find the cumulative distribution function of Y."},{"Start":"03:13.895 ","End":"03:15.470","Text":"In order to do that,"},{"Start":"03:15.470 ","End":"03:18.910","Text":"we start with the cumulative distribution of"},{"Start":"03:18.910 ","End":"03:27.170","Text":"W. We know that W is distributed with an exponential distribution,"},{"Start":"03:27.170 ","End":"03:30.650","Text":"so the probability of"},{"Start":"03:30.650 ","End":"03:37.175","Text":"the random variable W being less than or equal to some value, small w,"},{"Start":"03:37.175 ","End":"03:41.900","Text":"that\u0027s the cumulative distribution of W. What that equals to"},{"Start":"03:41.900 ","End":"03:50.150","Text":"1 minus e to the power of minus Lambda times W. We know what Lambda is,"},{"Start":"03:50.150 ","End":"03:59.540","Text":"Lambda is 1 so that equals to 1 minus e to the power of minus w. Let\u0027s"},{"Start":"03:59.540 ","End":"04:03.965","Text":"look at the probability that"},{"Start":"04:03.965 ","End":"04:10.640","Text":"the random variable Y being less than or equal to some value Y."},{"Start":"04:10.640 ","End":"04:15.070","Text":"Now, that equals 2. Now what\u0027s Y?"},{"Start":"04:15.070 ","End":"04:18.345","Text":"Y is e to the power of minus w,"},{"Start":"04:18.345 ","End":"04:21.720","Text":"that\u0027s the probability of e to the power of"},{"Start":"04:21.720 ","End":"04:27.805","Text":"minus w and that has to be less than or equal to some value Y."},{"Start":"04:27.805 ","End":"04:33.210","Text":"All we did is we switched from Y to the function of Y,"},{"Start":"04:33.210 ","End":"04:37.940","Text":"which is e to the power of minus w. We"},{"Start":"04:37.940 ","End":"04:45.750","Text":"know that y is between 0 and 1 we have to remember that."},{"Start":"04:46.180 ","End":"04:49.400","Text":"Let\u0027s try to isolate W,"},{"Start":"04:49.400 ","End":"04:51.860","Text":"we\u0027ll take the ln of both sides,"},{"Start":"04:51.860 ","End":"05:00.425","Text":"that\u0027ll be the probability of minus W that has to be less than or equal to ln of y."},{"Start":"05:00.425 ","End":"05:04.020","Text":"Let\u0027s get rid of this minus sign right here,"},{"Start":"05:04.020 ","End":"05:08.240","Text":"and when we do this, then we have to switch signs here, the inequality signs."},{"Start":"05:08.240 ","End":"05:16.620","Text":"That equals to the probability of w being greater or equal to minus ln of y."},{"Start":"05:18.830 ","End":"05:22.325","Text":"This probability right here,"},{"Start":"05:22.325 ","End":"05:31.560","Text":"what that equals to 1 minus the probability of w being less than minus ln of y."},{"Start":"05:33.290 ","End":"05:36.170","Text":"What\u0027s this probability?"},{"Start":"05:36.170 ","End":"05:38.330","Text":"Well, it\u0027s right here,"},{"Start":"05:38.330 ","End":"05:43.010","Text":"all we have to do is switch this expression for this expression,"},{"Start":"05:43.010 ","End":"05:45.110","Text":"that\u0027ll be equal to 1 minus,"},{"Start":"05:45.110 ","End":"05:47.090","Text":"here\u0027s the 1 minus."},{"Start":"05:47.090 ","End":"05:49.580","Text":"What\u0027s this expression?"},{"Start":"05:49.580 ","End":"05:55.350","Text":"That\u0027s 1 minus e to the power of minus something w. Well,"},{"Start":"05:55.350 ","End":"06:00.510","Text":"this value w equals to minus ln of y,"},{"Start":"06:00.510 ","End":"06:04.960","Text":"that has to be minus ln of y."},{"Start":"06:07.910 ","End":"06:10.710","Text":"Now, let\u0027s try to figure this out."},{"Start":"06:10.710 ","End":"06:13.305","Text":"Well, 1 cancels this 1 out,"},{"Start":"06:13.305 ","End":"06:15.525","Text":"minus cancels this out."},{"Start":"06:15.525 ","End":"06:18.230","Text":"We have e to the power of,"},{"Start":"06:18.230 ","End":"06:21.200","Text":"now, these minuses cancel each other out,"},{"Start":"06:21.200 ","End":"06:28.340","Text":"so that\u0027ll be e to the power of ln of y. e to the power of ln of y,"},{"Start":"06:28.340 ","End":"06:30.870","Text":"well that equals to y."},{"Start":"06:32.600 ","End":"06:37.625","Text":"Big F, the cumulative distribution of Y."},{"Start":"06:37.625 ","End":"06:44.180","Text":"That\u0027s the probability of big Y being less than or equal to some value y,"},{"Start":"06:44.180 ","End":"06:46.735","Text":"that equals to y."},{"Start":"06:46.735 ","End":"06:54.140","Text":"We started here and we ended up right here. What does that mean?"},{"Start":"06:54.140 ","End":"06:56.300","Text":"That means that the probability of Y,"},{"Start":"06:56.300 ","End":"07:01.770","Text":"for example, being less than 1/2, well that\u0027s1/2."},{"Start":"07:01.910 ","End":"07:11.420","Text":"The only distribution that has this characteristic is the uniform distribution."},{"Start":"07:11.420 ","End":"07:17.975","Text":"That means that Y is distributed with the uniform distribution where"},{"Start":"07:17.975 ","End":"07:25.470","Text":"a equals 0 and b that equals to 1."},{"Start":"07:26.030 ","End":"07:35.865","Text":"This thing right here is the cumulative distribution of y and that\u0027s answering section a."},{"Start":"07:35.865 ","End":"07:39.470","Text":"Here we\u0027ve identified this distribution as"},{"Start":"07:39.470 ","End":"07:46.560","Text":"the uniform distribution with its parameters and here we answered section b."}],"ID":13168},{"Watched":false,"Name":"Exercise 2","Duration":"4m 35s","ChapterTopicVideoID":12682,"CourseChapterTopicPlaylistID":245052,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12682.jpeg","UploadDate":"2019-01-31T23:36:45.0530000","DurationForVideoObject":"PT4M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.220","Text":"In this question, we assume that X is"},{"Start":"00:02.220 ","End":"00:05.640","Text":"distributed with a uniform distribution with 0 and 1."},{"Start":"00:05.640 ","End":"00:10.650","Text":"Now, a new variable, R is defined where R equals to X squared and we\u0027re asked to find"},{"Start":"00:10.650 ","End":"00:16.080","Text":"the density function of the new variable R. Let\u0027s write the information down,"},{"Start":"00:16.080 ","End":"00:18.930","Text":"we have X is distributed with"},{"Start":"00:18.930 ","End":"00:25.290","Text":"the uniform distribution where a equals 0 and b equals to 1."