Joint Probability Function
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Correlation between Variables
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- Tutorial 1
- Tutorial 2
- Tutorial 3
- Example
- Exercise 1 - Part a
- Exercise 1 - Parts b-c
- Exercise 1 - Parts d-f
- Exercise 2 - Parts a-b
- Exercise 2 - Parts c-e
- Exercise 3 - Part a
- Exercise 3 - Part b
- Exercise 4 - Part a
- Exercise 4 - Part b
- Exercise 4 - Part c
- Exercise 5 - Part a
- Exercise 5 - Parts b-c
- Exercise 5 - Part d
- Exercise 5 - Part e

Linear Combinations
0/8 completed

Summary Questions
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- Introduction
- Exercise 1 Part a
- Exercise 1 Part b
- Exercise 1 Parts c-d
- Exercise 2 Part a
- Exercise 2 Parts b-c
- Exercise 2 parts d-e
- Exercise 3 Part a
- Exercise 3 Parts b-d
- Exercise 4 Parts a-b
- Exercise 4 Parts c-d
- Exercise 5 Part a
- Exercise 5 Parts b-c
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9 Part a
- Exercise 9 Part b
- Exercise 10
- Exercise 11
- Exercise 12

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[{"Name":"Joint Probability Function","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial","Duration":"34s","ChapterTopicVideoID":12703,"CourseChapterTopicPlaylistID":245018,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12703.jpeg","UploadDate":"2018-07-30T14:12:14.8170000","DurationForVideoObject":"PT34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.820","Text":"In this chapter, we\u0027ll be talking about"},{"Start":"00:02.820 ","End":"00:08.112","Text":"a 2-dimensional discrete variable and a joint probability function."},{"Start":"00:08.112 ","End":"00:11.280","Text":"A 2-dimensional probability distribution is"},{"Start":"00:11.280 ","End":"00:15.915","Text":"a probability distribution involving 2 variables, X and Y."},{"Start":"00:15.915 ","End":"00:20.700","Text":"Our objective will be to construct a 2-dimensional table containing"},{"Start":"00:20.700 ","End":"00:26.775","Text":"the values of the 2 variables and within the table the joint probabilities."},{"Start":"00:26.775 ","End":"00:31.920","Text":"This table will be called the joint probability function of X and Y."},{"Start":"00:31.920 ","End":"00:35.320","Text":"Let\u0027s take a look at an example."}],"ID":13182},{"Watched":false,"Name":"Example","Duration":"14m 36s","ChapterTopicVideoID":12704,"CourseChapterTopicPlaylistID":245018,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12704.jpeg","UploadDate":"2018-07-30T14:13:33.4970000","DurationForVideoObject":"PT14M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"In our example, a student takes semester exams,"},{"Start":"00:03.240 ","End":"00:06.090","Text":"one in economics and one in statistics."},{"Start":"00:06.090 ","End":"00:09.930","Text":"Now, assume that the probability of passing the economics exam is"},{"Start":"00:09.930 ","End":"00:15.090","Text":"0.8 and the chances of passing the statistics exam is 0.9."},{"Start":"00:15.090 ","End":"00:19.185","Text":"The chances of passing both exams are 0.75."},{"Start":"00:19.185 ","End":"00:23.933","Text":"Now, let X be the number of courses that the student passes,"},{"Start":"00:23.933 ","End":"00:28.500","Text":"and Y is an indicator variable that receives the value 1"},{"Start":"00:28.500 ","End":"00:33.340","Text":"if the student passes the economics exam and 0 if he doesn\u0027t."},{"Start":"00:33.340 ","End":"00:39.200","Text":"Now, we\u0027re asked to construct the joint probability function of X and Y."},{"Start":"00:39.200 ","End":"00:43.370","Text":"First of all, let\u0027s describe our data."},{"Start":"00:43.370 ","End":"00:46.605","Text":"Let\u0027s say that A,"},{"Start":"00:46.605 ","End":"00:53.445","Text":"that\u0027s the event of passing economics,"},{"Start":"00:53.445 ","End":"01:00.580","Text":"and B would be passing statistics."},{"Start":"01:00.580 ","End":"01:06.041","Text":"Now, the probability of passing economics,"},{"Start":"01:06.041 ","End":"01:08.780","Text":"well, that equals to 0.8."},{"Start":"01:08.780 ","End":"01:11.839","Text":"Now, the probability of passing statistics,"},{"Start":"01:11.839 ","End":"01:14.725","Text":"that equals to 0.9."},{"Start":"01:14.725 ","End":"01:19.570","Text":"The probability of passing both economics and statistics,"},{"Start":"01:19.570 ","End":"01:23.030","Text":"well, that equals to 0.75."},{"Start":"01:23.390 ","End":"01:28.750","Text":"As I said, we\u0027re describing the data and I chose to"},{"Start":"01:28.750 ","End":"01:33.170","Text":"use the Venn diagram in order to describe this data."},{"Start":"01:33.170 ","End":"01:37.960","Text":"Let\u0027s just put all this information onto the Venn diagram."},{"Start":"01:37.960 ","End":"01:39.524","Text":"Well, let\u0027s start with this;"},{"Start":"01:39.524 ","End":"01:45.370","Text":"the probability of passing both economics and statistics that\u0027s 0.75,"},{"Start":"01:45.370 ","End":"01:49.570","Text":"so we\u0027ll put that here, that\u0027ll be 0.75."},{"Start":"01:49.570 ","End":"01:54.130","Text":"Now, the probability of passing economics,"},{"Start":"01:54.130 ","End":"02:01.340","Text":"that\u0027s 0.8 so this whole circle has to be equal to 0.8."},{"Start":"02:01.340 ","End":"02:03.860","Text":"Now, 0.75 is in the intersection,"},{"Start":"02:03.860 ","End":"02:06.305","Text":"so the rest of the circle,"},{"Start":"02:06.305 ","End":"02:10.640","Text":"that has to be equal to 0.05."},{"Start":"02:10.640 ","End":"02:14.840","Text":"Now, the same thing applies for circle B."},{"Start":"02:14.840 ","End":"02:17.335","Text":"B is passing statistic."},{"Start":"02:17.335 ","End":"02:22.895","Text":"We have 0.75 as the intersection between A and B,"},{"Start":"02:22.895 ","End":"02:28.440","Text":"so the rest of circle B has to be equal to 0.15."},{"Start":"02:28.700 ","End":"02:37.485","Text":"The total probability of A is 0.75 plus 0.05 plus 0.8,"},{"Start":"02:37.485 ","End":"02:41.415","Text":"and the total area for event B,"},{"Start":"02:41.415 ","End":"02:46.585","Text":"well, that\u0027s 0.75 plus 0.15 plus 0.9."},{"Start":"02:46.585 ","End":"02:50.105","Text":"Now, this whole rectangle area"},{"Start":"02:50.105 ","End":"02:53.450","Text":"that has to be equal to 1, so the area outside of the circle, well, what\u0027s that?"},{"Start":"02:53.450 ","End":"02:58.175","Text":"Well, this is"},{"Start":"02:58.175 ","End":"03:05.237","Text":"0.75 plus 0.15 plus 0.9 and this is 0.05,"},{"Start":"03:05.237 ","End":"03:10.785","Text":"so we have a total of 0.95 within the circles,"},{"Start":"03:10.785 ","End":"03:19.700","Text":"so the total area of the rectangle has to be equal to 0.05."},{"Start":"03:19.700 ","End":"03:26.010","Text":"That\u0027s the rest of the probability that remains outside of the circles."},{"Start":"03:26.030 ","End":"03:33.460","Text":"Now, let\u0027s take a look at the definition of X and Y again."},{"Start":"03:33.460 ","End":"03:40.730","Text":"Now, here, X is the number of courses that the student passes."},{"Start":"03:40.730 ","End":"03:45.156","Text":"Well, outside of the circles,"},{"Start":"03:45.156 ","End":"03:49.955","Text":"the student didn\u0027t pass economics and he didn\u0027t pass statistics,"},{"Start":"03:49.955 ","End":"03:53.605","Text":"so X here has to be equal to 0."},{"Start":"03:53.605 ","End":"03:58.220","Text":"What about in this area right here?"},{"Start":"03:58.220 ","End":"04:03.110","Text":"Well, he passed economics but didn\u0027t pass statistics,"},{"Start":"04:03.110 ","End":"04:06.340","Text":"so X has to be equal to 1."},{"Start":"04:06.340 ","End":"04:09.225","Text":"In this intersection area,"},{"Start":"04:09.225 ","End":"04:15.440","Text":"X equals to 2 and outside of the intersection on circle B,"},{"Start":"04:15.440 ","End":"04:18.590","Text":"well, that\u0027s X equaling 1 again."},{"Start":"04:18.590 ","End":"04:24.143","Text":"Here, X equals 1 because the student passed statistics,"},{"Start":"04:24.143 ","End":"04:27.735","Text":"and here X equals 1 because he passed economics."},{"Start":"04:27.735 ","End":"04:30.860","Text":"In the middle is X equals 2,"},{"Start":"04:30.860 ","End":"04:34.250","Text":"where he passed both economics and statistics,"},{"Start":"04:34.250 ","End":"04:35.990","Text":"and outside of the circles,"},{"Start":"04:35.990 ","End":"04:39.950","Text":"he didn\u0027t pass economics and he didn\u0027t pass statistics,"},{"Start":"04:39.950 ","End":"04:42.395","Text":"so X equals to 0."},{"Start":"04:42.395 ","End":"04:44.330","Text":"In the same manner,"},{"Start":"04:44.330 ","End":"04:47.900","Text":"we\u0027ll take a look at variable Y."},{"Start":"04:47.900 ","End":"04:50.900","Text":"Now, Y is an indicator variable that receives the value"},{"Start":"04:50.900 ","End":"04:54.650","Text":"1 if the student passes the economics exam."},{"Start":"04:54.650 ","End":"04:57.660","Text":"A is passing economics,"},{"Start":"04:57.660 ","End":"05:00.210","Text":"so Y here that equals to 1."},{"Start":"05:00.210 ","End":"05:02.385","Text":"Now, in the intersection,"},{"Start":"05:02.385 ","End":"05:05.055","Text":"Y also equals to 1."},{"Start":"05:05.055 ","End":"05:07.065","Text":"Now, what happens here?"},{"Start":"05:07.065 ","End":"05:12.020","Text":"Well, here, outside the intersection on circle B, well,"},{"Start":"05:12.020 ","End":"05:19.010","Text":"Y equals 0 because the student passed the statistics exam and not the economics."},{"Start":"05:19.010 ","End":"05:26.930","Text":"Here Y equals 0 because he didn\u0027t pass the economics or the statistics exam."},{"Start":"05:27.440 ","End":"05:30.575","Text":"Let\u0027s take a look at probabilities."},{"Start":"05:30.575 ","End":"05:35.450","Text":"Well, what\u0027s the probability of X equaling 0 and Y equaling 0?"},{"Start":"05:35.450 ","End":"05:42.340","Text":"That means that he didn\u0027t pass the economics exam and he didn\u0027t pass the statistics exam."},{"Start":"05:42.340 ","End":"05:44.040","Text":"Well, that\u0027s right here."},{"Start":"05:44.040 ","End":"05:47.450","Text":"The probability of being outside of the circle,"},{"Start":"05:47.450 ","End":"05:50.765","Text":"so that\u0027s 0.05, that\u0027s right here."},{"Start":"05:50.765 ","End":"05:55.220","Text":"Now, what about the probability of X being equal to 0 and Y equaling 1?"},{"Start":"05:55.220 ","End":"06:03.960","Text":"Well, X equals 0 means that he didn\u0027t pass either of the exams,"},{"Start":"06:03.960 ","End":"06:05.630","Text":"not the economics exam,"},{"Start":"06:05.630 ","End":"06:07.385","Text":"not the statistics exam."},{"Start":"06:07.385 ","End":"06:11.565","Text":"Y equaling 1 means that he passed the economics exam?"},{"Start":"06:11.565 ","End":"06:16.055","Text":"Well, no. Then there\u0027s no such event."},{"Start":"06:16.055 ","End":"06:19.760","Text":"Now, what happens when X equals 1 and Y equals 0?"},{"Start":"06:19.760 ","End":"06:21.945","Text":"Well, that\u0027s right here."},{"Start":"06:21.945 ","End":"06:27.530","Text":"He passed the statistics exam and he didn\u0027t pass the economics exams,"},{"Start":"06:27.530 ","End":"06:31.640","Text":"so the probability is 0.15, that\u0027s right here."},{"Start":"06:31.640 ","End":"06:34.895","Text":"What about X equals 1 and Y equals 1?"},{"Start":"06:34.895 ","End":"06:37.250","Text":"Well, that\u0027s right here,"},{"Start":"06:37.250 ","End":"06:41.540","Text":"and the probability is 0.05 right here."},{"Start":"06:41.540 ","End":"06:45.695","Text":"Now, what happens if X equals 2?"},{"Start":"06:45.695 ","End":"06:50.938","Text":"That means that the student passed both economics and the statistics exam,"},{"Start":"06:50.938 ","End":"06:53.567","Text":"but he didn\u0027t pass the economics exam."},{"Start":"06:53.567 ","End":"06:55.345","Text":"Well, there\u0027s no such event,"},{"Start":"06:55.345 ","End":"06:57.465","Text":"so he probability is 0."},{"Start":"06:57.465 ","End":"07:03.350","Text":"Now what happens when X equals 2 and Y equals 1?"},{"Start":"07:03.350 ","End":"07:07.905","Text":"Well, the student passed both the economics and the statistics exam."},{"Start":"07:07.905 ","End":"07:09.575","Text":"Y equals 1, well,"},{"Start":"07:09.575 ","End":"07:13.160","Text":"that\u0027s the student passing the economics exam,"},{"Start":"07:13.160 ","End":"07:14.768","Text":"well, that\u0027s right here."},{"Start":"07:14.768 ","End":"07:19.590","Text":"That equals to 0.75, that\u0027s the probability."},{"Start":"07:19.950 ","End":"07:24.460","Text":"The joint probability distribution can be displayed as"},{"Start":"07:24.460 ","End":"07:29.440","Text":"a 2-dimensional table where we have the variables x and y,"},{"Start":"07:29.440 ","End":"07:33.200","Text":"and we have the values of x. X can take on the value of 0,"},{"Start":"07:33.200 ","End":"07:36.520","Text":"1, and 2, we saw that here or here."},{"Start":"07:36.520 ","End":"07:39.970","Text":"Now, Y can take on the value of 0 and 1,"},{"Start":"07:39.970 ","End":"07:42.445","Text":"that\u0027s an indicator variable."},{"Start":"07:42.445 ","End":"07:45.659","Text":"Again, we see that here and here,"},{"Start":"07:45.659 ","End":"07:48.175","Text":"what do we have within the table?"},{"Start":"07:48.175 ","End":"07:56.770","Text":"Well, that\u0027s the probability of x equaling 0 and y equaling 0."},{"Start":"07:56.770 ","End":"07:58.555","Text":"Now, what\u0027s that?"},{"Start":"07:58.555 ","End":"08:02.065","Text":"Well, we saw that here, that\u0027s 0.05."},{"Start":"08:02.065 ","End":"08:04.750","Text":"We also saw that here x equals 0,"},{"Start":"08:04.750 ","End":"08:07.825","Text":"y equals 0 when the probability is 0.05."},{"Start":"08:07.825 ","End":"08:13.015","Text":"Now, what happens where y equals 1 and x equals 2?"},{"Start":"08:13.015 ","End":"08:15.100","Text":"Well, that\u0027s right here."},{"Start":"08:15.100 ","End":"08:17.620","Text":"The probability is 0.75,"},{"Start":"08:17.620 ","End":"08:24.040","Text":"so we fill that in and so on and so forth for all the other values of x and y."},{"Start":"08:24.040 ","End":"08:30.009","Text":"This 2-dimensional table is called the joint probability distribution."},{"Start":"08:30.009 ","End":"08:36.130","Text":"Now, one thing that you have to look at is that the sum of all the probabilities in"},{"Start":"08:36.130 ","End":"08:39.850","Text":"the common probability function is equal to 1 so if we"},{"Start":"08:39.850 ","End":"08:44.305","Text":"sum up all the probabilities in these 6 cells,"},{"Start":"08:44.305 ","End":"08:47.065","Text":"they\u0027ll come up to 1."},{"Start":"08:47.065 ","End":"08:50.170","Text":"Another thing that we have to consider when we\u0027re talking"},{"Start":"08:50.170 ","End":"08:52.480","Text":"about joint probability functions of x and"},{"Start":"08:52.480 ","End":"08:58.630","Text":"y is the marginal probabilities of x and the marginal probability of y."},{"Start":"08:58.630 ","End":"09:00.625","Text":"Now, what are we talking about?"},{"Start":"09:00.625 ","End":"09:07.855","Text":"Well, here, we constructed the margin or the joint probability function of x and y."},{"Start":"09:07.855 ","End":"09:12.250","Text":"We had the x values and the y values and within the table"},{"Start":"09:12.250 ","End":"09:17.260","Text":"was a joint probabilities of all the combinations of x and y\u0027s."},{"Start":"09:17.260 ","End":"09:20.455","Text":"Now, what\u0027s a marginal probability?"},{"Start":"09:20.455 ","End":"09:24.730","Text":"Well, as we can see here in this table right here,"},{"Start":"09:24.730 ","End":"09:29.365","Text":"the marginal probability is when I sum up"},{"Start":"09:29.365 ","End":"09:35.380","Text":"either the rows or the columns for the values of x or y."},{"Start":"09:35.380 ","End":"09:42.655","Text":"For example, the marginal probability of y is this column right here."},{"Start":"09:42.655 ","End":"09:43.870","Text":"Now, what\u0027s this column?"},{"Start":"09:43.870 ","End":"09:46.209","Text":"What\u0027s 0.2 for example,"},{"Start":"09:46.209 ","End":"09:51.580","Text":"that\u0027s the probability of y being equal to 0."},{"Start":"09:51.580 ","End":"09:54.430","Text":"I add up all the values,"},{"Start":"09:54.430 ","End":"10:00.610","Text":"all the probabilities of x where y equals to 0."},{"Start":"10:00.610 ","End":"10:04.270","Text":"I add up 0.05 plus 0.15 plus 0."},{"Start":"10:04.270 ","End":"10:08.350","Text":"That equals to 0.2. What\u0027s the probability of y equaling 1?"},{"Start":"10:08.350 ","End":"10:13.465","Text":"Well, that\u0027s 0 plus 0.05 plus 0.75 that\u0027s 0.8."},{"Start":"10:13.465 ","End":"10:20.860","Text":"Likewise, this then is the marginal probability of x,"},{"Start":"10:20.860 ","End":"10:22.495","Text":"so what did I do here?"},{"Start":"10:22.495 ","End":"10:28.375","Text":"I added up all the probabilities along the values of y,"},{"Start":"10:28.375 ","End":"10:30.385","Text":"where x equals 0,"},{"Start":"10:30.385 ","End":"10:34.960","Text":"so the probability of x is 0.05 plus 0."},{"Start":"10:34.960 ","End":"10:37.945","Text":"The probability of x being equal to 1,"},{"Start":"10:37.945 ","End":"10:42.044","Text":"well that\u0027s 0.15 plus 0.05 and that\u0027s 0.2."},{"Start":"10:42.044 ","End":"10:44.770","Text":"The probability of x equaling 2,"},{"Start":"10:44.770 ","End":"10:49.525","Text":"well that\u0027s 0 plus 0.75 and that equals to 0.75."},{"Start":"10:49.525 ","End":"10:58.010","Text":"These then right here and right here are the marginal probabilities of x and y."},{"Start":"10:58.200 ","End":"11:02.710","Text":"The last subject that I want to talk about is that of independent variables."},{"Start":"11:02.710 ","End":"11:07.135","Text":"Now, we\u0027re not talking about independent events, but independent variables."},{"Start":"11:07.135 ","End":"11:10.420","Text":"X and Y are independent variables if"},{"Start":"11:10.420 ","End":"11:13.690","Text":"the following is true for all possible values of x and y,"},{"Start":"11:13.690 ","End":"11:16.525","Text":"for all possible combinations of x and y."},{"Start":"11:16.525 ","End":"11:21.925","Text":"That means that the probability of x equaling k and y equaling l,"},{"Start":"11:21.925 ","End":"11:24.340","Text":"some values of x and y,"},{"Start":"11:24.340 ","End":"11:27.820","Text":"well then has to be equal to the probability of x equaling k"},{"Start":"11:27.820 ","End":"11:32.260","Text":"times the probability of y equaling l. We\u0027re talking"},{"Start":"11:32.260 ","End":"11:36.640","Text":"about the joint probability being equal to"},{"Start":"11:36.640 ","End":"11:41.275","Text":"the multiplication of the marginal probabilities."},{"Start":"11:41.275 ","End":"11:43.870","Text":"Now, what\u0027s this joint probability here?"},{"Start":"11:43.870 ","End":"11:47.533","Text":"Well, that\u0027s one of the values within the table right here."},{"Start":"11:47.533 ","End":"11:49.840","Text":"What\u0027s the marginal probabilities?"},{"Start":"11:49.840 ","End":"11:54.640","Text":"Well, that\u0027s one of the values for x and one of the values for y."},{"Start":"11:54.640 ","End":"11:59.725","Text":"Now, if the variables failed to meet this condition even once,"},{"Start":"11:59.725 ","End":"12:03.040","Text":"then they\u0027re dependent variables and not independent variables."},{"Start":"12:03.040 ","End":"12:04.570","Text":"Now let us take an example."},{"Start":"12:04.570 ","End":"12:07.840","Text":"The probability of x equaling 2 and y equal 1."},{"Start":"12:07.840 ","End":"12:13.600","Text":"Well, here\u0027s x equals 2 and y equaling 1 so the joint probability,"},{"Start":"12:13.600 ","End":"12:17.575","Text":"that\u0027s a value within the table right here, that\u0027s 0.75."},{"Start":"12:17.575 ","End":"12:19.225","Text":"That\u0027s this cell right here."},{"Start":"12:19.225 ","End":"12:26.170","Text":"That\u0027s the intersection of the column x equaling 2 and the row y equaling 1."},{"Start":"12:26.170 ","End":"12:30.325","Text":"Now, that equals to 0.75 as we said."},{"Start":"12:30.325 ","End":"12:34.180","Text":"Now, on the other hand, that has to equal to"},{"Start":"12:34.180 ","End":"12:38.200","Text":"the marginal probabilities of x equaling 2 and y equaling 1."},{"Start":"12:38.200 ","End":"12:43.270","Text":"Now, the marginal probability of x equaling 2 that\u0027s 0.75,"},{"Start":"12:43.270 ","End":"12:47.140","Text":"and the marginal probability of y equaling 1, that\u0027s 0.8."},{"Start":"12:47.140 ","End":"12:52.675","Text":"Now the multiplication of 0.75 times 0.8 gives us 0.6,"},{"Start":"12:52.675 ","End":"12:56.455","Text":"and that doesn\u0027t equal 0.75."},{"Start":"12:56.455 ","End":"13:00.040","Text":"Whenever we do not have an equality here,"},{"Start":"13:00.040 ","End":"13:02.635","Text":"when one doesn\u0027t equal the other,"},{"Start":"13:02.635 ","End":"13:10.690","Text":"then we\u0027re talking about the fact that x and y are dependent variables."},{"Start":"13:10.690 ","End":"13:16.375","Text":"Now, if there\u0027s a 0 in the joint probability distribution,"},{"Start":"13:16.375 ","End":"13:18.790","Text":"then we can conclude that the variables are"},{"Start":"13:18.790 ","End":"13:21.925","Text":"dependent because the above condition isn\u0027t met."},{"Start":"13:21.925 ","End":"13:23.965","Text":"Now, what do I mean by that?"},{"Start":"13:23.965 ","End":"13:28.300","Text":"Well, here\u0027s a 0 that\u0027s one of"},{"Start":"13:28.300 ","End":"13:33.085","Text":"the joint probabilities here when y equals 1 and x equals 0."},{"Start":"13:33.085 ","End":"13:35.710","Text":"We had a joint probability of 0."},{"Start":"13:35.710 ","End":"13:39.760","Text":"Now, what are the marginal probabilities where x equals 0,"},{"Start":"13:39.760 ","End":"13:42.940","Text":"that\u0027s 0.05, and when y equals 1,"},{"Start":"13:42.940 ","End":"13:45.775","Text":"that\u0027s 0.08 so the multiplication of"},{"Start":"13:45.775 ","End":"13:51.955","Text":"2 positive numbers will never give me a product of 0."},{"Start":"13:51.955 ","End":"13:58.870","Text":"I can conclude that if I have 0 within the table in the joint probabilities,"},{"Start":"13:58.870 ","End":"14:03.505","Text":"then the 2 variables x and y are dependent."},{"Start":"14:03.505 ","End":"14:07.975","Text":"On the other hand, if there\u0027s no 0 on the table,"},{"Start":"14:07.975 ","End":"14:11.755","Text":"it doesn\u0027t necessarily mean that the variables are independent."},{"Start":"14:11.755 ","End":"14:16.450","Text":"This has to be verified for each of the values of X and Y."},{"Start":"14:16.450 ","End":"14:22.015","Text":"We\u0027ve talked a lot about very subject of joint distributions"},{"Start":"14:22.015 ","End":"14:28.000","Text":"and independent variables and so on and so forth and marginal probabilities."},{"Start":"14:28.000 ","End":"14:36.050","Text":"But there\u0027s nothing like a good exercise to let this information sink in. Here we go."}],"ID":13183},{"Watched":false,"Name":"Exercise 1 - Part a","Duration":"9m 48s","ChapterTopicVideoID":12705,"CourseChapterTopicPlaylistID":245018,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12705.jpeg","UploadDate":"2018-07-30T14:14:16.5370000","DurationForVideoObject":"PT9M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.350","Text":"In this question, a person enters a casino with with 75 dollars in his pocket."},{"Start":"00:04.350 ","End":"00:08.295","Text":"He plays on a slot machine where his chances of winning are 0.3."},{"Start":"00:08.295 ","End":"00:09.540","Text":"If he wins the game,"},{"Start":"00:09.540 ","End":"00:12.360","Text":"the casino gives him 25 dollars and if he loses,"},{"Start":"00:12.360 ","End":"00:14.670","Text":"he pays the casino 25 dollars."},{"Start":"00:14.670 ","End":"00:18.990","Text":"The same person decides to stop playing the game if he has 100 dollars,"},{"Start":"00:18.990 ","End":"00:22.185","Text":"but in any case, he doesn\u0027t play more than 3 games."},{"Start":"00:22.185 ","End":"00:26.130","Text":"That we define x as the money he has when he leaves the casino,"},{"Start":"00:26.130 ","End":"00:28.750","Text":"and y as the number of games he plays."},{"Start":"00:28.750 ","End":"00:34.200","Text":"We\u0027re asked to construct the joint and marginal probability functions."},{"Start":"00:34.200 ","End":"00:41.130","Text":"The first thing that we want to do in these types of questions is to describe the data,"},{"Start":"00:41.130 ","End":"00:46.610","Text":"and the best way to describe this data right here is through a probability tree."},{"Start":"00:46.610 ","End":"00:49.399","Text":"Why is that? Well, we have stages,"},{"Start":"00:49.399 ","End":"00:52.070","Text":"a person plays the first game if he wins he stops,"},{"Start":"00:52.070 ","End":"00:54.230","Text":"if he doesn\u0027t go down to the second game,"},{"Start":"00:54.230 ","End":"00:57.830","Text":"if he gets to 100 dollars, then he stops."},{"Start":"00:57.830 ","End":"00:59.825","Text":"If he doesn\u0027t goes onto the third game,"},{"Start":"00:59.825 ","End":"01:01.775","Text":"and so on and so forth."},{"Start":"01:01.775 ","End":"01:06.030","Text":"Let\u0027s construct a probability tree."},{"Start":"01:06.380 ","End":"01:10.190","Text":"Let\u0027s start constructing a probability tree."},{"Start":"01:10.190 ","End":"01:12.319","Text":"Now, when he enters the casino,"},{"Start":"01:12.319 ","End":"01:15.410","Text":"he has 75 dollars in his pocket and he"},{"Start":"01:15.410 ","End":"01:18.545","Text":"goes to a slot machine and he plays the first game."},{"Start":"01:18.545 ","End":"01:20.660","Text":"Now y is the number of games,"},{"Start":"01:20.660 ","End":"01:23.715","Text":"so Y here has to be equal to 1."},{"Start":"01:23.715 ","End":"01:26.190","Text":"This is game number 1."},{"Start":"01:26.190 ","End":"01:28.400","Text":"Now, in game number 1,"},{"Start":"01:28.400 ","End":"01:31.160","Text":"he has a 30 percent chance of winning."},{"Start":"01:31.160 ","End":"01:34.115","Text":"That means that he has a 70 percent chance of losing."},{"Start":"01:34.115 ","End":"01:37.310","Text":"Now, if he wins, he wins 25 dollars."},{"Start":"01:37.310 ","End":"01:39.835","Text":"He has 100 dollars here."},{"Start":"01:39.835 ","End":"01:43.575","Text":"Now, if he has 100 dollars he stops,"},{"Start":"01:43.575 ","End":"01:45.410","Text":"but if he loses, well,"},{"Start":"01:45.410 ","End":"01:50.045","Text":"he loses 25 dollars and now he has 50 dollars in his pocket."},{"Start":"01:50.045 ","End":"01:54.650","Text":"If he has $50 in his pocket then he goes into game number 2."},{"Start":"01:54.650 ","End":"01:56.770","Text":"So y here equals 2."},{"Start":"01:56.770 ","End":"01:59.270","Text":"Now in game number 2, again,"},{"Start":"01:59.270 ","End":"02:04.250","Text":"he has a 30 percent chance of winning and a 70 percent chance of losing."},{"Start":"02:04.250 ","End":"02:05.940","Text":"Now, if he wins, well,"},{"Start":"02:05.940 ","End":"02:07.500","Text":"he wins 25 dollars."},{"Start":"02:07.500 ","End":"02:11.310","Text":"Now he has 75 dollars in his pocket and if he loses,"},{"Start":"02:11.310 ","End":"02:13.170","Text":"so when he loses 25 dollars,"},{"Start":"02:13.170 ","End":"02:17.050","Text":"so he has now 25 dollars in his pocket."},{"Start":"02:17.420 ","End":"02:21.030","Text":"Here he didn\u0027t get to 100 dollars,"},{"Start":"02:21.030 ","End":"02:25.640","Text":"so he has to go to the third and final game."},{"Start":"02:25.640 ","End":"02:29.600","Text":"Now, here, you can either win at"},{"Start":"02:29.600 ","End":"02:35.840","Text":"0.3 with a probability of 0.3 or lose with a probability of 0.7 or if he wins,"},{"Start":"02:35.840 ","End":"02:39.350","Text":"well, he wins 100 dollars and if he loses, well,"},{"Start":"02:39.350 ","End":"02:42.330","Text":"he loses 25 dollars,"},{"Start":"02:42.330 ","End":"02:45.185","Text":"so he has 50 dollars in his pocket."},{"Start":"02:45.185 ","End":"02:49.130","Text":"Here again, that\u0027s 0.3 and 0.7."},{"Start":"02:49.130 ","End":"02:52.250","Text":"If he wins then he has 50 dollars and if he loses,"},{"Start":"02:52.250 ","End":"02:55.435","Text":"then he doesn\u0027t have any money in his pocket."},{"Start":"02:55.435 ","End":"02:57.773","Text":"After the third game he quits,"},{"Start":"02:57.773 ","End":"02:59.840","Text":"so this then is the probability tree."},{"Start":"02:59.840 ","End":"03:03.860","Text":"Now let\u0027s see about the values of x and y."},{"Start":"03:03.860 ","End":"03:06.840","Text":"Well, for y, we have 1, 2,"},{"Start":"03:06.840 ","End":"03:09.850","Text":"and 3 for now, but what about the values of x?"},{"Start":"03:09.850 ","End":"03:16.550","Text":"Well, the values of x are determined by the end result of all the branches."},{"Start":"03:16.550 ","End":"03:19.300","Text":"This, for example,"},{"Start":"03:19.300 ","End":"03:27.110","Text":"is not a valid value for x because here he still has to continue 1,"},{"Start":"03:27.110 ","End":"03:31.115","Text":"so the values of x or this guy right here,"},{"Start":"03:31.115 ","End":"03:33.560","Text":"this guy right here."},{"Start":"03:33.560 ","End":"03:38.735","Text":"Same thing for this guy here and here,"},{"Start":"03:38.735 ","End":"03:46.895","Text":"so that means that x can have the values of 0, 50, and 100."},{"Start":"03:46.895 ","End":"03:48.425","Text":"Now what about y?"},{"Start":"03:48.425 ","End":"03:50.895","Text":"Well, he stops here."},{"Start":"03:50.895 ","End":"03:52.790","Text":"That\u0027s one of the branches where it stops."