},{"Start":"00:25.290 ","End":"00:27.270","Text":"That means that,"},{"Start":"00:27.270 ","End":"00:31.545","Text":"X is between 0 and 1."},{"Start":"00:31.545 ","End":"00:33.570","Text":"What about R?"},{"Start":"00:33.570 ","End":"00:40.030","Text":"Well, let\u0027s first plot this graph right here and see what R looks like."},{"Start":"00:40.030 ","End":"00:43.125","Text":"Here\u0027s the graph of R,"},{"Start":"00:43.125 ","End":"00:45.930","Text":"we have the X axis right here,"},{"Start":"00:45.930 ","End":"00:51.180","Text":"and we have the R axis right here and this is R equaling X squared."},{"Start":"00:51.180 ","End":"00:53.280","Text":"This is a regular parabola."},{"Start":"00:53.280 ","End":"00:56.970","Text":"Now, we\u0027ve defined X to be between 0 and 1,"},{"Start":"00:56.970 ","End":"00:59.970","Text":"so X right here that\u0027s 0,"},{"Start":"00:59.970 ","End":"01:02.090","Text":"and that\u0027s 1 right here."},{"Start":"01:02.090 ","End":"01:04.385","Text":"It won\u0027t have any other values."},{"Start":"01:04.385 ","End":"01:08.820","Text":"What happens to R when X is between 0 and 1?"},{"Start":"01:08.820 ","End":"01:11.410","Text":"Well, when X is 0,"},{"Start":"01:11.960 ","End":"01:16.045","Text":"R equals 0 squared, that\u0027s 0."},{"Start":"01:16.045 ","End":"01:18.590","Text":"What happens when X equals 1?"},{"Start":"01:18.590 ","End":"01:24.120","Text":"Well, then R equals 1 squared and that\u0027s 1. We\u0027re up here."},{"Start":"01:25.490 ","End":"01:33.370","Text":"R is between 1 and 0 or 0 and 1 and X is between 0 and 1."},{"Start":"01:33.770 ","End":"01:39.290","Text":"Let\u0027s now try to find the density function of the new variable"},{"Start":"01:39.290 ","End":"01:44.330","Text":"R. We\u0027ll start with the cumulative distribution function of X."},{"Start":"01:44.330 ","End":"01:45.740","Text":"We say that,"},{"Start":"01:45.740 ","End":"01:53.520","Text":"probability of X being less than or equal to some value. Now what is that?"},{"Start":"01:53.520 ","End":"01:57.335","Text":"Well, since X is distributed with the uniform distribution,"},{"Start":"01:57.335 ","End":"02:05.300","Text":"then this probability is x minus a divided by b minus a."},{"Start":"02:05.300 ","End":"02:08.115","Text":"Now, a is 0 and b is 1,"},{"Start":"02:08.115 ","End":"02:11.780","Text":"so we have x minus 0 divided by 1 minus 0,"},{"Start":"02:11.780 ","End":"02:14.510","Text":"and that equals to x."},{"Start":"02:14.510 ","End":"02:17.170","Text":"Now, what about R?"},{"Start":"02:17.170 ","End":"02:23.580","Text":"Well, the probability of R now being less than or equal to r,"},{"Start":"02:23.580 ","End":"02:30.590","Text":"a variable r being less than or equal to some value."},{"Start":"02:30.590 ","End":"02:33.235","Text":"Now, that equals to what?"},{"Start":"02:33.235 ","End":"02:35.415","Text":"Let\u0027s substitute now,"},{"Start":"02:35.415 ","End":"02:39.045","Text":"R with the transformation x squared."},{"Start":"02:39.045 ","End":"02:44.130","Text":"That\u0027ll be the probability of X squared being less than or equal"},{"Start":"02:44.130 ","End":"02:49.740","Text":"to some value r. Let\u0027s extract X now."},{"Start":"02:49.740 ","End":"02:52.140","Text":"That would be the probability."},{"Start":"02:52.140 ","End":"02:54.525","Text":"Now, let\u0027s take the square root of both sides,"},{"Start":"02:54.525 ","End":"02:58.946","Text":"so that\u0027ll be x being less than or equal to the square root of r. Now,"},{"Start":"02:58.946 ","End":"03:04.800","Text":"we can do that because X is between 0 and 1 only,"},{"Start":"03:04.800 ","End":"03:08.135","Text":"it doesn\u0027t have any negative values here."},{"Start":"03:08.135 ","End":"03:11.270","Text":"What does that equal to?"},{"Start":"03:11.270 ","End":"03:16.695","Text":"Well, that equals to the square root of r, and why is that?"},{"Start":"03:16.695 ","End":"03:21.500","Text":"Well, look at here, the probability of the random variable X being"},{"Start":"03:21.500 ","End":"03:26.270","Text":"less than or equal to some value is that value itself."},{"Start":"03:26.270 ","End":"03:29.420","Text":"Here, the probability of X,"},{"Start":"03:29.420 ","End":"03:32.720","Text":"a random variable X being less than or equal to some value."},{"Start":"03:32.720 ","End":"03:35.270","Text":"In our case, that\u0027s the square root of r,"},{"Start":"03:35.270 ","End":"03:37.130","Text":"well, that\u0027s that value itself."},{"Start":"03:37.130 ","End":"03:40.275","Text":"That\u0027s the square root of r. But hold on,"},{"Start":"03:40.275 ","End":"03:42.480","Text":"we\u0027re not finished yet, and why is that?"},{"Start":"03:42.480 ","End":"03:48.290","Text":"Because this is the cumulative distribution function"},{"Start":"03:48.290 ","End":"03:52.005","Text":"of R. That\u0027s F of R and we don\u0027t want that,"},{"Start":"03:52.005 ","End":"03:56.064","Text":"we want the density function of R. Now,"},{"Start":"03:56.064 ","End":"03:57.950","Text":"to get the density function of R,"},{"Start":"03:57.950 ","End":"04:02.780","Text":"we have to take the derivative of the cumulative distribution function."},{"Start":"04:02.780 ","End":"04:05.205","Text":"That means that F,"},{"Start":"04:05.205 ","End":"04:07.665","Text":"the derivative of F at R,"},{"Start":"04:07.665 ","End":"04:14.420","Text":"that equals to the density function of R and that equals to the derivative,"},{"Start":"04:14.420 ","End":"04:17.939","Text":"we derive by r the function,"},{"Start":"04:17.939 ","End":"04:21.480","Text":"the square root of r. This is what we want."},{"Start":"04:21.480 ","End":"04:26.270","Text":"Now, that equals to 1 divided by 2 times the square root of r,"},{"Start":"04:26.270 ","End":"04:28.880","Text":"where r, as we said,"},{"Start":"04:28.880 ","End":"04:31.310","Text":"is between 0 and 1."},{"Start":"04:31.310 ","End":"04:36.360","Text":"This then is the density function of r."