},{"Start":"03:52.790 ","End":"03:54.750","Text":"So y is 1,"},{"Start":"03:54.750 ","End":"03:57.405","Text":"1 is the legitimate value for y."},{"Start":"03:57.405 ","End":"03:59.250","Text":"Now, what about y equals 2?"},{"Start":"03:59.250 ","End":"04:01.440","Text":"Well, he didn\u0027t stop here at all."},{"Start":"04:01.440 ","End":"04:05.700","Text":"Y equaling 2 is not a valid value."},{"Start":"04:05.700 ","End":"04:10.845","Text":"Let\u0027s go to y equals 3 and here he stops 4 times."},{"Start":"04:10.845 ","End":"04:16.160","Text":"The valid values for y are 1 and 3 and for x are 0,"},{"Start":"04:16.160 ","End":"04:18.845","Text":"50, and 100. Great now."},{"Start":"04:18.845 ","End":"04:23.195","Text":"Now that we know what the values of x and y are,"},{"Start":"04:23.195 ","End":"04:31.310","Text":"and we can calculate the probabilities for each combination of x and y."},{"Start":"04:31.310 ","End":"04:36.630","Text":"Well, let\u0027s build our joint and marginal probability function."},{"Start":"04:36.970 ","End":"04:39.590","Text":"This is a table then,"},{"Start":"04:39.590 ","End":"04:45.880","Text":"but we\u0027ll input the probability values,"},{"Start":"04:45.880 ","End":"04:48.690","Text":"the joint probabilities of x and y here within"},{"Start":"04:48.690 ","End":"04:53.855","Text":"these 6 white cells and the marginal probabilities of y right here,"},{"Start":"04:53.855 ","End":"04:55.360","Text":"these 2 horn cells,"},{"Start":"04:55.360 ","End":"05:00.535","Text":"or the marginal probabilities of x in these 3 blue cells."},{"Start":"05:00.535 ","End":"05:06.430","Text":"Now, what are the combinations or the valid combinations of x and y?"},{"Start":"05:06.430 ","End":"05:10.060","Text":"Well, the first thing that I see here is this value of x right here,"},{"Start":"05:10.060 ","End":"05:12.940","Text":"where x equals 100 and y equals 1."},{"Start":"05:12.940 ","End":"05:19.560","Text":"So the probability of x equaling 100 and y equaling 1,"},{"Start":"05:19.560 ","End":"05:21.255","Text":"well, what\u0027s that probability?"},{"Start":"05:21.255 ","End":"05:24.460","Text":"That\u0027s 0.3, that\u0027s right here."},{"Start":"05:25.150 ","End":"05:29.925","Text":"Let\u0027s put that into the table,"},{"Start":"05:29.925 ","End":"05:32.805","Text":"y equals 1, that\u0027s this row right here,"},{"Start":"05:32.805 ","End":"05:34.170","Text":"x equals 100,"},{"Start":"05:34.170 ","End":"05:35.445","Text":"that\u0027s this column right here."},{"Start":"05:35.445 ","End":"05:37.060","Text":"The intersection is a cell."},{"Start":"05:37.060 ","End":"05:39.950","Text":"We\u0027ll put 0.3 here."},{"Start":"05:39.950 ","End":"05:43.950","Text":"Now, what about other values of x and y?"},{"Start":"05:43.950 ","End":"05:45.030","Text":"Well, y equals 2,"},{"Start":"05:45.030 ","End":"05:47.595","Text":"we said isn\u0027t a valid value."},{"Start":"05:47.595 ","End":"05:49.700","Text":"We\u0027ll go to y equals to 3."},{"Start":"05:49.700 ","End":"05:55.010","Text":"Now, there are 3 valid values of x for y equaling 3,"},{"Start":"05:55.010 ","End":"05:57.480","Text":"0, 50, and 100."},{"Start":"05:57.480 ","End":"06:01.020","Text":"Let\u0027s first of all take x being equal to 100,"},{"Start":"06:01.020 ","End":"06:02.910","Text":"x equals 100,"},{"Start":"06:02.910 ","End":"06:05.750","Text":"and y has to be equal to 3."},{"Start":"06:05.750 ","End":"06:07.430","Text":"Well, that equals to what?"},{"Start":"06:07.430 ","End":"06:11.060","Text":"Well, the person had to lose on"},{"Start":"06:11.060 ","End":"06:15.350","Text":"the first game and then win on the second game and win on the third game,"},{"Start":"06:15.350 ","End":"06:17.975","Text":"so the probabilities are 0.7,"},{"Start":"06:17.975 ","End":"06:19.370","Text":"that\u0027s this thing right here,"},{"Start":"06:19.370 ","End":"06:22.670","Text":"times 0.3 and again times 0.3."},{"Start":"06:22.670 ","End":"06:26.090","Text":"So times 0.3 times 0.3,"},{"Start":"06:26.090 ","End":"06:30.570","Text":"and that equals to 0.063."},{"Start":"06:30.570 ","End":"06:32.880","Text":"Let\u0027s just write that out here."},{"Start":"06:32.880 ","End":"06:37.680","Text":"That\u0027ll be x equaling 100 and y equaling 3,"},{"Start":"06:37.680 ","End":"06:43.320","Text":"so here it\u0027ll be 0.063."},{"Start":"06:43.320 ","End":"06:51.265","Text":"Now, what about the probability of x equaling 0 and y equaling 3?"},{"Start":"06:51.265 ","End":"06:54.145","Text":"Well, that\u0027s this branch right here,"},{"Start":"06:54.145 ","End":"06:55.810","Text":"he has to lose the first game,"},{"Start":"06:55.810 ","End":"06:57.130","Text":"lose on the second game,"},{"Start":"06:57.130 ","End":"06:58.870","Text":"and lose on the third game."},{"Start":"06:58.870 ","End":"07:03.320","Text":"That\u0027s 0.7 times 0.7 times"},{"Start":"07:03.320 ","End":"07:09.580","Text":"0.7 and that equals to 0.343."},{"Start":"07:09.580 ","End":"07:14.020","Text":"We\u0027ll put that here, 0.343."},{"Start":"07:14.020 ","End":"07:20.460","Text":"Now, what about the third cell right here?"},{"Start":"07:20.460 ","End":"07:27.960","Text":"That\u0027s the probability of x equaling 50 and y equaling 3."},{"Start":"07:27.960 ","End":"07:30.765","Text":"Well, we have 2 branches here."},{"Start":"07:30.765 ","End":"07:32.920","Text":"The first branch is this."},{"Start":"07:32.920 ","End":"07:34.900","Text":"He loses on his first game,"},{"Start":"07:34.900 ","End":"07:39.060","Text":"wins on a second game and loses on his third game."},{"Start":"07:39.060 ","End":"07:44.740","Text":"That\u0027s 0.7 times 0.3 times 0.7."},{"Start":"07:44.740 ","End":"07:46.180","Text":"Now we have another branch,"},{"Start":"07:46.180 ","End":"07:49.070","Text":"so we have to add this branch right here."},{"Start":"07:49.070 ","End":"07:50.770","Text":"He loses on the first game,"},{"Start":"07:50.770 ","End":"07:52.495","Text":"loses on a second game,"},{"Start":"07:52.495 ","End":"07:54.310","Text":"and wins on his third game."},{"Start":"07:54.310 ","End":"08:00.520","Text":"That means that 0.7 times 0.7 times 0.3,"},{"Start":"08:00.520 ","End":"08:05.840","Text":"and that comes out to 0.294."},{"Start":"08:05.840 ","End":"08:11.030","Text":"Let\u0027s put that in here, 0.294."},{"Start":"08:11.030 ","End":"08:18.015","Text":"Now, what about these 2 cells here where y equals 1 and x equals 0 or 50?"},{"Start":"08:18.015 ","End":"08:21.740","Text":"Well, we don\u0027t have this combination right here where y equals 1."},{"Start":"08:21.740 ","End":"08:23.180","Text":"That\u0027s the first game."},{"Start":"08:23.180 ","End":"08:26.570","Text":"We don\u0027t have x equaling 50."},{"Start":"08:26.570 ","End":"08:28.610","Text":"We can\u0027t use this thing right here,"},{"Start":"08:28.610 ","End":"08:34.270","Text":"because we have to continue until the end of this branch right here."},{"Start":"08:34.270 ","End":"08:38.175","Text":"The values here are 0 and 0."},{"Start":"08:38.175 ","End":"08:40.225","Text":"If that\u0027s the case,"},{"Start":"08:40.225 ","End":"08:46.460","Text":"let\u0027s calculate the valid marginal probabilities of x and y."},{"Start":"08:46.460 ","End":"08:49.850","Text":"Well, the marginal probability of y,"},{"Start":"08:49.850 ","End":"08:51.215","Text":"where y equals 1,"},{"Start":"08:51.215 ","End":"08:56.615","Text":"well that\u0027s 0 plus 0 plus 0.3, and that\u0027s 0.3."},{"Start":"08:56.615 ","End":"09:02.050","Text":"Here, the marginal probability of y,"},{"Start":"09:02.050 ","End":"09:04.600","Text":"where y equals to 3, well,"},{"Start":"09:04.600 ","End":"09:09.875","Text":"that\u0027s the sum of these values and that comes out to 0.7."},{"Start":"09:09.875 ","End":"09:12.420","Text":"Now, what about for x?"},{"Start":"09:12.420 ","End":"09:16.260","Text":"Well, that\u0027ll be 0.343."},{"Start":"09:16.260 ","End":"09:23.070","Text":"That\u0027ll be 0.294, and this will be 0.363."},{"Start":"09:23.070 ","End":"09:28.640","Text":"Now, if we add up the marginal probabilities of y, we\u0027ll get 1."},{"Start":"09:28.640 ","End":"09:32.110","Text":"If we add up the marginal probabilities of x,"},{"Start":"09:32.110 ","End":"09:37.325","Text":"then we\u0027ll get 1 and if we add up the joint probabilities of x and y,"},{"Start":"09:37.325 ","End":"09:40.197","Text":"the 6 white cells we\u0027ll also get 1,"},{"Start":"09:40.197 ","End":"09:42.795","Text":"so everything is fine here."},{"Start":"09:42.795 ","End":"09:48.040","Text":"This is then is the answer for this section."}],"ID":13184},{"Watched":false,"Name":"Exercise 1 - Parts - b-c","Duration":"6m 25s","ChapterTopicVideoID":12706,"CourseChapterTopicPlaylistID":245018,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12706.jpeg","UploadDate":"2018-07-30T14:14:34.0470000","DurationForVideoObject":"PT6M25S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.760","Text":"In this section, we\u0027re asked what\u0027s the expectation"},{"Start":"00:02.760 ","End":"00:06.270","Text":"of the number of games that the gambler played?"},{"Start":"00:06.270 ","End":"00:09.240","Text":"The number of games, that\u0027s Y,"},{"Start":"00:09.240 ","End":"00:10.905","Text":"so basically we\u0027re asked,"},{"Start":"00:10.905 ","End":"00:13.770","Text":"what\u0027s the expectation of Y?"},{"Start":"00:13.770 ","End":"00:15.750","Text":"The expectation of Y,"},{"Start":"00:15.750 ","End":"00:23.835","Text":"that\u0027s the sum of Y times the probability of Y."},{"Start":"00:23.835 ","End":"00:33.675","Text":"That equals to 1 times 0.3 plus 3 times 0.7,"},{"Start":"00:33.675 ","End":"00:37.905","Text":"and that comes out to 2.4,"},{"Start":"00:37.905 ","End":"00:44.550","Text":"so on average, the gambler plays 2.4 games."},{"Start":"00:44.710 ","End":"00:50.150","Text":"In this section, we\u0027re asked that if the gambler leaves a casino with $100,"},{"Start":"00:50.150 ","End":"00:55.265","Text":"what are the expectation and variance of the number of games that he played?"},{"Start":"00:55.265 ","End":"00:59.150","Text":"There are several stages to answer this question."},{"Start":"00:59.150 ","End":"01:08.110","Text":"The first stage is to calculate the conditional probability of Y given that X equals 100."},{"Start":"01:08.110 ","End":"01:10.745","Text":"That means that we want to calculate"},{"Start":"01:10.745 ","End":"01:15.245","Text":"the conditional probability of Y equaling 1 when X equals 100,"},{"Start":"01:15.245 ","End":"01:19.370","Text":"and what the probability of Y equaling 3 when X equals 100,"},{"Start":"01:19.370 ","End":"01:22.780","Text":"so that\u0027s the first thing that we need to do, so let\u0027s do that."},{"Start":"01:22.780 ","End":"01:30.225","Text":"The probability of Y equaling 1 given that X equals 100,"},{"Start":"01:30.225 ","End":"01:32.070","Text":"but that\u0027s a conditional probability,"},{"Start":"01:32.070 ","End":"01:33.555","Text":"so we know how to solve that."},{"Start":"01:33.555 ","End":"01:38.720","Text":"In the numerator that\u0027ll be the intersection of these 2 guys and into the denominator,"},{"Start":"01:38.720 ","End":"01:41.390","Text":"that\u0027ll be the probability of what\u0027s known,"},{"Start":"01:41.390 ","End":"01:49.035","Text":"so that\u0027s the probability of Y equaling 1 intersect X equaling 100."},{"Start":"01:49.035 ","End":"01:51.480","Text":"In the denominator,"},{"Start":"01:51.480 ","End":"01:55.490","Text":"that\u0027s the probability of X equaling 100."},{"Start":"01:55.490 ","End":"02:00.585","Text":"Now, the probability Y equaling 1 and X equaling 100."},{"Start":"02:00.585 ","End":"02:01.940","Text":"That\u0027s this row right here,"},{"Start":"02:01.940 ","End":"02:03.140","Text":"in this column right here,"},{"Start":"02:03.140 ","End":"02:04.846","Text":"so the intercept is 0.3,"},{"Start":"02:04.846 ","End":"02:08.300","Text":"so that would be 0.3 divided by,"},{"Start":"02:08.300 ","End":"02:11.395","Text":"what\u0027s the probability of X equaling 100?"},{"Start":"02:11.395 ","End":"02:13.320","Text":"That\u0027s this guy right here."},{"Start":"02:13.320 ","End":"02:22.450","Text":"That\u0027s 0.363, so that equals to 0.826."},{"Start":"02:24.800 ","End":"02:32.340","Text":"This is the probability of Y equaling 1 would given that X equals 100,"},{"Start":"02:32.340 ","End":"02:34.785","Text":"what about when Y equals 3?"},{"Start":"02:34.785 ","End":"02:36.450","Text":"Let\u0027s calculate that."},{"Start":"02:36.450 ","End":"02:40.360","Text":"As we\u0027re looking for the probability of Y equaling 3,"},{"Start":"02:40.360 ","End":"02:44.205","Text":"given that X equals 100."},{"Start":"02:44.205 ","End":"02:46.130","Text":"So let\u0027s do the same thing here."},{"Start":"02:46.130 ","End":"02:52.405","Text":"That\u0027s the probability of Y equaling 3 intersect X"},{"Start":"02:52.405 ","End":"02:59.790","Text":"equals 100 divided by the probability of X equaling 100."},{"Start":"02:59.960 ","End":"03:04.305","Text":"The probability Y equaling 3 and X equaling 100,"},{"Start":"03:04.305 ","End":"03:06.735","Text":"Y equals 3, that\u0027s this right here,"},{"Start":"03:06.735 ","End":"03:09.195","Text":"and 100 that\u0027s this column right here."},{"Start":"03:09.195 ","End":"03:12.740","Text":"The intercept is 0.063."},{"Start":"03:12.740 ","End":"03:14.195","Text":"That\u0027s this guy right here,"},{"Start":"03:14.195 ","End":"03:18.860","Text":"divided by the probability of X equaling 100, that\u0027s this guy,"},{"Start":"03:18.860 ","End":"03:27.880","Text":"that\u0027s 0.363 and that equals to 0.174."},{"Start":"03:27.880 ","End":"03:31.575","Text":"We\u0027re not done yet, this is just the first stage."},{"Start":"03:31.575 ","End":"03:40.320","Text":"Let\u0027s now write the probability distribution of Y given X equals 100."},{"Start":"03:40.850 ","End":"03:49.965","Text":"We\u0027re looking for Y given X equaling 100 on one hand,"},{"Start":"03:49.965 ","End":"03:57.114","Text":"and on the other hand, we\u0027re looking for the probability of Y given that X equals 100."},{"Start":"03:57.114 ","End":"04:07.540","Text":"Now, the probability of Y given X equals 100 can have the values of 1 or 3."},{"Start":"04:07.540 ","End":"04:10.910","Text":"Now, what\u0027s that probability right here?"},{"Start":"04:10.910 ","End":"04:15.485","Text":"For the probability of Y equaling 1 when X equals 100,"},{"Start":"04:15.485 ","End":"04:16.700","Text":"that\u0027s this guy right here,"},{"Start":"04:16.700 ","End":"04:21.785","Text":"that\u0027s 0.826, and when Y equals 3,"},{"Start":"04:21.785 ","End":"04:26.015","Text":"that conditional probability is 0.174."},{"Start":"04:26.015 ","End":"04:27.785","Text":"Let\u0027s do a quick check."},{"Start":"04:27.785 ","End":"04:32.540","Text":"The sum of the probabilities have to equal 1, and they do."},{"Start":"04:32.540 ","End":"04:39.200","Text":"Now that we have this probability distribution, what do we ask?"},{"Start":"04:39.200 ","End":"04:44.560","Text":"We\u0027re asked for the expectation of Y and the variance of Y."},{"Start":"04:44.560 ","End":"04:46.840","Text":"Let\u0027s calculate that."},{"Start":"04:46.840 ","End":"04:53.545","Text":"The expectation of Y given X equals 100."},{"Start":"04:53.545 ","End":"05:04.680","Text":"That equals to the sum of Y times the probability of Y given X equals 100."},{"Start":"05:04.680 ","End":"05:08.224","Text":"That equals to 1 times"},{"Start":"05:08.224 ","End":"05:17.385","Text":"0.826 plus 3 times 0.174."},{"Start":"05:17.385 ","End":"05:22.410","Text":"Now that equals to 1.347,"},{"Start":"05:22.410 ","End":"05:26.150","Text":"so this is the expectation of the number of games"},{"Start":"05:26.150 ","End":"05:31.875","Text":"played given that the gambler won $100,"},{"Start":"05:31.875 ","End":"05:33.890","Text":"so that\u0027s right here."},{"Start":"05:33.890 ","End":"05:35.990","Text":"Now, what about the variance?"},{"Start":"05:35.990 ","End":"05:41.445","Text":"When the variance of Y given X equaling 100,"},{"Start":"05:41.445 ","End":"05:46.355","Text":"that equals to the expectation of X squared,"},{"Start":"05:46.355 ","End":"05:52.080","Text":"minus the expectation squared of X or Y in our case."},{"Start":"05:52.080 ","End":"05:54.060","Text":"We\u0027ll have just have to sum that up."},{"Start":"05:54.060 ","End":"06:00.090","Text":"That will be 1 squared times 0.826 plus 3 squared"},{"Start":"06:00.090 ","End":"06:07.305","Text":"times 0.174 minus 1.347 squared,"},{"Start":"06:07.305 ","End":"06:15.075","Text":"and that comes out to 0.575 proximately."},{"Start":"06:15.075 ","End":"06:17.555","Text":"This then is the variance,"},{"Start":"06:17.555 ","End":"06:25.350","Text":"and this is the expectation of Y given that X equals 100."}],"ID":13185},{"Watched":false,"Name":"Exercise 2","Duration":"4m 3s","ChapterTopicVideoID":12707,"CourseChapterTopicPlaylistID":245018,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12707.jpeg","UploadDate":"2018-07-30T14:14:45.9900000","DurationForVideoObject":"PT4M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.770","Text":"In this question, we\u0027re given"},{"Start":"00:01.770 ","End":"00:03.930","Text":"the joint and marginal probability function with"},{"Start":"00:03.930 ","End":"00:06.735","Text":"2 discrete variables in this table right here,"},{"Start":"00:06.735 ","End":"00:09.015","Text":"with probabilities that are missing."},{"Start":"00:09.015 ","End":"00:13.310","Text":"We\u0027re asked first to complete the missing probabilities in the table."},{"Start":"00:13.310 ","End":"00:15.480","Text":"Let\u0027s just quickly do that."},{"Start":"00:15.480 ","End":"00:20.595","Text":"We know that the sum of the probabilities have to add up to 1 right here,"},{"Start":"00:20.595 ","End":"00:24.150","Text":"so if that\u0027s the case, here that\u0027ll be 0.15."},{"Start":"00:24.150 ","End":"00:28.170","Text":"This here would be 0.4,"},{"Start":"00:28.170 ","End":"00:32.170","Text":"this here would be 0.27."},{"Start":"00:34.150 ","End":"00:40.915","Text":"Here we\u0027ll have 0.2 that means it\u0027s 0.1."},{"Start":"00:40.915 ","End":"00:46.961","Text":"That means that here that\u0027s 0 and here that\u0027ll be 0.08,"},{"Start":"00:46.961 ","End":"00:54.485","Text":"so that\u0027s the 4 joint and marginal probability function of x and y."},{"Start":"00:54.485 ","End":"00:58.775","Text":"Next, we\u0027re asked whether x and y are dependent variables."},{"Start":"00:58.775 ","End":"01:00.500","Text":"In order to answer that,"},{"Start":"01:00.500 ","End":"01:04.745","Text":"let\u0027s see whether they meet the criteria for independence."},{"Start":"01:04.745 ","End":"01:07.475","Text":"Now, what\u0027s the criteria for independence?"},{"Start":"01:07.475 ","End":"01:12.380","Text":"We know that if we have the joint probability of x and y,"},{"Start":"01:12.380 ","End":"01:16.130","Text":"that has to be equal to the multiplication of"},{"Start":"01:16.130 ","End":"01:23.010","Text":"the marginal probabilities of x times the marginal probability of y."},{"Start":"01:23.010 ","End":"01:26.975","Text":"Now, if this expression is true,"},{"Start":"01:26.975 ","End":"01:29.866","Text":"then x and y are independent,"},{"Start":"01:29.866 ","End":"01:35.800","Text":"so the best way to see whether this comes true or not is to look for a 0 value,"},{"Start":"01:35.800 ","End":"01:38.115","Text":"in the joint probability."},{"Start":"01:38.115 ","End":"01:42.525","Text":"That\u0027s the cells within the table right here."},{"Start":"01:42.525 ","End":"01:45.185","Text":"We have a 0 value right here,"},{"Start":"01:45.185 ","End":"01:49.330","Text":"where x equals 2 and y equals 3."},{"Start":"01:49.330 ","End":"01:51.590","Text":"We have 0 on 1 hand,"},{"Start":"01:51.590 ","End":"01:53.420","Text":"that\u0027s a joint probability here."},{"Start":"01:53.420 ","End":"01:56.210","Text":"Now what\u0027s the marginal probability of x and y?"},{"Start":"01:56.210 ","End":"01:58.300","Text":"Well, when x equals 2,"},{"Start":"01:58.300 ","End":"02:03.740","Text":"the marginal probability of x is 0.2."},{"Start":"02:03.740 ","End":"02:10.640","Text":"When y equals 3 the marginal probability y, that\u0027s 0.15."},{"Start":"02:10.640 ","End":"02:14.359","Text":"We have here 2 positive integers,"},{"Start":"02:14.359 ","End":"02:16.490","Text":"and when multiplied together,"},{"Start":"02:16.490 ","End":"02:20.315","Text":"that will never give us a product of 0."},{"Start":"02:20.315 ","End":"02:26.040","Text":"That means that x and y are dependent,"},{"Start":"02:26.840 ","End":"02:29.805","Text":"and the answer are x and y dependent?"},{"Start":"02:29.805 ","End":"02:33.140","Text":"Oh yes they are. In this section,"},{"Start":"02:33.140 ","End":"02:38.525","Text":"we\u0027re asked to find the probability that y equals 3 if it\u0027s known that x equals 1."},{"Start":"02:38.525 ","End":"02:45.635","Text":"We\u0027re looking for the conditional probability of y equaling 3,"},{"Start":"02:45.635 ","End":"02:48.710","Text":"given that x equals to 1."},{"Start":"02:48.710 ","End":"02:51.995","Text":"Now we know how to solve this condition probability."},{"Start":"02:51.995 ","End":"02:55.490","Text":"In the numerator we have the intersect of these guys,"},{"Start":"02:55.490 ","End":"02:59.485","Text":"and in the denominator we have the probability what\u0027s given."},{"Start":"02:59.485 ","End":"03:01.070","Text":"Let\u0027s just write that down."},{"Start":"03:01.070 ","End":"03:04.820","Text":"That\u0027s the probability of y equaling 3,"},{"Start":"03:04.820 ","End":"03:10.985","Text":"intersect x equaling 1"},{"Start":"03:10.985 ","End":"03:16.295","Text":"divided by the probability of x equaling 1."},{"Start":"03:16.295 ","End":"03:20.705","Text":"What\u0027s the probability of x equals 1?"},{"Start":"03:20.705 ","End":"03:22.645","Text":"Well, that\u0027s right here."},{"Start":"03:22.645 ","End":"03:25.945","Text":"That\u0027s 0.4, when x equals 1,"},{"Start":"03:25.945 ","End":"03:29.135","Text":"then the probability of that is 0.4."},{"Start":"03:29.135 ","End":"03:34.400","Text":"Now what about the intersecting probability where y equals 3 and x equals 1?"},{"Start":"03:34.400 ","End":"03:38.240","Text":"Well, this is the column x equals 1,"},{"Start":"03:38.240 ","End":"03:40.520","Text":"and this is a row y equals 3."},{"Start":"03:40.520 ","End":"03:45.900","Text":"The intersecting cell right here, that\u0027s 0.05."},{"Start":"03:46.090 ","End":"03:54.360","Text":"That means that the ratio here comes out to 0.125,"},{"Start":"03:54.360 ","End":"04:03.150","Text":"so this then is the conditional probability of y equaling 3 given that x equals 1."}],"ID":13186},{"Watched":false,"Name":"Exercise 3 - Parts a-c","Duration":"5m 30s","ChapterTopicVideoID":12709,"CourseChapterTopicPlaylistID":245018,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12709.jpeg","UploadDate":"2018-07-30T14:15:56.6370000","DurationForVideoObject":"PT5M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.240","Text":"In this question, we have a factory that markets a product that is"},{"Start":"00:03.240 ","End":"00:07.110","Text":"packed in packages containing 2 products or 3 products."},{"Start":"00:07.110 ","End":"00:09.780","Text":"The number of packages containing 2 products"},{"Start":"00:09.780 ","End":"00:13.065","Text":"equals the number of packages containing 3 products."},{"Start":"00:13.065 ","End":"00:16.935","Text":"The probability that a product is faulty is 1/10."},{"Start":"00:16.935 ","End":"00:21.540","Text":"The production engineer randomly selects a package for quality assurance."},{"Start":"00:21.540 ","End":"00:24.090","Text":"Let X be the number of products in"},{"Start":"00:24.090 ","End":"00:28.560","Text":"a package and Y the number of faulty products in a package."},{"Start":"00:28.560 ","End":"00:35.140","Text":"We\u0027re asked what\u0027s the probability distribution of Y given that X equals 3?"},{"Start":"00:35.270 ","End":"00:37.590","Text":"We have this,"},{"Start":"00:37.590 ","End":"00:40.755","Text":"Y given X equals 3,"},{"Start":"00:40.755 ","End":"00:45.589","Text":"and we want to know what the probability distribution of this expression."},{"Start":"00:45.589 ","End":"00:47.600","Text":"Now, what do we have?"},{"Start":"00:47.600 ","End":"00:49.020","Text":"Well, what is Y?"},{"Start":"00:49.020 ","End":"00:54.160","Text":"Y is the number of faulty products in a package and what\u0027s X equaling 3?"},{"Start":"00:54.160 ","End":"01:01.835","Text":"Well, that equals to the number of products in a package that we chose."},{"Start":"01:01.835 ","End":"01:05.120","Text":"Again, we\u0027re interested in the number of"},{"Start":"01:05.120 ","End":"01:09.410","Text":"faulty products in this specific package right here."},{"Start":"01:09.410 ","End":"01:14.810","Text":"The number of packages containing 2 products equals the number of packages containing"},{"Start":"01:14.810 ","End":"01:20.345","Text":"3 products and we can assume independence here all over the place,"},{"Start":"01:20.345 ","End":"01:24.560","Text":"so in essence, we\u0027re interested again in"},{"Start":"01:24.560 ","End":"01:31.190","Text":"the number of faulty products within this type of a package."},{"Start":"01:31.190 ","End":"01:37.075","Text":"That means that we\u0027re looking for the number of successes."},{"Start":"01:37.075 ","End":"01:40.685","Text":"This is the number of successes, so that\u0027s a y,"},{"Start":"01:40.685 ","End":"01:48.779","Text":"where success is defined as a faulty product."},{"Start":"01:48.779 ","End":"01:53.405","Text":"What\u0027s the probability of success?"},{"Start":"01:53.405 ","End":"01:58.040","Text":"That\u0027s like asking, what\u0027s the probability of a faulty product?"},{"Start":"01:58.040 ","End":"01:59.420","Text":"We\u0027re given that,"},{"Start":"01:59.420 ","End":"02:01.460","Text":"that\u0027s 1 divided by 10."},{"Start":"02:01.460 ","End":"02:07.180","Text":"If that\u0027s the case, then y,"},{"Start":"02:07.180 ","End":"02:10.080","Text":"given that x equals 3,"},{"Start":"02:10.080 ","End":"02:17.340","Text":"we\u0027re looking for the number of faulty products in a package that has 3 products in it."},{"Start":"02:17.340 ","End":"02:22.370","Text":"That\u0027s distributed with a binomial distribution where n equals 3 and"},{"Start":"02:22.370 ","End":"02:27.890","Text":"the probability of success of the probability of the product being faulty,"},{"Start":"02:27.890 ","End":"02:30.315","Text":"that equals to 1 divided by 10."},{"Start":"02:30.315 ","End":"02:34.055","Text":"This then is the answer to this section."},{"Start":"02:34.055 ","End":"02:36.080","Text":"In this section we\u0027re asked what\u0027s"},{"Start":"02:36.080 ","End":"02:40.800","Text":"the probability distribution of Y given that X is any K?"},{"Start":"02:40.800 ","End":"02:43.940","Text":"If we recall from Section A,"},{"Start":"02:43.940 ","End":"02:48.050","Text":"we had y given x equals to 3, well,"},{"Start":"02:48.050 ","End":"02:56.244","Text":"that had a binomial distribution where n equals to 3 and p equals 1 divided by 10."},{"Start":"02:56.244 ","End":"02:58.549","Text":"What was this expression?"},{"Start":"02:58.549 ","End":"03:04.160","Text":"That was the number of faulty products in a package,"},{"Start":"03:04.160 ","End":"03:08.570","Text":"where the package contained here 3 products."},{"Start":"03:08.570 ","End":"03:10.865","Text":"Now, by the same logic,"},{"Start":"03:10.865 ","End":"03:14.735","Text":"y given x equals to K,"},{"Start":"03:14.735 ","End":"03:21.410","Text":"we\u0027re looking for the number of faulty products in a package that contains k products."},{"Start":"03:21.410 ","End":"03:25.310","Text":"Now, by the same logic that we used in Section A,"},{"Start":"03:25.310 ","End":"03:31.290","Text":"that has a binomial distribution where n equals to k now,"},{"Start":"03:31.290 ","End":"03:34.745","Text":"and p equals to 1 divided by 10."},{"Start":"03:34.745 ","End":"03:37.175","Text":"Now, the next question is,"},{"Start":"03:37.175 ","End":"03:40.940","Text":"what are the valid values for k?"},{"Start":"03:40.940 ","End":"03:43.850","Text":"Let\u0027s get back to our question here."},{"Start":"03:43.850 ","End":"03:48.380","Text":"We\u0027re given that a factory markets a product that is packaged in"},{"Start":"03:48.380 ","End":"03:52.829","Text":"packages containing 2 or 3 products."},{"Start":"03:52.829 ","End":"03:58.965","Text":"Packages can either have 2 or 3 products in them,"},{"Start":"03:58.965 ","End":"04:03.215","Text":"so k then will be equal to this,"},{"Start":"04:03.215 ","End":"04:07.570","Text":"k will be equal to either 2 or 3,"},{"Start":"04:07.570 ","End":"04:11.330","Text":"so if we\u0027ve chosen a package containing 2 products,"},{"Start":"04:11.330 ","End":"04:14.300","Text":"then this expression would have"},{"Start":"04:14.300 ","End":"04:18.275","Text":"a binomial distribution where n equals 2 and p equals 1/10."},{"Start":"04:18.275 ","End":"04:22.025","Text":"If we\u0027ve chosen a package containing 3 products,"},{"Start":"04:22.025 ","End":"04:24.140","Text":"then this expression would have"},{"Start":"04:24.140 ","End":"04:30.005","Text":"a binomial distribution where n equals 3 and p equals 1 divided by 10."},{"Start":"04:30.005 ","End":"04:33.470","Text":"In this section, we\u0027re asked what\u0027s the expectation of the number of"},{"Start":"04:33.470 ","End":"04:36.965","Text":"faulty products in a package containing 3 products?"},{"Start":"04:36.965 ","End":"04:43.325","Text":"We\u0027re looking at the expectation of y given that x equals to 3."},{"Start":"04:43.325 ","End":"04:48.620","Text":"We know how this expression is distributed."