}],"ID":13169},{"Watched":false,"Name":"Exercise 3","Duration":"5m 32s","ChapterTopicVideoID":12683,"CourseChapterTopicPlaylistID":245052,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12683.jpeg","UploadDate":"2019-01-31T23:37:11.9200000","DurationForVideoObject":"PT5M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.670","Text":"In this question, we assume that"},{"Start":"00:02.670 ","End":"00:05.625","Text":"x has an exponential distribution with parameter Lambda,"},{"Start":"00:05.625 ","End":"00:09.460","Text":"and y our new variable that equals to ln of x,"},{"Start":"00:09.460 ","End":"00:11.220","Text":"and we\u0027re asked to prove that"},{"Start":"00:11.220 ","End":"00:14.550","Text":"the density function of y is given by the following formulas,"},{"Start":"00:14.550 ","End":"00:18.210","Text":"f of y equals to this expression right here."},{"Start":"00:18.210 ","End":"00:21.595","Text":"First of all, let\u0027s write down what we know."},{"Start":"00:21.595 ","End":"00:29.435","Text":"We know that x is distributed with an exponential distribution with parameter Lambda."},{"Start":"00:29.435 ","End":"00:32.908","Text":"That means that the cumulative distribution function of x,"},{"Start":"00:32.908 ","End":"00:41.330","Text":"that\u0027s F of x equals to the probability of x being less than or equal to some value."},{"Start":"00:41.330 ","End":"00:47.180","Text":"That equals to 1 minus e to the power of minus Lambda times x."},{"Start":"00:47.180 ","End":"00:49.730","Text":"That\u0027s the cumulative distribution of"},{"Start":"00:49.730 ","End":"00:52.400","Text":"x when is distributed with an exponential distribution."},{"Start":"00:52.400 ","End":"00:53.875","Text":"That\u0027s the definition."},{"Start":"00:53.875 ","End":"01:00.245","Text":"What are the values that x can have in this distribution?"},{"Start":"01:00.245 ","End":"01:04.400","Text":"We know that x can have only positive values,"},{"Start":"01:04.400 ","End":"01:08.160","Text":"from 0 to plus infinity."},{"Start":"01:08.180 ","End":"01:11.190","Text":"What about y?"},{"Start":"01:11.190 ","End":"01:16.505","Text":"How does y behave when x is a positive integer?"},{"Start":"01:16.505 ","End":"01:21.110","Text":"For that, we need to draw this function right here."},{"Start":"01:21.110 ","End":"01:24.545","Text":"Let\u0027s do that. Here we go."},{"Start":"01:24.545 ","End":"01:26.405","Text":"This is our plot."},{"Start":"01:26.405 ","End":"01:29.760","Text":"Here we see y equals ln of x."},{"Start":"01:29.760 ","End":"01:32.555","Text":"That\u0027s this graph right here."},{"Start":"01:32.555 ","End":"01:36.350","Text":"We have our x-axis and we have a y-axis."},{"Start":"01:36.350 ","End":"01:41.420","Text":"We said that x can only have positive values."},{"Start":"01:41.420 ","End":"01:46.685","Text":"That means that x is from 0 to plus infinity."},{"Start":"01:46.685 ","End":"01:49.375","Text":"What happens to y?"},{"Start":"01:49.375 ","End":"01:55.950","Text":"As x approaches 0,"},{"Start":"01:55.950 ","End":"01:59.329","Text":"y approaches to minus infinity,"},{"Start":"01:59.329 ","End":"02:01.205","Text":"that\u0027ll be minus infinity here."},{"Start":"02:01.205 ","End":"02:04.759","Text":"What happens when x goes to plus infinity?"},{"Start":"02:04.759 ","End":"02:12.575","Text":"So does y. Y is defined from minus infinity to plus infinity."},{"Start":"02:12.575 ","End":"02:15.360","Text":"Basically all the numbers of y."},{"Start":"02:15.940 ","End":"02:22.400","Text":"Having understood that, let\u0027s see how we can calculate the density function of y."},{"Start":"02:22.400 ","End":"02:23.765","Text":"In order to do that,"},{"Start":"02:23.765 ","End":"02:29.660","Text":"let\u0027s first see what the cumulative distribution function of y is."},{"Start":"02:29.660 ","End":"02:40.880","Text":"That\u0027s F of y that equals to the probability of y being less than or equal to some value."},{"Start":"02:40.880 ","End":"02:43.130","Text":"What\u0027s y?"},{"Start":"02:43.130 ","End":"02:47.150","Text":"The random variable of y equals to ln of x."},{"Start":"02:47.150 ","End":"02:50.870","Text":"We can substitute ln of x instead of y."},{"Start":"02:50.870 ","End":"02:57.245","Text":"That\u0027ll be the probability of ln of x being less than or equal to y."},{"Start":"02:57.245 ","End":"03:00.410","Text":"We need to extract x here."},{"Start":"03:00.410 ","End":"03:02.215","Text":"How do we do that?"},{"Start":"03:02.215 ","End":"03:04.440","Text":"In order to extract x,"},{"Start":"03:04.440 ","End":"03:11.940","Text":"we need to take the inverse function of the ln operator, and what\u0027s that?"},{"Start":"03:11.940 ","End":"03:20.780","Text":"That\u0027s e. That equals to the probability of e to the power of ln x,"},{"Start":"03:20.780 ","End":"03:24.350","Text":"and that has to be less than or equal to e to the power of"},{"Start":"03:24.350 ","End":"03:30.250","Text":"y. E to the power of ln x, that\u0027s x."},{"Start":"03:30.250 ","End":"03:37.225","Text":"That\u0027s the probability of x being less than or equal to e to the power of y."},{"Start":"03:37.225 ","End":"03:39.885","Text":"What do we have here?"},{"Start":"03:39.885 ","End":"03:43.550","Text":"This is the definition of the cumulative distribution of"},{"Start":"03:43.550 ","End":"03:47.370","Text":"x when x is less than or equal to some value."},{"Start":"03:47.370 ","End":"03:49.620","Text":"That\u0027s this expression right here,"},{"Start":"03:49.620 ","End":"03:51.075","Text":"where instead of x,"},{"Start":"03:51.075 ","End":"03:56.865","Text":"we have to substitute x for this value right here. Let\u0027s do that."},{"Start":"03:56.865 ","End":"04:03.335","Text":"That equals to 1 minus e to the power of minus Lambda, that\u0027s right here."},{"Start":"04:03.335 ","End":"04:07.050","Text":"Instead of x, we\u0027ll put an e to the power of y."},{"Start":"04:07.510 ","End":"04:14.550","Text":"This then is the cumulative distribution function of y,"},{"Start":"04:14.570 ","End":"04:17.420","Text":"but we weren\u0027t asked that."},{"Start":"04:17.420 ","End":"04:21.005","Text":"We were asked about the density function of y."},{"Start":"04:21.005 ","End":"04:22.685","Text":"In order to do that,"},{"Start":"04:22.685 ","End":"04:26.210","Text":"we need to derive or take the derivative of"},{"Start":"04:26.210 ","End":"04:30.395","Text":"the cumulative distribution in order to get the density function."},{"Start":"04:30.395 ","End":"04:36.150","Text":"Let\u0027s just take the derivative of this expression right here."