},{"Start":"04:48.620 ","End":"04:51.905","Text":"Y given that x equals to 3,"},{"Start":"04:51.905 ","End":"04:59.056","Text":"that has a binomial distribution where n equals to 3 and p equals to 1 divided by 10."},{"Start":"04:59.056 ","End":"05:01.777","Text":"What else do we know?"},{"Start":"05:01.777 ","End":"05:05.810","Text":"We know that whenever we have a binomial distribution,"},{"Start":"05:05.810 ","End":"05:13.925","Text":"the expectation of a random variable will be equal to n times p. In our case,"},{"Start":"05:13.925 ","End":"05:17.840","Text":"n is 3 and p is 1 divided by 10,"},{"Start":"05:17.840 ","End":"05:21.986","Text":"so the expectation here would be 0.3,"},{"Start":"05:21.986 ","End":"05:25.280","Text":"so 0.3 then is the expectation of the number of"},{"Start":"05:25.280 ","End":"05:30.880","Text":"faulty products in a package containing 3 products."}],"ID":13187},{"Watched":false,"Name":"Exercise 3 - Part d","Duration":"11m 16s","ChapterTopicVideoID":12708,"CourseChapterTopicPlaylistID":245018,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12708.jpeg","UploadDate":"2018-07-30T14:15:38.3300000","DurationForVideoObject":"PT11M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.065","Text":"In this section, we\u0027re asked to construct"},{"Start":"00:02.065 ","End":"00:08.340","Text":"a joint probability function of x and y. Let\u0027s just do that."},{"Start":"00:08.340 ","End":"00:11.835","Text":"Here then is the table connecting x and y."},{"Start":"00:11.835 ","End":"00:14.100","Text":"Here we have the values of x,"},{"Start":"00:14.100 ","End":"00:15.660","Text":"x being equal to 2 or 3."},{"Start":"00:15.660 ","End":"00:19.065","Text":"Now, remember, x is the number of products in a package,"},{"Start":"00:19.065 ","End":"00:23.040","Text":"and that can be only 2 or 3 products in a package."},{"Start":"00:23.040 ","End":"00:27.945","Text":"Now, y is the number of faulty products in a package,"},{"Start":"00:27.945 ","End":"00:31.680","Text":"y can be either 0 or 1, or 2 or 3."},{"Start":"00:31.680 ","End":"00:34.770","Text":"Now, what we\u0027re asked to do is we\u0027re asked to"},{"Start":"00:34.770 ","End":"00:39.450","Text":"calculate the joint probabilities of x and y."},{"Start":"00:39.450 ","End":"00:44.110","Text":"That means we\u0027re looking at the white cells right here,"},{"Start":"00:44.110 ","End":"00:46.015","Text":"so how do we do that?"},{"Start":"00:46.015 ","End":"00:48.005","Text":"First of all, what do we know?"},{"Start":"00:48.005 ","End":"00:54.290","Text":"We know that y given x equals 2,"},{"Start":"00:54.290 ","End":"00:59.600","Text":"that has a binomial distribution where n equals to 2 and k equals"},{"Start":"00:59.600 ","End":"01:04.970","Text":"to 1 divided by 10 and y given x equals to 3,"},{"Start":"01:04.970 ","End":"01:12.560","Text":"that has a binomial distribution where n equals to 3 and P equals to 1 divided by 10."},{"Start":"01:12.560 ","End":"01:17.690","Text":"We also know that the probability of any random variable,"},{"Start":"01:17.690 ","End":"01:20.015","Text":"for example, y,"},{"Start":"01:20.015 ","End":"01:23.150","Text":"being equal to some value k,"},{"Start":"01:23.150 ","End":"01:26.510","Text":"well that equals to n over k,"},{"Start":"01:26.510 ","End":"01:34.520","Text":"p to the power of k times 1 minus p to the power of n minus k. Now,"},{"Start":"01:34.520 ","End":"01:36.800","Text":"what else do we need to know?"},{"Start":"01:36.800 ","End":"01:44.270","Text":"We know that the probability of x or y,"},{"Start":"01:44.270 ","End":"01:45.995","Text":"let\u0027s do it this way,"},{"Start":"01:45.995 ","End":"01:49.535","Text":"y given x,"},{"Start":"01:49.535 ","End":"01:59.510","Text":"that equals to the probability of y intersect x divided by the probability of x."},{"Start":"01:59.510 ","End":"02:02.050","Text":"Now, having said all that,"},{"Start":"02:02.050 ","End":"02:06.690","Text":"let\u0027s see if we take 1 of the cells,"},{"Start":"02:06.690 ","End":"02:09.540","Text":"for example when y equals 1,"},{"Start":"02:09.540 ","End":"02:14.090","Text":"and x equals 3, and see what the joint probability is."},{"Start":"02:14.090 ","End":"02:21.950","Text":"The joint probability is the intersect of x equaling 3 and y equaling 1."},{"Start":"02:21.950 ","End":"02:23.960","Text":"That\u0027s right here."},{"Start":"02:23.960 ","End":"02:32.515","Text":"We can say that the probability of y equaling 1,"},{"Start":"02:32.515 ","End":"02:36.240","Text":"given x equaling 3,"},{"Start":"02:36.240 ","End":"02:42.620","Text":"that equals to the probability of y equaling 1,"},{"Start":"02:42.620 ","End":"02:50.835","Text":"intersect x equaling 3 divided by the probability of x equaling 3."},{"Start":"02:50.835 ","End":"02:54.120","Text":"We want to know what this probability is."},{"Start":"02:54.120 ","End":"02:59.525","Text":"That means that we\u0027re looking at the probability of y equals 1"},{"Start":"02:59.525 ","End":"03:06.110","Text":"given x equals 3 times the probability of x equaling 3."},{"Start":"03:06.110 ","End":"03:08.510","Text":"All I did is take this and move that over here."},{"Start":"03:08.510 ","End":"03:12.710","Text":"I multiplied both sides by the probability of x equaling 3,"},{"Start":"03:12.710 ","End":"03:19.620","Text":"and that equals to the probability of y equaling 1 intersect x equaling 3."},{"Start":"03:19.620 ","End":"03:22.575","Text":"This is what I\u0027m looking for right here."},{"Start":"03:22.575 ","End":"03:25.125","Text":"Before it can continue,"},{"Start":"03:25.125 ","End":"03:30.890","Text":"let\u0027s just take a look at the marginal probabilities of x."},{"Start":"03:30.890 ","End":"03:34.010","Text":"Where x equals 2 and x equals 3."},{"Start":"03:34.010 ","End":"03:37.850","Text":"Now we were given that the number of packages containing"},{"Start":"03:37.850 ","End":"03:42.890","Text":"2 products is equal to the number of packages containing 3 products."},{"Start":"03:42.890 ","End":"03:45.275","Text":"The probabilities are the same."},{"Start":"03:45.275 ","End":"03:50.495","Text":"That means that this is 0.5 right here and 0.5 right here."},{"Start":"03:50.495 ","End":"03:53.665","Text":"Now, what else?"},{"Start":"03:53.665 ","End":"04:00.535","Text":"Now we know also that the probability of y equaling 1,"},{"Start":"04:00.535 ","End":"04:04.470","Text":"y, given that x equals 3,"},{"Start":"04:04.470 ","End":"04:07.815","Text":"that\u0027s the distribution right here."},{"Start":"04:07.815 ","End":"04:09.885","Text":"What is that?"},{"Start":"04:09.885 ","End":"04:11.820","Text":"Let\u0027s calculate that."},{"Start":"04:11.820 ","End":"04:15.390","Text":"They\u0027ll be 3 over what,"},{"Start":"04:15.390 ","End":"04:23.820","Text":"over 1 times 0.1 to the power of 1 times 0.9 to the power of 2."},{"Start":"04:23.820 ","End":"04:33.735","Text":"Now, that equals to 0.243. We\u0027re not done yet."},{"Start":"04:33.735 ","End":"04:37.230","Text":"This then is this guy right here."},{"Start":"04:37.230 ","End":"04:44.800","Text":"That will be 0.243 times now what\u0027s the probability of x being equal to 3?"},{"Start":"04:44.800 ","End":"04:46.535","Text":"That\u0027s this guy right here."},{"Start":"04:46.535 ","End":"04:48.750","Text":"That\u0027s 0.5."},{"Start":"04:48.750 ","End":"04:54.280","Text":"That means that this equals to 0.1215."},{"Start":"04:56.840 ","End":"05:01.420","Text":"We\u0027ve just input that right here."},{"Start":"05:01.420 ","End":"05:12.710","Text":"0.1215 and that is the joint probability of y equaling 1 and x equaling 3."},{"Start":"05:13.400 ","End":"05:15.455","Text":"Let\u0027s do another one."},{"Start":"05:15.455 ","End":"05:19.465","Text":"Let\u0027s take y equals 2 now and x equals 3."},{"Start":"05:19.465 ","End":"05:23.540","Text":"Again, we\u0027re using this distribution right here,"},{"Start":"05:23.540 ","End":"05:25.550","Text":"y given x equals 3,"},{"Start":"05:25.550 ","End":"05:29.840","Text":"that\u0027s distributed with a binomial distribution with n equals 3."},{"Start":"05:29.840 ","End":"05:40.025","Text":"We\u0027re looking for the probability of y equaling to 2 given x equals 3."},{"Start":"05:40.025 ","End":"05:45.965","Text":"Now that equals to the probability of the intersect equaling 2 intersect,"},{"Start":"05:45.965 ","End":"05:53.970","Text":"x equaling 3 divided by the probability of x equaling 3."},{"Start":"05:53.970 ","End":"05:59.930","Text":"Let\u0027s just multiply both sides by the probability x equals 3."},{"Start":"05:59.930 ","End":"06:03.725","Text":"That means that we\u0027re looking for the probability y equaling 2,"},{"Start":"06:03.725 ","End":"06:09.480","Text":"given x equals 3 times the probability of x equaling 3,"},{"Start":"06:09.480 ","End":"06:11.550","Text":"and that equals to the intersect."},{"Start":"06:11.550 ","End":"06:16.880","Text":"That\u0027s the probability of x equal 2 intersect x equaling 3."},{"Start":"06:16.880 ","End":"06:22.400","Text":"Now this is a joint probability that we\u0027re looking for and once we calculate that,"},{"Start":"06:22.400 ","End":"06:24.710","Text":"we\u0027ll put that in here."},{"Start":"06:24.710 ","End":"06:30.680","Text":"What\u0027s now the probability of y equaling 2 given x equals 3?"},{"Start":"06:30.680 ","End":"06:36.020","Text":"We\u0027ll use this distribution and we\u0027ll use that here,"},{"Start":"06:36.020 ","End":"06:42.140","Text":"this expression for the binomial distribution to calculate that probability,"},{"Start":"06:42.140 ","End":"06:47.760","Text":"that equals to n, that\u0027s 3."},{"Start":"06:48.220 ","End":"06:58.830","Text":"That will be 3 over 2 times 0.1 to the power of 2 times 0.9 to the power of 1."},{"Start":"06:58.830 ","End":"07:03.640","Text":"Now, that equals to 0.027."},{"Start":"07:04.730 ","End":"07:07.860","Text":"That\u0027s this guy right here."},{"Start":"07:07.860 ","End":"07:16.625","Text":"That will be 0.027 and that has to be multiplied by the probability of x equaling 3,"},{"Start":"07:16.625 ","End":"07:17.900","Text":"that\u0027s this guy right here,"},{"Start":"07:17.900 ","End":"07:21.145","Text":"that\u0027s 0.5 and that equals"},{"Start":"07:21.145 ","End":"07:28.530","Text":"to 0.0135."},{"Start":"07:28.530 ","End":"07:39.195","Text":"This value right here is the joint probability where x equals 3 and y equals to 2."},{"Start":"07:39.195 ","End":"07:43.895","Text":"That will be 0.0135."},{"Start":"07:43.895 ","End":"07:49.055","Text":"Now, likewise for all other values where x equals 3,"},{"Start":"07:49.055 ","End":"07:51.485","Text":"for y equals 0 and y equals 3."},{"Start":"07:51.485 ","End":"07:52.910","Text":"Let\u0027s just fill that in."},{"Start":"07:52.910 ","End":"08:03.620","Text":"That\u0027s 0.3645 and here that will be 0.0005."},{"Start":"08:03.620 ","End":"08:08.910","Text":"What about when x equals 2?"},{"Start":"08:09.320 ","End":"08:15.095","Text":"Let\u0027s take the example now where y equals 1 and x equals 2."},{"Start":"08:15.095 ","End":"08:22.385","Text":"We\u0027re looking for the probability of y equaling 1,"},{"Start":"08:22.385 ","End":"08:26.225","Text":"where x equals to give an x equals 2."},{"Start":"08:26.225 ","End":"08:30.620","Text":"That equals to the intersect the probability of y equaling"},{"Start":"08:30.620 ","End":"08:37.880","Text":"1 intersect x equal 2 divided by the probability of x equals 2."},{"Start":"08:37.880 ","End":"08:41.780","Text":"Now, let\u0027s multiply both sides by the probability of x equals 2."},{"Start":"08:41.780 ","End":"08:46.460","Text":"That means that we\u0027re looking for the probability of y equals 1 given x"},{"Start":"08:46.460 ","End":"08:51.635","Text":"equals 2 times the probability of x equaling 2."},{"Start":"08:51.635 ","End":"08:56.445","Text":"That equals to the probability of the intersect y equals 1,"},{"Start":"08:56.445 ","End":"09:01.475","Text":"intersect x equals 2. Just a reminder."},{"Start":"09:01.475 ","End":"09:05.150","Text":"This then is the joint probability that we\u0027re"},{"Start":"09:05.150 ","End":"09:09.190","Text":"looking for and we\u0027re looking to input that,"},{"Start":"09:09.190 ","End":"09:10.940","Text":"where y equals 1,"},{"Start":"09:10.940 ","End":"09:15.050","Text":"x equals 2, that\u0027s in this cell right here."},{"Start":"09:15.050 ","End":"09:23.795","Text":"Now, we know that the probability of y equals 1 given x equals 2."},{"Start":"09:23.795 ","End":"09:26.510","Text":"That\u0027s this distribution right here,"},{"Start":"09:26.510 ","End":"09:30.430","Text":"the binomial distribution now with n equaling 2."},{"Start":"09:30.430 ","End":"09:34.325","Text":"Let\u0027s calculate this probability using this expression."},{"Start":"09:34.325 ","End":"09:38.690","Text":"That\u0027ll be 2 over"},{"Start":"09:38.690 ","End":"09:47.060","Text":"1 times 0.1 to the power of 1 times 0.9 to the power of 1,"},{"Start":"09:47.060 ","End":"09:49.370","Text":"that equals to 0.18."},{"Start":"09:49.370 ","End":"09:51.805","Text":"Now, as we said,"},{"Start":"09:51.805 ","End":"09:58.470","Text":"this value right here is equal to this probability right here."},{"Start":"09:58.470 ","End":"10:01.150","Text":"That\u0027ll be 0.18 times."},{"Start":"10:01.150 ","End":"10:04.080","Text":"Now what\u0027s the probability of x equaling 2?"},{"Start":"10:04.080 ","End":"10:08.705","Text":"The marginal probability here, that\u0027s 0.5."},{"Start":"10:08.705 ","End":"10:16.525","Text":"That means that the joint probability where y equals 1 and x equals 2,"},{"Start":"10:16.525 ","End":"10:20.540","Text":"that equals to 0.09."},{"Start":"10:20.540 ","End":"10:22.680","Text":"We\u0027ll put that right here."},{"Start":"10:22.680 ","End":"10:26.945","Text":"They\u0027ll be 0.09. Now,"},{"Start":"10:26.945 ","End":"10:29.080","Text":"what about the other probabilities?"},{"Start":"10:29.080 ","End":"10:33.130","Text":"The thing that stands out here is this cell right here,"},{"Start":"10:33.130 ","End":"10:36.785","Text":"where we have 2 products in the package,"},{"Start":"10:36.785 ","End":"10:40.400","Text":"that we have 3 faulty products,"},{"Start":"10:40.400 ","End":"10:41.630","Text":"and that can\u0027t be,"},{"Start":"10:41.630 ","End":"10:45.400","Text":"the probability here has to be 0."},{"Start":"10:45.400 ","End":"10:51.915","Text":"Let\u0027s just fill out the rest of the cells using this technique."},{"Start":"10:51.915 ","End":"10:55.270","Text":"This would be 0.405,"},{"Start":"10:55.520 ","End":"11:01.575","Text":"and here there will be 0.005."},{"Start":"11:01.575 ","End":"11:05.270","Text":"Here we go. That\u0027s the joint probability function."