},{"Start":"04:37.490 ","End":"04:42.200","Text":"The derivative of the cumulative distribution of y,"},{"Start":"04:42.200 ","End":"04:44.975","Text":"that\u0027s f of y,"},{"Start":"04:44.975 ","End":"04:49.280","Text":"that equals to the derivative of this thing right here."},{"Start":"04:49.280 ","End":"04:55.880","Text":"That equals to minus e to the power of minus Lambda e to the power of y times"},{"Start":"04:55.880 ","End":"05:03.490","Text":"the internal derivative times minus Lambda e to the power of y."},{"Start":"05:07.670 ","End":"05:10.620","Text":"That equals to, minus and minus cancel each other out,"},{"Start":"05:10.620 ","End":"05:14.025","Text":"so that\u0027ll be Lambda e to the power"},{"Start":"05:14.025 ","End":"05:22.745","Text":"of minus Lambda e to the power of y plus 1."},{"Start":"05:22.745 ","End":"05:27.110","Text":"This then is the density function of y,"},{"Start":"05:27.110 ","End":"05:29.375","Text":"and this is what we were meant to prove."},{"Start":"05:29.375 ","End":"05:32.880","Text":"This thing equals to this thing."}],"ID":13170},{"Watched":false,"Name":"Exercise 4","Duration":"10m 41s","ChapterTopicVideoID":12684,"CourseChapterTopicPlaylistID":245052,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12684.jpeg","UploadDate":"2019-01-31T23:37:43.2770000","DurationForVideoObject":"PT10M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.370","Text":"In this function, we assume that X has"},{"Start":"00:02.370 ","End":"00:05.640","Text":"an exponential distribution with lambda equaling 1."},{"Start":"00:05.640 ","End":"00:11.760","Text":"Let Y a new random variable be equal to 1 minus 2 times e to the power of minus x."},{"Start":"00:11.760 ","End":"00:15.500","Text":"We\u0027re asked to find the cumulative distribution function of Y"},{"Start":"00:15.500 ","End":"00:20.340","Text":"and also to see if we can identify the probability distribution of Y."},{"Start":"00:20.340 ","End":"00:24.005","Text":"The first thing we want to do is write down what we know."},{"Start":"00:24.005 ","End":"00:27.560","Text":"Well, we have x and it\u0027s distributed with"},{"Start":"00:27.560 ","End":"00:31.865","Text":"an exponential distribution where lambda equals to 1."},{"Start":"00:31.865 ","End":"00:36.485","Text":"Now, with for an exponential distribution,"},{"Start":"00:36.485 ","End":"00:41.670","Text":"the valid values of X is all the positive values."},{"Start":"00:41.670 ","End":"00:46.220","Text":"X is defined on the positive range of values."},{"Start":"00:46.220 ","End":"00:47.600","Text":"X can\u0027t be negative,"},{"Start":"00:47.600 ","End":"00:50.180","Text":"it\u0027s from 0 to plus infinity."},{"Start":"00:50.180 ","End":"00:57.265","Text":"Now, that also means that the cumulative distribution function of X,"},{"Start":"00:57.265 ","End":"01:03.785","Text":"that equals to the probability of X being less than or equal to some value."},{"Start":"01:03.785 ","End":"01:11.515","Text":"Well that equals to 1 minus e to the power of minus lambda times X."},{"Start":"01:11.515 ","End":"01:13.350","Text":"Now in our case,"},{"Start":"01:13.350 ","End":"01:18.080","Text":"that equals to 1 minus e to the power of minus X."},{"Start":"01:18.080 ","End":"01:20.900","Text":"Now why is that? Because lambda equals 1."},{"Start":"01:20.900 ","End":"01:29.678","Text":"The cumulative distribution function of X is 1 minus e to the power of minus X."},{"Start":"01:29.678 ","End":"01:34.375","Text":"Now let\u0027s take a look at Y."},{"Start":"01:34.375 ","End":"01:40.655","Text":"Now, what\u0027s the behavior of Y when X is positive,"},{"Start":"01:40.655 ","End":"01:43.715","Text":"when X goes from 0 to plus infinity."},{"Start":"01:43.715 ","End":"01:50.955","Text":"Well, let\u0027s just research this function right here."},{"Start":"01:50.955 ","End":"01:57.146","Text":"Now we know that y equals 1 minus e 2 times e to the power of minus X."},{"Start":"01:57.146 ","End":"01:59.285","Text":"The first thing that we want to know is,"},{"Start":"01:59.285 ","End":"02:02.360","Text":"does this function have a maximum or"},{"Start":"02:02.360 ","End":"02:07.055","Text":"minimum or is it monotonously increasing, decreasing?"},{"Start":"02:07.055 ","End":"02:09.020","Text":"Let\u0027s find that out first."},{"Start":"02:09.020 ","End":"02:12.050","Text":"Let\u0027s take the derivative of Y."},{"Start":"02:12.050 ","End":"02:21.520","Text":"That equals to minus 2 times e to the power of minus X times minus 1."},{"Start":"02:21.520 ","End":"02:29.030","Text":"Now, that equals to 2 times e to the power of minus x, and that\u0027s positive."},{"Start":"02:29.030 ","End":"02:34.760","Text":"That means that the function is monotonously increasing."},{"Start":"02:34.760 ","End":"02:40.900","Text":"Now, let\u0027s take a look at what happens to Y when X equals 0."},{"Start":"02:40.900 ","End":"02:43.035","Text":"Well, when X equals 0,"},{"Start":"02:43.035 ","End":"02:44.630","Text":"let\u0027s just plug in 0 here."},{"Start":"02:44.630 ","End":"02:49.700","Text":"There\u0027ll be 1 minus 2 times e to the power of minus 0."},{"Start":"02:49.700 ","End":"02:52.950","Text":"Well, that equals to 1 minus,"},{"Start":"02:52.950 ","End":"02:55.530","Text":"now e to the power of 0 or minus 0 equals 1."},{"Start":"02:55.530 ","End":"02:57.000","Text":"It\u0027ll be 1 minus 2,"},{"Start":"02:57.000 ","End":"02:58.935","Text":"that\u0027ll be minus 1."},{"Start":"02:58.935 ","End":"03:04.835","Text":"What happens to Y when X equals plus infinity?"},{"Start":"03:04.835 ","End":"03:13.535","Text":"Well, that\u0027s 1 minus 2 to the power of e. 2 times e to the power of minus infinity."},{"Start":"03:13.535 ","End":"03:17.640","Text":"Well, that means that this expression right here goes to 0,"},{"Start":"03:17.640 ","End":"03:18.950","Text":"so it\u0027d be 1 minus 1,"},{"Start":"03:18.950 ","End":"03:20.797","Text":"that\u0027ll be equal to 1."},{"Start":"03:20.797 ","End":"03:28.860","Text":"Now we know that Y is between minus 1 and 1,"},{"Start":"03:28.860 ","End":"03:36.560","Text":"minus 1 when X equals 0 and 1 when X goes to plus infinity."},{"Start":"03:36.560 ","End":"03:40.105","Text":"Let\u0027s just see how this function looks."},{"Start":"03:40.105 ","End":"03:42.380","Text":"Let\u0027s just draw this out."},{"Start":"03:42.380 ","End":"03:45.520","Text":"Here we go. This is what a function looks like."