},{"Start":"11:05.270 ","End":"11:07.249","Text":"We weren\u0027t even asked to calculate"},{"Start":"11:07.249 ","End":"11:11.345","Text":"the marginal distribution of y. I\u0027m not going to do that."},{"Start":"11:11.345 ","End":"11:16.350","Text":"Here. We just answered the question for Section D."}],"ID":13188},{"Watched":false,"Name":"Exercise 4 - Part a","Duration":"6m 12s","ChapterTopicVideoID":12710,"CourseChapterTopicPlaylistID":245018,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12710.jpeg","UploadDate":"2018-07-30T14:16:11.9800000","DurationForVideoObject":"PT6M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.090","Text":"In this question, we\u0027re given that a container has 3 balls numbered 2, 4, and 8."},{"Start":"00:05.090 ","End":"00:08.475","Text":"Now 2 balls are removed without returns."},{"Start":"00:08.475 ","End":"00:13.830","Text":"We define X as the smallest number on the balls and Y as the largest number."},{"Start":"00:13.830 ","End":"00:17.670","Text":"We\u0027re asked to calculate the probability distribution of X and Y,"},{"Start":"00:17.670 ","End":"00:20.395","Text":"or the joint distribution."},{"Start":"00:20.395 ","End":"00:24.915","Text":"The first thing that we need to do is we need to describe the data."},{"Start":"00:24.915 ","End":"00:29.330","Text":"The best way to do that is through a probability tree. Now, why is that?"},{"Start":"00:29.330 ","End":"00:31.340","Text":"Well, because we have stages."},{"Start":"00:31.340 ","End":"00:35.120","Text":"The first stage is that we choose a ball,"},{"Start":"00:35.120 ","End":"00:37.400","Text":"1 of 3 balls."},{"Start":"00:37.400 ","End":"00:43.310","Text":"The next stage is where we choose 1 of the 2 remaining balls."},{"Start":"00:43.310 ","End":"00:47.020","Text":"Let\u0027s just draw the probability tree."},{"Start":"00:47.020 ","End":"00:50.810","Text":"This is a probability tree and this is the first stage."},{"Start":"00:50.810 ","End":"00:53.690","Text":"Now we can choose ball number 2,"},{"Start":"00:53.690 ","End":"00:54.980","Text":"ball number 4,"},{"Start":"00:54.980 ","End":"00:56.030","Text":"or ball number 8,"},{"Start":"00:56.030 ","End":"00:59.540","Text":"and the probability of choosing 1 of the balls is 1/3;"},{"Start":"00:59.540 ","End":"01:04.245","Text":"1/3 here, 1/3 here, and 1/3 here."},{"Start":"01:04.245 ","End":"01:09.885","Text":"Now, the next stage is to choose another ball,"},{"Start":"01:09.885 ","End":"01:12.315","Text":"1 of the 2 remaining balls."},{"Start":"01:12.315 ","End":"01:15.860","Text":"If we\u0027ve chosen 8 on the first stage,"},{"Start":"01:15.860 ","End":"01:20.705","Text":"well here we\u0027ll choose either 2 or 4."},{"Start":"01:20.705 ","End":"01:24.250","Text":"The probability of doing that is 1/2."},{"Start":"01:24.250 ","End":"01:27.905","Text":"There will be 1/2 here and 1/2 here."},{"Start":"01:27.905 ","End":"01:30.395","Text":"Now, if we\u0027ve chosen 4,"},{"Start":"01:30.395 ","End":"01:35.980","Text":"then the next stage is to choose either a 2 or an 8."},{"Start":"01:35.980 ","End":"01:40.070","Text":"Here, again, the probability is 1/2 and 1/2."},{"Start":"01:40.070 ","End":"01:43.505","Text":"If we\u0027ve chosen 2 in the first stage,"},{"Start":"01:43.505 ","End":"01:47.980","Text":"then we have to choose between 4 and 8,"},{"Start":"01:47.980 ","End":"01:52.840","Text":"again with a probability of 1/2 and 1/2."},{"Start":"01:52.840 ","End":"01:58.790","Text":"What are the X values and what are the Y values here?"},{"Start":"01:58.790 ","End":"02:02.375","Text":"Well, in this branch right here,"},{"Start":"02:02.375 ","End":"02:04.550","Text":"X is the smallest number,"},{"Start":"02:04.550 ","End":"02:09.495","Text":"so X here, what\u0027s the smallest number between 4 and 8?"},{"Start":"02:09.495 ","End":"02:10.935","Text":"Well, that\u0027s 4,"},{"Start":"02:10.935 ","End":"02:15.030","Text":"and what\u0027s the largest number? That\u0027s 8."},{"Start":"02:15.030 ","End":"02:17.710","Text":"In this branch right here, well,"},{"Start":"02:17.710 ","End":"02:20.015","Text":"the smallest number equals to 2,"},{"Start":"02:20.015 ","End":"02:22.940","Text":"and the largest number equals to 8."},{"Start":"02:22.940 ","End":"02:27.635","Text":"Here, the smallest number equals 4,"},{"Start":"02:27.635 ","End":"02:30.845","Text":"and the largest number equals 8."},{"Start":"02:30.845 ","End":"02:38.240","Text":"Here, the smallest number equals to 2 and the largest number equals to 4,"},{"Start":"02:38.240 ","End":"02:44.895","Text":"and here, the smallest number equals 2 and the largest number equals to 8."},{"Start":"02:44.895 ","End":"02:52.120","Text":"Here, the smallest number equals 2 and the largest number equals to 4."},{"Start":"02:52.190 ","End":"02:54.635","Text":"What we have here,"},{"Start":"02:54.635 ","End":"02:58.985","Text":"what are the relevant values for X?"},{"Start":"02:58.985 ","End":"03:04.285","Text":"Well, X will be either 2 or 4,"},{"Start":"03:04.285 ","End":"03:07.125","Text":"and the relevant values for Y,"},{"Start":"03:07.125 ","End":"03:10.155","Text":"well that\u0027ll be 4 and 8."},{"Start":"03:10.155 ","End":"03:12.030","Text":"Once we have that,"},{"Start":"03:12.030 ","End":"03:19.320","Text":"let\u0027s draw the joint probability distribution function."},{"Start":"03:19.850 ","End":"03:22.605","Text":"This is our table right here."},{"Start":"03:22.605 ","End":"03:25.690","Text":"We have our X-values and our Y-values."},{"Start":"03:25.690 ","End":"03:30.965","Text":"We have our marginal probabilities for Y and for X."},{"Start":"03:30.965 ","End":"03:33.920","Text":"We know that the sum of the probabilities is equal 1."},{"Start":"03:33.920 ","End":"03:36.575","Text":"What we want to do is we want to calculate"},{"Start":"03:36.575 ","End":"03:40.675","Text":"the joint probabilities of these 4 white cells."},{"Start":"03:40.675 ","End":"03:51.350","Text":"Let\u0027s get to it. We\u0027re looking for the probability of X equaling 2 and Y equaling 4."},{"Start":"03:51.350 ","End":"03:53.390","Text":"Now, where\u0027s that?"},{"Start":"03:53.390 ","End":"03:55.010","Text":"X equals 2, Y equals 4,"},{"Start":"03:55.010 ","End":"03:56.795","Text":"that\u0027s this right here,"},{"Start":"03:56.795 ","End":"03:58.565","Text":"and this right here."},{"Start":"03:58.565 ","End":"04:02.870","Text":"Now, what\u0027s the probability of landing on this branch?"},{"Start":"04:02.870 ","End":"04:07.940","Text":"Well, that\u0027s 1/3 times 1/2, 1/3 times 1/2."},{"Start":"04:07.940 ","End":"04:09.545","Text":"We have that twice."},{"Start":"04:09.545 ","End":"04:11.240","Text":"We have that here,"},{"Start":"04:11.240 ","End":"04:13.614","Text":"and we have that here,"},{"Start":"04:13.614 ","End":"04:14.965","Text":"1/3 times 1/2,"},{"Start":"04:14.965 ","End":"04:22.020","Text":"plus 1/3 times 1/2 that equals to 2/6 or 1/3."},{"Start":"04:22.020 ","End":"04:24.795","Text":"X equals 2, Y equals 4,"},{"Start":"04:24.795 ","End":"04:27.625","Text":"that\u0027s right here, that\u0027ll be 1/3."},{"Start":"04:27.625 ","End":"04:31.145","Text":"Now let\u0027s do that for the other cells as well,"},{"Start":"04:31.145 ","End":"04:39.380","Text":"where X equals 2 and Y equals 8 now."},{"Start":"04:39.380 ","End":"04:41.750","Text":"That\u0027s right here, this cell right here."},{"Start":"04:41.750 ","End":"04:43.220","Text":"Well, where\u0027s that?"},{"Start":"04:43.220 ","End":"04:47.610","Text":"That\u0027s right here and right here."},{"Start":"04:47.610 ","End":"04:54.675","Text":"Now, again, that\u0027ll be 1/3 times 1/2 plus 1/3 times 1/2."},{"Start":"04:54.675 ","End":"04:58.365","Text":"That equals to 2/6 and that equals to 1/3."},{"Start":"04:58.365 ","End":"05:00.230","Text":"When X equals 2 and Y equals 8,"},{"Start":"05:00.230 ","End":"05:05.990","Text":"the joint probability would be 1/3."},{"Start":"05:05.990 ","End":"05:09.770","Text":"Now, what happens when Y equals 4 and X equals 4?"},{"Start":"05:09.770 ","End":"05:11.976","Text":"Well, we don\u0027t have that."},{"Start":"05:11.976 ","End":"05:14.660","Text":"that\u0027s 0."},{"Start":"05:14.660 ","End":"05:17.990","Text":"What about the last cell right here?"},{"Start":"05:17.990 ","End":"05:24.410","Text":"That\u0027s the probability of X equaling 4 and Y equaling 8."},{"Start":"05:24.410 ","End":"05:27.995","Text":"Well, again, that\u0027s this thing right here,"},{"Start":"05:27.995 ","End":"05:29.735","Text":"and this thing right here."},{"Start":"05:29.735 ","End":"05:36.140","Text":"Again, that\u0027ll be 1/3 times 1/2 plus 1/3 times 1/2,"},{"Start":"05:36.140 ","End":"05:38.090","Text":"which equals 2/6,"},{"Start":"05:38.090 ","End":"05:40.650","Text":"and that equals to 1/3."},{"Start":"05:40.840 ","End":"05:45.545","Text":"What are the marginal probabilities right now?"},{"Start":"05:45.545 ","End":"05:49.190","Text":"Well, the marginal probability of y, when Y equals 4,"},{"Start":"05:49.190 ","End":"05:54.825","Text":"that\u0027s 1/3, and when Y equals 8, well that\u0027s 2/3."},{"Start":"05:54.825 ","End":"05:56.670","Text":"Now when X equals 2,"},{"Start":"05:56.670 ","End":"06:01.515","Text":"that\u0027ll be 2/3 and when X equals 4, that\u0027ll be 1/3."},{"Start":"06:01.515 ","End":"06:07.810","Text":"This then is the joint probability distribution of X and Y,"},{"Start":"06:07.810 ","End":"06:12.420","Text":"which includes the marginal probabilities of X and Y."}],"ID":13189},{"Watched":false,"Name":"Exercise 4 - Parts b-c","Duration":"3m 41s","ChapterTopicVideoID":12711,"CourseChapterTopicPlaylistID":245018,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12711.jpeg","UploadDate":"2018-07-30T14:16:24.0900000","DurationForVideoObject":"PT3M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.480","Text":"In this section, we\u0027re asked that if the lowest number selected is"},{"Start":"00:03.480 ","End":"00:07.940","Text":"2 what is the chances that the largest number is 8?"},{"Start":"00:07.940 ","End":"00:11.805","Text":"Basically we\u0027re looking at a conditional probability"},{"Start":"00:11.805 ","End":"00:16.420","Text":"and we\u0027re looking for the probability of y equaling 8,"},{"Start":"00:16.420 ","End":"00:20.380","Text":"given that x equals 2."},{"Start":"00:20.380 ","End":"00:24.575","Text":"Now, how do we calculate the conditional probability?"},{"Start":"00:24.575 ","End":"00:27.530","Text":"Well, that\u0027s the probability of the intercept."},{"Start":"00:27.530 ","End":"00:38.225","Text":"y equals 8 intersect x equaling 2 divided by the probability of what\u0027s known, x equals 2."},{"Start":"00:38.225 ","End":"00:41.780","Text":"Now, that equals to the following."},{"Start":"00:41.780 ","End":"00:45.140","Text":"Now y equals 8 and x equals 2."},{"Start":"00:45.140 ","End":"00:50.720","Text":"Well that\u0027s the joint probability where y equals 8 and x equals 2."},{"Start":"00:50.720 ","End":"00:58.055","Text":"Well, that\u0027s 1/3 divided by the probability of x equals 2."},{"Start":"00:58.055 ","End":"01:04.635","Text":"Well, that\u0027s the marginal probability where x equals 2, that\u0027s 2/3."},{"Start":"01:04.635 ","End":"01:08.300","Text":"That means that here,"},{"Start":"01:08.300 ","End":"01:11.395","Text":"this comes out to 1/2."},{"Start":"01:11.395 ","End":"01:20.040","Text":"That\u0027s the probability that the largest number will be 8 when the lowest number equals 2."},{"Start":"01:20.050 ","End":"01:22.610","Text":"In this section, we\u0027re asked to calculate"},{"Start":"01:22.610 ","End":"01:26.000","Text":"the probability distribution of x given that y equals"},{"Start":"01:26.000 ","End":"01:32.190","Text":"4 and then to find the expectation of x given that y equals to 4."},{"Start":"01:32.240 ","End":"01:38.698","Text":"We\u0027re looking here for a conditional probability,"},{"Start":"01:38.698 ","End":"01:40.805","Text":"when y equals 4,"},{"Start":"01:40.805 ","End":"01:48.290","Text":"x can only equal to 2 so x equaling 2 given that y equals to 4."},{"Start":"01:48.290 ","End":"01:51.665","Text":"Now, what does that equal to since this is a conditional probability?"},{"Start":"01:51.665 ","End":"01:56.985","Text":"Well, in the numerator we have the intersect that\u0027s x equaling 2"},{"Start":"01:56.985 ","End":"02:02.630","Text":"and y equaling 4 divided by the probability of what\u0027s known."},{"Start":"02:02.630 ","End":"02:05.245","Text":"That\u0027s y equaling 4."},{"Start":"02:05.245 ","End":"02:08.195","Text":"Now, let\u0027s go to our table."},{"Start":"02:08.195 ","End":"02:12.715","Text":"The joint probability where x equals 2 and y equals 4."},{"Start":"02:12.715 ","End":"02:14.690","Text":"This is x equals 2, y equals 4."},{"Start":"02:14.690 ","End":"02:16.940","Text":"Well that\u0027s 1/3, that\u0027s right here."},{"Start":"02:16.940 ","End":"02:20.300","Text":"That equals to a third divided by,"},{"Start":"02:20.300 ","End":"02:22.820","Text":"now the probability of y equals 4."},{"Start":"02:22.820 ","End":"02:28.900","Text":"Well that\u0027s a marginal probability where y equals 4 and that equals also to a third."},{"Start":"02:28.900 ","End":"02:32.960","Text":"That means that we have a probability of 1."},{"Start":"02:32.960 ","End":"02:34.445","Text":"This is a certainty."},{"Start":"02:34.445 ","End":"02:40.005","Text":"The probability of x equaling 2 when y equals to 4, what certainty?"},{"Start":"02:40.005 ","End":"02:42.905","Text":"It\u0027s certain that x will equal to 2."},{"Start":"02:42.905 ","End":"02:48.980","Text":"Now, for those of you who want to be very formal so what,"},{"Start":"02:48.980 ","End":"02:54.590","Text":"we need to do is we need to write the probability distribution so let\u0027s do that."},{"Start":"02:54.590 ","End":"02:59.325","Text":"That\u0027s x given y equals to 4."},{"Start":"02:59.325 ","End":"03:04.310","Text":"Those are the values that x can have or you can only have 1 value that\u0027s 2."},{"Start":"03:04.310 ","End":"03:09.470","Text":"This is a probability of x given y equals to 4."},{"Start":"03:09.470 ","End":"03:11.950","Text":"We\u0027ve calculated that to be 1."},{"Start":"03:11.950 ","End":"03:14.239","Text":"That\u0027s the probability distribution."},{"Start":"03:14.239 ","End":"03:15.800","Text":"Now, if that\u0027s the case,"},{"Start":"03:15.800 ","End":"03:21.020","Text":"then what\u0027s the expectation of x given y equals 4?"},{"Start":"03:21.020 ","End":"03:24.380","Text":"Well, that\u0027s x times its probability,"},{"Start":"03:24.380 ","End":"03:28.004","Text":"so it\u0027ll be 2 times 1 and that equals to 2,"},{"Start":"03:28.004 ","End":"03:29.840","Text":"so here we go."},{"Start":"03:29.840 ","End":"03:34.070","Text":"The expectation of x given y equals to 4 equals to 2,"},{"Start":"03:34.070 ","End":"03:38.990","Text":"and the probability of x equaling 2 given y equals 4,"},{"Start":"03:38.990 ","End":"03:41.490","Text":"well, that equals to 1."