},{"Start":"03:45.520 ","End":"03:48.170","Text":"Here we have our x-axis right here and here,"},{"Start":"03:48.170 ","End":"03:49.775","Text":"we have our y-axis."},{"Start":"03:49.775 ","End":"03:51.800","Text":"Now, this function right here,"},{"Start":"03:51.800 ","End":"03:57.320","Text":"that\u0027s Y equaling 1 minus 2 times e to the power of minus X."},{"Start":"03:57.320 ","End":"03:58.670","Text":"Now, as we see,"},{"Start":"03:58.670 ","End":"04:03.350","Text":"when x goes to 0,"},{"Start":"04:03.350 ","End":"04:09.420","Text":"right here, then Y equals minus 1."},{"Start":"04:09.420 ","End":"04:14.495","Text":"When X goes to plus infinity,"},{"Start":"04:14.495 ","End":"04:16.700","Text":"that goes to plus 1."},{"Start":"04:16.700 ","End":"04:18.080","Text":"This is 1 right here,"},{"Start":"04:18.080 ","End":"04:20.410","Text":"this is minus 1 right here."},{"Start":"04:20.410 ","End":"04:27.125","Text":"Now, this is defined for all positive values of X. X doesn\u0027t have any negative values."},{"Start":"04:27.125 ","End":"04:31.850","Text":"Great. Now that we know what Y looks like,"},{"Start":"04:31.850 ","End":"04:37.230","Text":"Let\u0027s now calculate the cumulative distribution function of Y."},{"Start":"04:37.640 ","End":"04:42.560","Text":"Let\u0027s calculate the cumulative distribution of Y."},{"Start":"04:42.560 ","End":"04:44.465","Text":"Well, first of all,"},{"Start":"04:44.465 ","End":"04:48.185","Text":"let\u0027s recall that the cumulative distribution of X,"},{"Start":"04:48.185 ","End":"04:52.670","Text":"that equal to 1 minus e to the power of minus X. Now why is that?"},{"Start":"04:52.670 ","End":"04:55.736","Text":"Well, that was because lambda equal to 1."},{"Start":"04:55.736 ","End":"05:00.905","Text":"What\u0027s the cumulative distribution function of Y?"},{"Start":"05:00.905 ","End":"05:06.080","Text":"That\u0027s the probability of Y being less than or equal to some value Y."},{"Start":"05:06.080 ","End":"05:09.110","Text":"Now, instead of Y, well,"},{"Start":"05:09.110 ","End":"05:13.730","Text":"let\u0027s just substitute that by this expression right here."},{"Start":"05:13.730 ","End":"05:18.200","Text":"That\u0027s the probability of 1 minus 2 times e to"},{"Start":"05:18.200 ","End":"05:23.300","Text":"the power of minus X well then has to be less than or equal to some value of Y."},{"Start":"05:23.300 ","End":"05:30.935","Text":"Now, let\u0027s just do a little bit of manipulation to see if we can isolate X."},{"Start":"05:30.935 ","End":"05:36.950","Text":"That equals to the probability of minus 2 times e"},{"Start":"05:36.950 ","End":"05:43.070","Text":"to the power of minus X that\u0027s less than or equal to Y minus 1."},{"Start":"05:43.070 ","End":"05:50.150","Text":"Now, that equals to probability of e to the power of minus x."},{"Start":"05:50.150 ","End":"05:55.730","Text":"Now that has to be greater or equal to y minus 1 divided by minus 2."},{"Start":"05:55.730 ","End":"05:57.170","Text":"Since we divided by minus 2,"},{"Start":"05:57.170 ","End":"06:00.020","Text":"we switched the inequality sign."},{"Start":"06:00.020 ","End":"06:03.815","Text":"Now, let\u0027s take ln of both sides."},{"Start":"06:03.815 ","End":"06:10.475","Text":"That\u0027s the probability of ln times e to the power of minus x."},{"Start":"06:10.475 ","End":"06:20.625","Text":"That\u0027s greater or equal to ln of y minus 1 divided by minus 2."},{"Start":"06:20.625 ","End":"06:26.194","Text":"Now that equals to ln of e of something."},{"Start":"06:26.194 ","End":"06:28.235","Text":"Well, that\u0027s something itself,"},{"Start":"06:28.235 ","End":"06:32.730","Text":"so that\u0027s the probability of minus x being greater or"},{"Start":"06:32.730 ","End":"06:38.680","Text":"equal to ln of y minus 1 divided by minus 2."},{"Start":"06:38.840 ","End":"06:46.940","Text":"That equals to the probability of x being less than or equal"},{"Start":"06:46.940 ","End":"06:55.545","Text":"to ln minus ln of y minus 1 divided by minus 2."},{"Start":"06:55.545 ","End":"07:01.205","Text":"Let\u0027s just get rid of this minus right here and that\u0027ll be"},{"Start":"07:01.205 ","End":"07:07.950","Text":"the probability of x being less than or equal to minus ln."},{"Start":"07:07.950 ","End":"07:11.040","Text":"Now, instead of y minus 1 divided by minus 2,"},{"Start":"07:11.040 ","End":"07:14.690","Text":"we\u0027ll say 1 minus y divided by 2."},{"Start":"07:14.690 ","End":"07:16.910","Text":"What do we have here?"},{"Start":"07:16.910 ","End":"07:22.220","Text":"Well, this is the probability of X being less than some value right here."},{"Start":"07:22.220 ","End":"07:27.760","Text":"This is the cumulative distribution function of X,"},{"Start":"07:27.760 ","End":"07:30.230","Text":"and this is this guy right here."},{"Start":"07:30.230 ","End":"07:37.835","Text":"All we need to do is to substitute this value for this x right here."},{"Start":"07:37.835 ","End":"07:39.170","Text":"Let\u0027s do that."},{"Start":"07:39.170 ","End":"07:44.375","Text":"That\u0027ll be equal to 1 minus e to the power of minus."},{"Start":"07:44.375 ","End":"07:46.010","Text":"Now, this expression right here,"},{"Start":"07:46.010 ","End":"07:52.710","Text":"that\u0027ll be minus ln of 1 minus y divided by 2."},{"Start":"07:52.960 ","End":"07:56.420","Text":"Now, let\u0027s simplify that,"},{"Start":"07:56.420 ","End":"08:00.290","Text":"that\u0027s 1 minus e to the power of minus, minus,"},{"Start":"08:00.290 ","End":"08:08.610","Text":"that\u0027s plus e to the power of ln of 1 minus y divided by 2."},{"Start":"08:10.760 ","End":"08:14.490","Text":"That equals to now,"},{"Start":"08:14.490 ","End":"08:16.775","Text":"e to the power of ln of something."},{"Start":"08:16.775 ","End":"08:18.470","Text":"Well that\u0027s something."},{"Start":"08:18.470 ","End":"08:20.900","Text":"We have 1 minus."},{"Start":"08:20.900 ","End":"08:25.835","Text":"Now, there\u0027ll be 1 minus y divided by 2."},{"Start":"08:25.835 ","End":"08:29.330","Text":"Now, let\u0027s take a common denominator."},{"Start":"08:29.330 ","End":"08:40.815","Text":"This is equal to 2 minus 1 plus y divided by 2."},{"Start":"08:40.815 ","End":"08:43.785","Text":"Now that equals to what?"},{"Start":"08:43.785 ","End":"08:45.390","Text":"What 2 minus 1,"},{"Start":"08:45.390 ","End":"08:46.530","Text":"that\u0027s 1 plus y."},{"Start":"08:46.530 ","End":"08:50.347","Text":"It will be y plus 1 divided by 2."},{"Start":"08:50.347 ","End":"08:56.150","Text":"This then is the cumulative distribution function of Y,"},{"Start":"08:56.150 ","End":"08:57.770","Text":"but in what range?"},{"Start":"08:57.770 ","End":"09:01.129","Text":"Let\u0027s just write this out formally."},{"Start":"09:01.129 ","End":"09:04.190","Text":"The cumulative distribution function of Y."},{"Start":"09:04.190 ","End":"09:09.770","Text":"Well, that equals to y plus 1 divided by 2,"},{"Start":"09:09.770 ","End":"09:15.655","Text":"where y is between minus 1 and 1."},{"Start":"09:15.655 ","End":"09:20.915","Text":"Where y is less than minus 1,"},{"Start":"09:20.915 ","End":"09:22.700","Text":"well that\u0027ll be equal to 1,"},{"Start":"09:22.700 ","End":"09:26.790","Text":"and it\u0027ll be 1 where y is greater than 1."},{"Start":"09:26.860 ","End":"09:31.880","Text":"In this section, we\u0027re asked to identify the probability distribution of Y."},{"Start":"09:31.880 ","End":"09:34.460","Text":"In section a, we found what that was,"},{"Start":"09:34.460 ","End":"09:38.205","Text":"now we need to identify we want to know what this is."},{"Start":"09:38.205 ","End":"09:47.060","Text":"I maintain that Y has a uniform distribution where a equals to minus 1,"},{"Start":"09:47.060 ","End":"09:50.660","Text":"and b is equal to 1."},{"Start":"09:50.660 ","End":"09:53.600","Text":"Now why is that? Well, we know that"},{"Start":"09:53.600 ","End":"09:55.770","Text":"the cumulative distribution function of"},{"Start":"09:55.770 ","End":"10:00.140","Text":"a random variable that has a uniform distribution,"},{"Start":"10:00.140 ","End":"10:06.395","Text":"that equals to y minus a divided by b minus a."},{"Start":"10:06.395 ","End":"10:09.455","Text":"Now, in our case,"},{"Start":"10:09.455 ","End":"10:11.705","Text":"that\u0027ll be y minus."},{"Start":"10:11.705 ","End":"10:14.240","Text":"Now a is minus 1,"},{"Start":"10:14.240 ","End":"10:20.540","Text":"so that\u0027ll be minus 1 divided by b minus minus 1."},{"Start":"10:20.540 ","End":"10:26.560","Text":"Now that equals to y plus 1 divided by 2."},{"Start":"10:26.560 ","End":"10:33.205","Text":"This thing is the same as this expression right here."},{"Start":"10:33.205 ","End":"10:41.099","Text":"Y then has a uniform distribution where a equals minus 1 and b equals 1."}],"ID":13171},{"Watched":false,"Name":"Exercise 5","Duration":"4m 28s","ChapterTopicVideoID":12685,"CourseChapterTopicPlaylistID":245052,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12685.jpeg","UploadDate":"2019-01-31T23:37:55.8270000","DurationForVideoObject":"PT4M28S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"In this question, we\u0027re given that the length of the side of a die has"},{"Start":"00:03.210 ","End":"00:06.525","Text":"a uniform probability distribution between 1 and 2 and we\u0027re"},{"Start":"00:06.525 ","End":"00:10.875","Text":"asked to find the density function of the die\u0027s volume."},{"Start":"00:10.875 ","End":"00:17.790","Text":"Let\u0027s write down what we know that x is distributed with a uniform distribution,"},{"Start":"00:17.790 ","End":"00:22.500","Text":"where a equals 1 and b equals to 2."},{"Start":"00:22.500 ","End":"00:27.330","Text":"That means that x then is between 1 and 2."},{"Start":"00:27.330 ","End":"00:33.080","Text":"Now, our cumulative distribution function, that\u0027s big F of x,"},{"Start":"00:33.080 ","End":"00:38.990","Text":"just the probability of x being less than or equal to some value,"},{"Start":"00:38.990 ","End":"00:44.075","Text":"that equals to x minus a divided by b minus a."},{"Start":"00:44.075 ","End":"00:45.890","Text":"Now, in our case,"},{"Start":"00:45.890 ","End":"00:50.840","Text":"that\u0027s x minus 1 divided by 2 minus 1,"},{"Start":"00:50.840 ","End":"00:54.930","Text":"and that equals to x minus 1."},{"Start":"00:54.940 ","End":"00:58.310","Text":"Now, let\u0027s take a look at y,"},{"Start":"00:58.310 ","End":"01:04.545","Text":"we\u0027ll define y as the volume of the die."},{"Start":"01:04.545 ","End":"01:09.310","Text":"Now that equals to x cubed."},{"Start":"01:09.310 ","End":"01:14.645","Text":"How does y behave when x is between 1 and 2?"},{"Start":"01:14.645 ","End":"01:20.100","Text":"Well, let\u0027s just draw this function and see what\u0027s the range of y."},{"Start":"01:20.390 ","End":"01:22.620","Text":"Here\u0027s our graph."},{"Start":"01:22.620 ","End":"01:29.780","Text":"We see here our x-axis and our y-axis and we see here the graph of y equals x cubed."},{"Start":"01:29.780 ","End":"01:34.955","Text":"Now, what happens to y when x equals 1?"},{"Start":"01:34.955 ","End":"01:38.530","Text":"Well, when x equals 1, that\u0027s right here."},{"Start":"01:38.530 ","End":"01:42.825","Text":"When x equals 1, that\u0027ll be 1 cubed,"},{"Start":"01:42.825 ","End":"01:45.285","Text":"so y would be equal to 1."},{"Start":"01:45.285 ","End":"01:48.780","Text":"Now, what happens when x equals 2?"},{"Start":"01:48.780 ","End":"01:51.105","Text":"Well, when x equals to 2,"},{"Start":"01:51.105 ","End":"01:54.405","Text":"2 cubed would be equal to 8."},{"Start":"01:54.405 ","End":"02:01.630","Text":"We know now that y is between 1 and 8."},{"Start":"02:02.060 ","End":"02:06.185","Text":"Having understood that, let\u0027s try to figure out"},{"Start":"02:06.185 ","End":"02:09.560","Text":"what the cumulative distribution function of y is,"},{"Start":"02:09.560 ","End":"02:12.350","Text":"that\u0027s big F of y."},{"Start":"02:12.350 ","End":"02:17.600","Text":"That equals to the probability of y being less than or equal to some value."},{"Start":"02:17.600 ","End":"02:20.580","Text":"Now, y equals to x cubed,"},{"Start":"02:20.580 ","End":"02:22.100","Text":"so let\u0027s just substitute that."},{"Start":"02:22.100 ","End":"02:28.235","Text":"That\u0027ll be the probability of x cubed being less than y."},{"Start":"02:28.235 ","End":"02:31.150","Text":"Now, let\u0027s extract x,"},{"Start":"02:31.150 ","End":"02:38.290","Text":"that\u0027ll be equal to the probability of x being less than or equal to the cube root of y."