}],"ID":13190},{"Watched":false,"Name":"Exercise 5 - Part a","Duration":"7m 12s","ChapterTopicVideoID":12712,"CourseChapterTopicPlaylistID":245018,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12712.jpeg","UploadDate":"2018-07-30T14:16:50.0570000","DurationForVideoObject":"PT7M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.874","Text":"In this question, we\u0027re given that there are 2 bank branches in the community,"},{"Start":"00:03.874 ","End":"00:06.375","Text":"City Bank and the Union Bank."},{"Start":"00:06.375 ","End":"00:09.885","Text":"The following data applies to the adult population in the community."},{"Start":"00:09.885 ","End":"00:12.810","Text":"Now, 60 percent have an account in City Bank,"},{"Start":"00:12.810 ","End":"00:15.540","Text":"50 percent have an account in the Union Bank,"},{"Start":"00:15.540 ","End":"00:19.635","Text":"95 percent have an account in at least 1 branch."},{"Start":"00:19.635 ","End":"00:22.530","Text":"Now, let X be the number of branches in which"},{"Start":"00:22.530 ","End":"00:25.660","Text":"an adult resident of the community has an account."},{"Start":"00:25.660 ","End":"00:28.710","Text":"Let Y be an indicator variable,"},{"Start":"00:28.710 ","End":"00:33.218","Text":"where Y equals 1 if the person has an account at the City Bank,"},{"Start":"00:33.218 ","End":"00:35.970","Text":"and Y equals 0 otherwise."},{"Start":"00:35.970 ","End":"00:38.510","Text":"The first thing that we\u0027re asked to do is to construct"},{"Start":"00:38.510 ","End":"00:41.030","Text":"a joint probability function of X and Y,"},{"Start":"00:41.030 ","End":"00:45.355","Text":"and afterwards, add the marginal probability function."},{"Start":"00:45.355 ","End":"00:49.820","Text":"The first thing that we need to do is we need to describe the data,"},{"Start":"00:49.820 ","End":"00:51.770","Text":"and we\u0027ll be using the Venn diagram for that."},{"Start":"00:51.770 ","End":"00:57.049","Text":"But before that, let\u0027s define specific variables or events."},{"Start":"00:57.049 ","End":"01:03.020","Text":"We\u0027ll define A as having an account in the City Bank,"},{"Start":"01:03.020 ","End":"01:07.980","Text":"and B,"},{"Start":"01:07.980 ","End":"01:14.050","Text":"that will be having an account in the Union Bank."},{"Start":"01:14.740 ","End":"01:21.200","Text":"Now, the probability of A having an account in the City Bank,"},{"Start":"01:21.200 ","End":"01:24.140","Text":"that\u0027s given to us at 0.6."},{"Start":"01:24.140 ","End":"01:30.110","Text":"The probability of B having an account in the Union Bank, that equals to 0.5."},{"Start":"01:30.110 ","End":"01:33.530","Text":"Now, we\u0027re also given that the probability of"},{"Start":"01:33.530 ","End":"01:38.090","Text":"having a bank account in at least 1 of the banks,"},{"Start":"01:38.090 ","End":"01:40.985","Text":"that equals to 0.95."},{"Start":"01:40.985 ","End":"01:46.700","Text":"What\u0027s missing? Well, we can use this formula right here."},{"Start":"01:46.700 ","End":"01:50.270","Text":"That\u0027s the probability of A union B,"},{"Start":"01:50.270 ","End":"01:55.880","Text":"but that equals to the probability of A plus the probability of B,"},{"Start":"01:55.880 ","End":"02:00.005","Text":"minus the probability of A intersect B."},{"Start":"02:00.005 ","End":"02:01.700","Text":"That\u0027s what\u0027s missing."},{"Start":"02:01.700 ","End":"02:03.785","Text":"Now, let\u0027s just plug in the numbers."},{"Start":"02:03.785 ","End":"02:08.110","Text":"The probability of A union B, well that\u0027s 0.95."},{"Start":"02:08.110 ","End":"02:12.035","Text":"That has to equal to the probability of A that\u0027s 0.6,"},{"Start":"02:12.035 ","End":"02:18.980","Text":"plus the probability of B that\u0027s 0.5 minus the probability of A intersect B."},{"Start":"02:18.980 ","End":"02:25.115","Text":"Now, that means that the probability of A intersect B,"},{"Start":"02:25.115 ","End":"02:29.360","Text":"that equals to 0.15."},{"Start":"02:29.360 ","End":"02:32.810","Text":"Excellent. Now we have all the probabilities."},{"Start":"02:32.810 ","End":"02:36.155","Text":"Let\u0027s just use our Venn diagram now."},{"Start":"02:36.155 ","End":"02:38.835","Text":"Here\u0027s our Venn diagram."},{"Start":"02:38.835 ","End":"02:41.600","Text":"Let\u0027s start plugging in the probabilities."},{"Start":"02:41.600 ","End":"02:44.010","Text":"First, let\u0027s look at the intersection,"},{"Start":"02:44.010 ","End":"02:46.490","Text":"or the probability of the intersect between A and B,"},{"Start":"02:46.490 ","End":"02:49.280","Text":"that\u0027s 0.15. We\u0027ll put that here."},{"Start":"02:49.280 ","End":"02:51.515","Text":"That would be 0.15."},{"Start":"02:51.515 ","End":"02:55.235","Text":"Now, what about the probability of all of A?"},{"Start":"02:55.235 ","End":"02:57.290","Text":"Well, that\u0027s 0.6."},{"Start":"02:57.290 ","End":"03:04.400","Text":"The remainder of A that\u0027s not in the intersect, that\u0027s 0.45."},{"Start":"03:04.400 ","End":"03:06.485","Text":"Now, what about B?"},{"Start":"03:06.485 ","End":"03:10.145","Text":"Well, all of B, well that\u0027s 0.5."},{"Start":"03:10.145 ","End":"03:12.935","Text":"The remainder of B,"},{"Start":"03:12.935 ","End":"03:15.620","Text":"the probability not in the intersect, well,"},{"Start":"03:15.620 ","End":"03:19.705","Text":"that equals to 0.35."},{"Start":"03:19.705 ","End":"03:22.115","Text":"Now, what else?"},{"Start":"03:22.115 ","End":"03:28.165","Text":"Well, we have here 0.45 plus 0.15,"},{"Start":"03:28.165 ","End":"03:33.570","Text":"that\u0027s 0.6 plus 0.35, that\u0027s 0.95."},{"Start":"03:33.570 ","End":"03:37.425","Text":"Outside of the events,"},{"Start":"03:37.425 ","End":"03:42.880","Text":"we have a probability of 0.05."},{"Start":"03:42.880 ","End":"03:47.075","Text":"Those are the people that don\u0027t have an account at all."},{"Start":"03:47.075 ","End":"03:53.740","Text":"Now, let\u0027s take a look at the definition of X and Y."},{"Start":"03:53.740 ","End":"03:57.320","Text":"Now, X is the number of branches in which"},{"Start":"03:57.320 ","End":"04:01.025","Text":"an adult resident in the community has an account."},{"Start":"04:01.025 ","End":"04:05.605","Text":"Well then, here in the intersect,"},{"Start":"04:05.605 ","End":"04:08.415","Text":"X has to be equal to 2."},{"Start":"04:08.415 ","End":"04:15.075","Text":"This is the number of branches in which a person has an account."},{"Start":"04:15.075 ","End":"04:18.495","Text":"He has an account both in the Union Bank and in the City Bank."},{"Start":"04:18.495 ","End":"04:22.565","Text":"Here, X equals to 1."},{"Start":"04:22.565 ","End":"04:26.065","Text":"Is the person having an account in the City Bank only?"},{"Start":"04:26.065 ","End":"04:28.750","Text":"Here, X equals 1."},{"Start":"04:28.750 ","End":"04:32.485","Text":"That\u0027s a person having an account in the Union Bank only."},{"Start":"04:32.485 ","End":"04:36.285","Text":"Here, X equals 0,"},{"Start":"04:36.285 ","End":"04:43.000","Text":"that\u0027s the number of accounts that a person has that\u0027s not in the Union Bank,"},{"Start":"04:43.000 ","End":"04:44.365","Text":"not in the City Bank."},{"Start":"04:44.365 ","End":"04:46.450","Text":"That\u0027s that."},{"Start":"04:46.450 ","End":"04:48.170","Text":"Now, what about Y?"},{"Start":"04:48.170 ","End":"04:55.855","Text":"Y equals 1 if the person has an account in the City Bank right here."},{"Start":"04:55.855 ","End":"04:58.450","Text":"Y here equals 1."},{"Start":"04:58.450 ","End":"05:02.765","Text":"Here, Y equals 1, here,"},{"Start":"05:02.765 ","End":"05:10.205","Text":"Y equals 0, because this is the Union Bank, not the City Bank."},{"Start":"05:10.205 ","End":"05:13.980","Text":"Here also, Y equals 0."},{"Start":"05:14.450 ","End":"05:22.125","Text":"Now that we\u0027ve calculated all the probabilities and we know the values of X and Y,"},{"Start":"05:22.125 ","End":"05:29.850","Text":"we can now draw or calculate the joint probability function of X and Y."},{"Start":"05:29.850 ","End":"05:35.893","Text":"This is a table that should contain the joint probabilities of X and Y,"},{"Start":"05:35.893 ","End":"05:38.780","Text":"and the marginal probabilities as well."},{"Start":"05:38.780 ","End":"05:41.900","Text":"That\u0027s what we\u0027re asked to do in Section B,"},{"Start":"05:41.900 ","End":"05:44.375","Text":"but let\u0027s not get ahead of ourselves."},{"Start":"05:44.375 ","End":"05:46.375","Text":"X, as we said,"},{"Start":"05:46.375 ","End":"05:49.260","Text":"has the values of 0, 1, and 2,"},{"Start":"05:49.260 ","End":"05:55.575","Text":"and Y is an indicator variable with values of 0 and 1."},{"Start":"05:55.575 ","End":"05:58.770","Text":"Let\u0027s just plug in the joint probabilities."},{"Start":"05:58.770 ","End":"06:02.090","Text":"Now, when X equals 0 and Y equals 0,"},{"Start":"06:02.090 ","End":"06:04.625","Text":"the probability is 0.05."},{"Start":"06:04.625 ","End":"06:08.330","Text":"Let\u0027s put that in, that\u0027s 0.05."},{"Start":"06:08.330 ","End":"06:14.660","Text":"Now, what about when X equals 1 and Y equals 0? That\u0027s right here."},{"Start":"06:14.660 ","End":"06:17.735","Text":"That equals to 0.35."},{"Start":"06:17.735 ","End":"06:23.430","Text":"Now, what about when X equals 2 and Y equals 1?"},{"Start":"06:23.430 ","End":"06:26.590","Text":"Well, that equals to 0.15."},{"Start":"06:26.660 ","End":"06:30.100","Text":"When X equals 1 and Y equals 1,"},{"Start":"06:30.100 ","End":"06:34.235","Text":"that\u0027s right here, that\u0027s 0.45."},{"Start":"06:34.235 ","End":"06:42.170","Text":"Now, do we have any values for X equals 0 and Y equals 1?"},{"Start":"06:42.170 ","End":"06:45.590","Text":"Well, that doesn\u0027t make sense because if I don\u0027t have an account,"},{"Start":"06:45.590 ","End":"06:48.260","Text":"well, how can I have an account in the City Bank?"},{"Start":"06:48.260 ","End":"06:50.255","Text":"That has to be 0."},{"Start":"06:50.255 ","End":"06:54.799","Text":"Likewise, if I have 2 accounts,"},{"Start":"06:54.799 ","End":"07:00.320","Text":"then there\u0027s no way that I don\u0027t have an account in the City Bank."},{"Start":"07:00.320 ","End":"07:03.390","Text":"That also has to be equal to 0."},{"Start":"07:03.650 ","End":"07:07.610","Text":"Now that we have the joint probabilities,"},{"Start":"07:07.610 ","End":"07:12.510","Text":"we can go ahead and calculate the marginal probabilities."}],"ID":13191},{"Watched":false,"Name":"Exercise 5 - Parts b-c","Duration":"2m 18s","ChapterTopicVideoID":12713,"CourseChapterTopicPlaylistID":245018,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12713.jpeg","UploadDate":"2018-07-30T14:16:57.1970000","DurationForVideoObject":"PT2M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.855","Text":"Let\u0027s take a look at the marginal probabilities now."},{"Start":"00:03.855 ","End":"00:11.010","Text":"That\u0027s just adding up all the probabilities on the columns and on the rows."},{"Start":"00:11.010 ","End":"00:15.150","Text":"The marginal probability of X when X equals 0,"},{"Start":"00:15.150 ","End":"00:19.290","Text":"that\u0027s 0.05 plus 0, that\u0027ll be 0.05."},{"Start":"00:19.290 ","End":"00:21.885","Text":"Here, that\u0027ll be 0.8,"},{"Start":"00:21.885 ","End":"00:24.765","Text":"and here that\u0027ll be 0.15."},{"Start":"00:24.765 ","End":"00:28.275","Text":"Now, what\u0027s the marginal probability of Y?"},{"Start":"00:28.275 ","End":"00:29.880","Text":"When Y equals 0,"},{"Start":"00:29.880 ","End":"00:32.880","Text":"that\u0027ll be 0.05 plus 0.35,"},{"Start":"00:32.880 ","End":"00:34.705","Text":"that\u0027ll be 0.4,"},{"Start":"00:34.705 ","End":"00:38.585","Text":"and here that\u0027ll be 0.6."},{"Start":"00:38.585 ","End":"00:41.990","Text":"Just to make sure that everything adds up to 1,"},{"Start":"00:41.990 ","End":"00:44.545","Text":"that adds up to 1,"},{"Start":"00:44.545 ","End":"00:46.170","Text":"this adds up to 1,"},{"Start":"00:46.170 ","End":"00:50.925","Text":"and in the matrix itself,"},{"Start":"00:50.925 ","End":"00:52.620","Text":"in the table itself,"},{"Start":"00:52.620 ","End":"00:57.210","Text":"these 6 probabilities add up to 1 as well, we\u0027re okay."},{"Start":"00:57.210 ","End":"01:02.360","Text":"In this section, it\u0027s known that a given resident has an account at City Bank,"},{"Start":"01:02.360 ","End":"01:06.890","Text":"and we\u0027re asked what\u0027s the probability that he has no other account."},{"Start":"01:06.890 ","End":"01:11.210","Text":"We\u0027re looking for the probability of X equaling 1."},{"Start":"01:11.210 ","End":"01:13.760","Text":"That\u0027s the number of accounts that a person has,"},{"Start":"01:13.760 ","End":"01:17.160","Text":"given that Y equals 1,"},{"Start":"01:17.160 ","End":"01:21.375","Text":"that he has an account at City Bank."},{"Start":"01:21.375 ","End":"01:24.560","Text":"That equals to the probability,"},{"Start":"01:24.560 ","End":"01:27.785","Text":"since this is a conditional probability in the numerator,"},{"Start":"01:27.785 ","End":"01:31.400","Text":"that\u0027ll be the joint probability of the intersect."},{"Start":"01:31.400 ","End":"01:34.015","Text":"That\u0027ll be the probability of X equaling 1,"},{"Start":"01:34.015 ","End":"01:36.690","Text":"intersect Y equals 1,"},{"Start":"01:36.690 ","End":"01:41.265","Text":"divided by the probability of Y equaling 1,"},{"Start":"01:41.265 ","End":"01:43.830","Text":"that\u0027s the probability of what\u0027s known."},{"Start":"01:43.830 ","End":"01:46.664","Text":"That equals to the following."},{"Start":"01:46.664 ","End":"01:50.490","Text":"When X equals 1 and when Y equals 1,"},{"Start":"01:50.490 ","End":"01:54.150","Text":"that\u0027s right here, that\u0027s Y equals 1, X equals 1."},{"Start":"01:54.150 ","End":"02:01.035","Text":"We have 0.45 divided by the probability of Y equaling 1."},{"Start":"02:01.035 ","End":"02:03.020","Text":"That\u0027s this thing right here,"},{"Start":"02:03.020 ","End":"02:09.605","Text":"that\u0027s 0.6 and that equals to 0.75."},{"Start":"02:09.605 ","End":"02:14.240","Text":"That\u0027s the probability that a person has no other account"},{"Start":"02:14.240 ","End":"02:19.260","Text":"if it\u0027s known that he has an account at City Bank."}],"ID":13192}],"Thumbnail":null,"ID":245018},{"Name":"Correlation between Variables","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"6m 26s","ChapterTopicVideoID":12722,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12722.jpeg","UploadDate":"2018-07-30T15:01:53.2770000","DurationForVideoObject":"PT6M26S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"In this chapter, we\u0027ll be talking about a 2-dimensional discrete variables."