},{"Start":"02:38.290 ","End":"02:43.910","Text":"Here we have the probability of x being less than or equal to some value right here."},{"Start":"02:43.910 ","End":"02:46.925","Text":"Now we know what that is."},{"Start":"02:46.925 ","End":"02:50.570","Text":"This is the probability of x being less than or equal to"},{"Start":"02:50.570 ","End":"02:53.810","Text":"some value that equals to x minus 1."},{"Start":"02:53.810 ","End":"02:55.430","Text":"Now instead of x minus 1,"},{"Start":"02:55.430 ","End":"02:56.990","Text":"we\u0027ll just substitute for x,"},{"Start":"02:56.990 ","End":"02:58.760","Text":"this value right here."},{"Start":"02:58.760 ","End":"03:06.610","Text":"That\u0027ll be equal to the cube root of y minus 1."},{"Start":"03:06.710 ","End":"03:12.665","Text":"This then is the cumulative distribution function of y this right here."},{"Start":"03:12.665 ","End":"03:15.340","Text":"But we were asked about the density function,"},{"Start":"03:15.340 ","End":"03:22.890","Text":"so we have to take the derivative of this expression right here. Let\u0027s do that."},{"Start":"03:23.750 ","End":"03:26.730","Text":"Small f of y,"},{"Start":"03:26.730 ","End":"03:32.510","Text":"that equals to the derivative of the cumulative distribution of y."},{"Start":"03:32.510 ","End":"03:39.020","Text":"Now, that means that we have to do take the derivative of this expression right here."},{"Start":"03:39.020 ","End":"03:41.930","Text":"Now we can write this a little bit better."},{"Start":"03:41.930 ","End":"03:47.645","Text":"This will be y^1/3 minus 1."},{"Start":"03:47.645 ","End":"03:49.805","Text":"This is the same thing."},{"Start":"03:49.805 ","End":"03:58.170","Text":"The derivative of this will be 1/3y^minus 2/3."},{"Start":"03:59.740 ","End":"04:04.385","Text":"What\u0027s this? Well, let\u0027s write this out a little bit better."},{"Start":"04:04.385 ","End":"04:13.210","Text":"That equals to 1 divided by 3 times the cube root of y squared."},{"Start":"04:13.210 ","End":"04:18.360","Text":"That\u0027s for y between 1 and 8,"},{"Start":"04:18.360 ","End":"04:23.155","Text":"so this then is the density function of y,"},{"Start":"04:23.155 ","End":"04:28.180","Text":"where y is defined between 1 and 8."}],"ID":13172},{"Watched":false,"Name":"Exercise 6 - Parts a-b","Duration":"2m 26s","ChapterTopicVideoID":12687,"CourseChapterTopicPlaylistID":245052,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12687.jpeg","UploadDate":"2019-01-31T23:38:15.7030000","DurationForVideoObject":"PT2M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.795","Text":"In this question, we assume that the following cumulative distribution function,"},{"Start":"00:03.795 ","End":"00:10.341","Text":"F sub x of t equals Theta to the power of t minus 1 for t defined between 0 and 1."},{"Start":"00:10.341 ","End":"00:14.655","Text":"We\u0027re asked to find the value of the parameter Theta."},{"Start":"00:14.655 ","End":"00:19.110","Text":"What do we know about cumulative distribution functions?"},{"Start":"00:19.110 ","End":"00:22.905","Text":"Well, F sub x of t, well,"},{"Start":"00:22.905 ","End":"00:31.905","Text":"it has values between 0 and 1 where t is between 0 and 1."},{"Start":"00:31.905 ","End":"00:36.470","Text":"F sub x of 0, for example, well,"},{"Start":"00:36.470 ","End":"00:41.750","Text":"that has to be equal to 0 and F sub x of 1,"},{"Start":"00:41.750 ","End":"00:45.470","Text":"where t equals 1, that has to be equal to 1."},{"Start":"00:45.470 ","End":"00:52.900","Text":"Knowing that, let\u0027s extract Theta using t equaling 1."},{"Start":"00:52.900 ","End":"00:57.005","Text":"So 1, that\u0027s F of x at 1,"},{"Start":"00:57.005 ","End":"01:01.385","Text":"that equals to Theta to the power of 1,"},{"Start":"01:01.385 ","End":"01:03.170","Text":"t equals 1 minus 1."},{"Start":"01:03.170 ","End":"01:07.795","Text":"That means that Theta equals to 2."},{"Start":"01:07.795 ","End":"01:11.360","Text":"What does that mean regarding"},{"Start":"01:11.360 ","End":"01:18.280","Text":"our cumulative distribution function of F sub x of t? Well, that equals 2."},{"Start":"01:18.280 ","End":"01:22.915","Text":"Well, 2 to the power of t minus 1,"},{"Start":"01:22.915 ","End":"01:26.980","Text":"where t is between 0 and 1."},{"Start":"01:26.980 ","End":"01:30.870","Text":"It\u0027s 0 when t is less than 0,"},{"Start":"01:30.870 ","End":"01:35.240","Text":"and it\u0027s 1 when t is greater than 1."},{"Start":"01:35.240 ","End":"01:39.190","Text":"In this section, we\u0027re asked to find the density function of x."},{"Start":"01:39.190 ","End":"01:44.995","Text":"That means that we need to take the derivative of the cumulative distribution function."},{"Start":"01:44.995 ","End":"01:49.495","Text":"Small f of t,"},{"Start":"01:49.495 ","End":"01:52.285","Text":"that equals the derivative of"},{"Start":"01:52.285 ","End":"01:55.030","Text":"the cumulative distribution function of t"},{"Start":"01:55.030 ","End":"01:58.515","Text":"and that means that we\u0027re looking for the derivative of"},{"Start":"01:58.515 ","End":"02:03.555","Text":"this expression right here and that equals to 2 to the power of"},{"Start":"02:03.555 ","End":"02:10.575","Text":"t times ln of 2 where t is between 0 and 1,"},{"Start":"02:10.575 ","End":"02:16.560","Text":"or if we want to write this out as a function of x, small x, well,"},{"Start":"02:16.560 ","End":"02:20.955","Text":"that equals to 2 to the power of x ln 2,"},{"Start":"02:20.955 ","End":"02:26.980","Text":"now where x is between 0 and 1."}],"ID":13173},{"Watched":false,"Name":"Exercise 6 - Part c","Duration":"4m 34s","ChapterTopicVideoID":12686,"CourseChapterTopicPlaylistID":245052,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12686.jpeg","UploadDate":"2019-01-31T23:38:08.9900000","DurationForVideoObject":"PT4M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.415","Text":"In this section, we\u0027re given Y."},{"Start":"00:02.415 ","End":"00:08.100","Text":"A new random variable that equals to 2^X minus 1."},{"Start":"00:08.100 ","End":"00:10.440","Text":"We\u0027re asked to find the density function of Y,"},{"Start":"00:10.440 ","End":"00:13.630","Text":"and identify the probability."},{"Start":"00:13.670 ","End":"00:20.060","Text":"We know the range of X. X is between 0 and 1."},{"Start":"00:20.060 ","End":"00:21.925","Text":"What about Y?"},{"Start":"00:21.925 ","End":"00:32.535","Text":"Well, Y, let\u0027s substitute 0 and 1 for X right here and see what values of Y we get."},{"Start":"00:32.535 ","End":"00:37.620","Text":"Well, what\u0027s Y when X equals 0?"},{"Start":"00:37.620 ","End":"00:40.750","Text":"Well, that\u0027s 2^0 minus 1,"},{"Start":"00:40.750 ","End":"00:42.530","Text":"and that equals to 0."},{"Start":"00:42.530 ","End":"00:46.230","Text":"What about when X equals to 1?"},{"Start":"00:46.230 ","End":"00:49.200","Text":"Well, that\u0027s 2^1 minus 1,"},{"Start":"00:49.200 ","End":"00:50.820","Text":"that equals to 1."},{"Start":"00:50.820 ","End":"00:55.905","Text":"Here we know that Y is between 0 and 1 as well."},{"Start":"00:55.905 ","End":"01:02.565","Text":"Now, let\u0027s just remember the probability of X being less than or equal to some value,"},{"Start":"01:02.565 ","End":"01:06.465","Text":"well that equals to 2^X minus 1."},{"Start":"01:06.465 ","End":"01:12.315","Text":"What about the probability now of Y being less than or equal to some value?"},{"Start":"01:12.315 ","End":"01:15.120","Text":"Well, that equals,"},{"Start":"01:15.120 ","End":"01:16.590","Text":"now instead of Y,"},{"Start":"01:16.590 ","End":"01:18.585","Text":"let\u0027s substitute this value."},{"Start":"01:18.585 ","End":"01:19.850","Text":"That\u0027s the transformation,"},{"Start":"01:19.850 ","End":"01:23.945","Text":"that\u0027s the probability of 2^X minus 1,"},{"Start":"01:23.945 ","End":"01:26.495","Text":"and that has to be less than or equal to Y."},{"Start":"01:26.495 ","End":"01:29.805","Text":"Now, let\u0027s extract our X."},{"Start":"01:29.805 ","End":"01:33.810","Text":"That\u0027s the probability of 2^X,"},{"Start":"01:33.810 ","End":"01:37.280","Text":"that\u0027s less than or equal to Y plus 1,"},{"Start":"01:37.280 ","End":"01:39.875","Text":"and that equals to the probability, now,"},{"Start":"01:39.875 ","End":"01:45.305","Text":"let\u0027s take log base 2 from both sides in order to isolate X."},{"Start":"01:45.305 ","End":"01:49.945","Text":"That\u0027s log base 2 of 2^X."},{"Start":"01:49.945 ","End":"01:56.350","Text":"That has to be less than or equal to log base 2 of Y plus 1."},{"Start":"01:56.780 ","End":"02:03.525","Text":"Now, the log base 2 of 2 to the power of something,"},{"Start":"02:03.525 ","End":"02:05.640","Text":"well, that\u0027s that something itself,"},{"Start":"02:05.640 ","End":"02:07.425","Text":"so that\u0027ll be X,"},{"Start":"02:07.425 ","End":"02:14.835","Text":"and that has to be less than or equal to log base 2 of Y plus 1."},{"Start":"02:14.835 ","End":"02:20.120","Text":"Now we have the probability of X being less than or equal to some value here."},{"Start":"02:20.120 ","End":"02:21.505","Text":"Now we know what that is."},{"Start":"02:21.505 ","End":"02:25.460","Text":"The probability of X being less than or equal to some value, well,"},{"Start":"02:25.460 ","End":"02:27.880","Text":"that\u0027s 2^X minus 1,"},{"Start":"02:27.880 ","End":"02:29.445","Text":"so instead of X,"},{"Start":"02:29.445 ","End":"02:31.955","Text":"we\u0027ll just substitute for this value."},{"Start":"02:31.955 ","End":"02:35.085","Text":"Now, that\u0027ll be equal then to 2 to the power of,"},{"Start":"02:35.085 ","End":"02:37.005","Text":"and now instead of X, we\u0027ll write,"},{"Start":"02:37.005 ","End":"02:43.570","Text":"log base 2 of Y plus 1 minus 1."},{"Start":"02:43.580 ","End":"02:46.560","Text":"Now, that equals to what?"},{"Start":"02:46.560 ","End":"02:49.920","Text":"2 to the power of log base 2 of something, well,"},{"Start":"02:49.920 ","End":"02:51.360","Text":"that\u0027s that something itself,"},{"Start":"02:51.360 ","End":"02:54.840","Text":"so that\u0027ll be Y plus 1."},{"Start":"02:54.840 ","End":"02:57.465","Text":"Let\u0027s not forget our minus 1 here,"},{"Start":"02:57.465 ","End":"03:00.580","Text":"and that equals Y."},{"Start":"03:01.010 ","End":"03:05.780","Text":"This then is our cumulative distribution function."},{"Start":"03:05.780 ","End":"03:06.980","Text":"Now we weren\u0027t asked about that."},{"Start":"03:06.980 ","End":"03:09.110","Text":"We were asked about the density function."},{"Start":"03:09.110 ","End":"03:12.050","Text":"Now, the density function of Y, well,"},{"Start":"03:12.050 ","End":"03:15.935","Text":"that\u0027s the derivative of the cumulative distribution function."},{"Start":"03:15.935 ","End":"03:19.120","Text":"That means that we have to take the derivative of Y,"},{"Start":"03:19.120 ","End":"03:21.090","Text":"and that equals to 1,"},{"Start":"03:21.090 ","End":"03:25.815","Text":"where Y is defined between 0 and 1."},{"Start":"03:25.815 ","End":"03:28.050","Text":"This means then,"},{"Start":"03:28.050 ","End":"03:33.190","Text":"that Y has a uniform distribution,"},{"Start":"03:33.190 ","End":"03:38.705","Text":"where a equals to 0 and b equals to 1."},{"Start":"03:38.705 ","End":"03:41.815","Text":"Now, let\u0027s just check that."},{"Start":"03:41.815 ","End":"03:49.145","Text":"We know that the cumulative distribution function of a uniform distribution,"},{"Start":"03:49.145 ","End":"03:54.220","Text":"well that equals to Y minus a divided by b minus a."},{"Start":"03:54.220 ","End":"03:56.970","Text":"Again, that\u0027s for Y,"},{"Start":"03:56.970 ","End":"03:59.100","Text":"now between 0 and 1,"},{"Start":"03:59.100 ","End":"04:04.725","Text":"so that means that we\u0027re looking at F at Y."},{"Start":"04:04.725 ","End":"04:09.780","Text":"Big F of Y, that equals to Y minus 0,"},{"Start":"04:09.780 ","End":"04:12.990","Text":"that\u0027s a divided by b,"},{"Start":"04:12.990 ","End":"04:15.920","Text":"1 minus a, that\u0027s also 0,"},{"Start":"04:15.920 ","End":"04:17.600","Text":"so that equals to Y."},{"Start":"04:17.600 ","End":"04:25.975","Text":"Here we go. This then is the cumulative distribution function that we got right here."},{"Start":"04:25.975 ","End":"04:29.420","Text":"That means that Y is really distributed with"},{"Start":"04:29.420 ","End":"04:34.740","Text":"a uniform distribution where a equals 0 and b equals 1."}],"ID":13174}],"Thumbnail":null,"ID":245052}]

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