},{"Start":"00:03.540 ","End":"00:06.660","Text":"Specifically the correlation between variables."},{"Start":"00:06.660 ","End":"00:10.110","Text":"Now, we want to examine the degree of linear correlation"},{"Start":"00:10.110 ","End":"00:14.190","Text":"between 2 variables using the linear correlation coefficient,"},{"Start":"00:14.190 ","End":"00:18.510","Text":"and we denote that as the Greek letter Rho,"},{"Start":"00:18.510 ","End":"00:22.545","Text":"or sometimes you\u0027ll be seeing that as the letter r. Now,"},{"Start":"00:22.545 ","End":"00:26.820","Text":"the correlation coefficient has values between minus 1,"},{"Start":"00:26.820 ","End":"00:28.880","Text":"and 1, and here,"},{"Start":"00:28.880 ","End":"00:34.560","Text":"we can see examples of connections between 2 variables, x,"},{"Start":"00:34.560 ","End":"00:38.495","Text":"and y, this axis is x and this axis is y,"},{"Start":"00:38.495 ","End":"00:41.230","Text":"for different values of Rho."},{"Start":"00:41.230 ","End":"00:44.375","Text":"Here, Rho equals minus 1,"},{"Start":"00:44.375 ","End":"00:48.450","Text":"and we have here a perfect negative correlation."},{"Start":"00:48.450 ","End":"00:50.630","Text":"Here, Rho equals 0,"},{"Start":"00:50.630 ","End":"00:53.975","Text":"and there\u0027s no correlation between x and y,"},{"Start":"00:53.975 ","End":"00:56.750","Text":"and here Rho equals to 1,"},{"Start":"00:56.750 ","End":"01:02.910","Text":"and we have a perfect positive correlation between x and y."},{"Start":"01:03.470 ","End":"01:06.980","Text":"Now, what does this mean?"},{"Start":"01:06.980 ","End":"01:12.295","Text":"Well, a correlation coefficient equaling minus 1 or 1,"},{"Start":"01:12.295 ","End":"01:13.870","Text":"so we\u0027re talking about the edges here,"},{"Start":"01:13.870 ","End":"01:16.555","Text":"we\u0027re talking about Rho here equaling minus 1,"},{"Start":"01:16.555 ","End":"01:19.350","Text":"or here Rho equaling 1."},{"Start":"01:19.350 ","End":"01:21.710","Text":"Well, that means that there\u0027s a complete,"},{"Start":"01:21.710 ","End":"01:25.300","Text":"and absolute linear dependency between the variables x,"},{"Start":"01:25.300 ","End":"01:30.910","Text":"and y that can be expressed through the formula y equals a x plus b."},{"Start":"01:30.910 ","End":"01:36.555","Text":"Here we have y equaling a x plus b,"},{"Start":"01:36.555 ","End":"01:42.130","Text":"and again here also we have y equaling a x plus b."},{"Start":"01:42.130 ","End":"01:45.700","Text":"Now, a complete positive correlation,"},{"Start":"01:45.700 ","End":"01:47.470","Text":"that means we\u0027re here,"},{"Start":"01:47.470 ","End":"01:54.570","Text":"that\u0027s Rho equaling 1 means that there\u0027s a complete linear dependency where here,"},{"Start":"01:54.570 ","End":"01:56.760","Text":"the value a is positive,"},{"Start":"01:56.760 ","End":"02:00.000","Text":"so a here has to be positive,"},{"Start":"02:00.000 ","End":"02:03.020","Text":"while a complete negative correlation means that there\u0027s"},{"Start":"02:03.020 ","End":"02:09.695","Text":"a complete linear dependency where the slope a here now is negative,"},{"Start":"02:09.695 ","End":"02:14.100","Text":"so that means that a is less than 0."},{"Start":"02:14.830 ","End":"02:18.180","Text":"Let\u0027s move on now."},{"Start":"02:18.580 ","End":"02:24.560","Text":"A partial positive correlation means that as 1 variable increases,"},{"Start":"02:24.560 ","End":"02:27.905","Text":"the other has a tendency to increase in value,"},{"Start":"02:27.905 ","End":"02:32.150","Text":"but there\u0027s no linear formula absolutely connecting x and, y."},{"Start":"02:32.150 ","End":"02:37.985","Text":"Similarly, a partial negative correlation means that as 1 variable increases,"},{"Start":"02:37.985 ","End":"02:41.465","Text":"the other has a tendency to decrease in value,"},{"Start":"02:41.465 ","End":"02:46.160","Text":"but there\u0027s no linear formula absolutely connecting x and y."},{"Start":"02:46.160 ","End":"02:49.535","Text":"Now, let\u0027s take a look at an example here."},{"Start":"02:49.535 ","End":"02:57.350","Text":"Here we see examples of partial positive correlations,"},{"Start":"02:57.350 ","End":"02:59.240","Text":"and partial negative correlations."},{"Start":"02:59.240 ","End":"03:00.445","Text":"Let\u0027s take a look at this,"},{"Start":"03:00.445 ","End":"03:05.460","Text":"where Rho equals to 0.3 or 0.8."},{"Start":"03:05.460 ","End":"03:09.575","Text":"Well, here we have a low positive correlation between x, and y,"},{"Start":"03:09.575 ","End":"03:13.640","Text":"and here we have a high positive correlation between x, and y,"},{"Start":"03:13.640 ","End":"03:17.670","Text":"but we still can\u0027t describe the connection between x,"},{"Start":"03:17.670 ","End":"03:20.960","Text":"and y as a mathematical formula."},{"Start":"03:20.960 ","End":"03:25.010","Text":"Similarly here, that\u0027s the mirror image here between x,"},{"Start":"03:25.010 ","End":"03:29.010","Text":"and y where Rho equals minus 0.3,"},{"Start":"03:29.010 ","End":"03:32.805","Text":"and Rho equals minus 0.8 or x and y."},{"Start":"03:32.805 ","End":"03:39.360","Text":"We can see that there\u0027s a negative connection between x,"},{"Start":"03:39.360 ","End":"03:42.060","Text":"and y, as x increases,"},{"Start":"03:42.060 ","End":"03:49.490","Text":"y decreases, but we still can\u0027t describe the connection between x,"},{"Start":"03:49.490 ","End":"03:53.215","Text":"and y as a mathematical formula."},{"Start":"03:53.215 ","End":"03:55.595","Text":"Now, what else?"},{"Start":"03:55.595 ","End":"04:02.215","Text":"Well, we can see then that as we move away from 0,"},{"Start":"04:02.215 ","End":"04:04.590","Text":"and come closer to the edges,"},{"Start":"04:04.590 ","End":"04:09.080","Text":"and it doesn\u0027t matter whether we come closer to minus 1 or 1,"},{"Start":"04:09.080 ","End":"04:12.740","Text":"then the connection between x and y becomes stronger."},{"Start":"04:12.740 ","End":"04:15.550","Text":"From 0, that\u0027s no correlation,"},{"Start":"04:15.550 ","End":"04:19.580","Text":"to a perfect negative correlation here,"},{"Start":"04:19.580 ","End":"04:21.725","Text":"perfect positive correlation,"},{"Start":"04:21.725 ","End":"04:25.820","Text":"and similarly, as you move away from the edges and come"},{"Start":"04:25.820 ","End":"04:30.664","Text":"closer to the value of Rho equaling 0,"},{"Start":"04:30.664 ","End":"04:33.020","Text":"then the connection between x,"},{"Start":"04:33.020 ","End":"04:35.315","Text":"and y become weaker."},{"Start":"04:35.315 ","End":"04:38.795","Text":"Now, how do we calculate the correlation coefficient?"},{"Start":"04:38.795 ","End":"04:42.110","Text":"Well, the formula for the correlation coefficient is this,"},{"Start":"04:42.110 ","End":"04:45.840","Text":"that\u0027s Rho equaling covariance of x,"},{"Start":"04:45.840 ","End":"04:51.260","Text":"and y divided by sigma x times sigma y. Sigma x is a standard deviation of x,"},{"Start":"04:51.260 ","End":"04:54.260","Text":"and sigma y, that\u0027s the standard deviation of y."},{"Start":"04:54.260 ","End":"04:56.630","Text":"Now, what\u0027s the covariance?"},{"Start":"04:56.630 ","End":"05:01.520","Text":"Well, the covariance is the joint variance between x and y,"},{"Start":"05:01.520 ","End":"05:09.800","Text":"and that\u0027s calculated as the expectation of x minus mu of x times y minus mu of y."},{"Start":"05:09.800 ","End":"05:14.330","Text":"Now, after a little bit of algebraic manipulation,"},{"Start":"05:14.330 ","End":"05:18.165","Text":"that becomes the expectation of x times y"},{"Start":"05:18.165 ","End":"05:22.305","Text":"minus the expectation of x times the expectation of y."},{"Start":"05:22.305 ","End":"05:26.240","Text":"It\u0027s a little bit easier when working with covariances."},{"Start":"05:26.240 ","End":"05:31.055","Text":"Now, there are 2 major characteristics of the covariance that we need to know."},{"Start":"05:31.055 ","End":"05:33.920","Text":"1 is that the covariance of x,"},{"Start":"05:33.920 ","End":"05:35.060","Text":"and y, well,"},{"Start":"05:35.060 ","End":"05:37.700","Text":"that equals to the covariance of y, and x,"},{"Start":"05:37.700 ","End":"05:41.280","Text":"so it really doesn\u0027t matter whether you keep x,"},{"Start":"05:41.280 ","End":"05:42.705","Text":"and y in the same order,"},{"Start":"05:42.705 ","End":"05:44.525","Text":"or you switch them around,"},{"Start":"05:44.525 ","End":"05:48.770","Text":"it doesn\u0027t affect the connection between x,"},{"Start":"05:48.770 ","End":"05:57.165","Text":"and y. Secondly, the covariance of x with itself,"},{"Start":"05:57.165 ","End":"05:59.610","Text":"that equals to the variance of x,"},{"Start":"05:59.610 ","End":"06:02.989","Text":"and we can see that right here using this formula."},{"Start":"06:02.989 ","End":"06:06.165","Text":"If we substitute y equaling x,"},{"Start":"06:06.165 ","End":"06:13.565","Text":"that would be equal to the expectation of x squared minus the expectation squared of x,"},{"Start":"06:13.565 ","End":"06:16.025","Text":"and that equals to the variance."},{"Start":"06:16.025 ","End":"06:22.205","Text":"Let\u0027s take a look now at an example of how do we work with covariances,"},{"Start":"06:22.205 ","End":"06:26.550","Text":"and calculating the correlation coefficient."}],"ID":13193},{"Watched":false,"Name":"Tutorial 2","Duration":"2m 1s","ChapterTopicVideoID":12723,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12723.jpeg","UploadDate":"2018-07-30T15:02:02.0330000","DurationForVideoObject":"PT2M1S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.360","Text":"Here, I\u0027d like to talk about uncorrelated variables."},{"Start":"00:03.360 ","End":"00:08.550","Text":"Uncorrelated variables are variables whose correlation coefficient is equal to 0."},{"Start":"00:08.550 ","End":"00:10.635","Text":"That means Rho equals to 0."},{"Start":"00:10.635 ","End":"00:12.675","Text":"Now in order for this to be the case,"},{"Start":"00:12.675 ","End":"00:17.370","Text":"then their covariance must be equal to 0 as well. Now, why is that?"},{"Start":"00:17.370 ","End":"00:22.110","Text":"Well, Rho is defined as the covariance of x,"},{"Start":"00:22.110 ","End":"00:26.025","Text":"and y divided by Sigma x times Sigma y."},{"Start":"00:26.025 ","End":"00:28.335","Text":"Now, if that equals to 0,"},{"Start":"00:28.335 ","End":"00:34.515","Text":"then that implies that the covariance between x and y have to be equal to 0."},{"Start":"00:34.515 ","End":"00:37.890","Text":"Now, uncorrelated variables are variables"},{"Start":"00:37.890 ","End":"00:41.400","Text":"with no linear correlation whatsoever between them."},{"Start":"00:41.400 ","End":"00:43.670","Text":"That means when we are talking about correlation,"},{"Start":"00:43.670 ","End":"00:46.010","Text":"we\u0027re talking about linear dependence."},{"Start":"00:46.010 ","End":"00:49.130","Text":"Now, independent variables, on the other hand,"},{"Start":"00:49.130 ","End":"00:52.235","Text":"are variables with no connection whatsoever between them."},{"Start":"00:52.235 ","End":"00:55.820","Text":"Independent variables are therefore uncorrelated."},{"Start":"00:55.820 ","End":"01:00.065","Text":"Uncorrelated variables, however, are not necessarily independent."},{"Start":"01:00.065 ","End":"01:02.735","Text":"Now, let\u0027s just clarify that."},{"Start":"01:02.735 ","End":"01:05.210","Text":"Let\u0027s take 2 variables, for example,"},{"Start":"01:05.210 ","End":"01:07.875","Text":"y equals to x squared."},{"Start":"01:07.875 ","End":"01:09.975","Text":"This is a connection between them,"},{"Start":"01:09.975 ","End":"01:12.225","Text":"are they independent? Well, no."},{"Start":"01:12.225 ","End":"01:18.350","Text":"Because we have a definite mathematical formula that connects the 2."},{"Start":"01:18.350 ","End":"01:21.890","Text":"But Rho here is equal to 0."},{"Start":"01:21.890 ","End":"01:23.990","Text":"That means that they\u0027re uncorrelated."},{"Start":"01:23.990 ","End":"01:26.645","Text":"We can conclude that if we have 2 variables,"},{"Start":"01:26.645 ","End":"01:35.490","Text":"and they\u0027re independent, then obviously they\u0027re uncorrelated."},{"Start":"01:37.570 ","End":"01:40.760","Text":"But if they\u0027re uncorrelated,"},{"Start":"01:40.760 ","End":"01:42.880","Text":"well, are they independent?"},{"Start":"01:42.880 ","End":"01:45.285","Text":"Well, no. That\u0027s not true."},{"Start":"01:45.285 ","End":"01:48.090","Text":"Again, we saw this example right here."},{"Start":"01:48.090 ","End":"01:55.335","Text":"In our example, we had Rho equaling 0.853,"},{"Start":"01:55.335 ","End":"02:00.220","Text":"so x and y were correlated."}],"ID":13194},{"Watched":false,"Name":"Tutorial 3","Duration":"3m 37s","ChapterTopicVideoID":12724,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12724.jpeg","UploadDate":"2018-07-30T15:02:14.9270000","DurationForVideoObject":"PT3M37S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.710","Text":"Here I\u0027ll be talking about the effect of"},{"Start":"00:01.710 ","End":"00:04.830","Text":"a linear transformation to correlation coefficient."},{"Start":"00:04.830 ","End":"00:07.455","Text":"If we have a random variable X,"},{"Start":"00:07.455 ","End":"00:10.470","Text":"we have a linear transformation on X,"},{"Start":"00:10.470 ","End":"00:13.890","Text":"ax plus b, and similarly,"},{"Start":"00:13.890 ","End":"00:16.050","Text":"we have a random variable Y,"},{"Start":"00:16.050 ","End":"00:18.000","Text":"and we have a linear transformation on Y,"},{"Start":"00:18.000 ","End":"00:20.370","Text":"that\u0027ll be cy plus d,"},{"Start":"00:20.370 ","End":"00:26.195","Text":"then what\u0027s the correlation coefficient of the transformation"},{"Start":"00:26.195 ","End":"00:33.330","Text":"of ax plus b and cy plus d?"},{"Start":"00:33.330 ","End":"00:40.760","Text":"Well, we see here that that equals to the correlation coefficient of X and Y."},{"Start":"00:40.760 ","End":"00:48.620","Text":"The linear transformation doesn\u0027t affect the strength of the connection between X and Y."},{"Start":"00:48.620 ","End":"00:51.470","Text":"Now, what it affects,"},{"Start":"00:51.470 ","End":"00:57.845","Text":"it could affect the sign or the direction of the connection. Why is that?"},{"Start":"00:57.845 ","End":"01:01.835","Text":"Well, when a times c is greater than 0,"},{"Start":"01:01.835 ","End":"01:04.160","Text":"this A right here and this C right here,"},{"Start":"01:04.160 ","End":"01:07.355","Text":"that\u0027s the multipliers of the linear transformation."},{"Start":"01:07.355 ","End":"01:11.450","Text":"When the multiplication of these multipliers is positive,"},{"Start":"01:11.450 ","End":"01:14.200","Text":"then nothing happens to the correlation coefficient,"},{"Start":"01:14.200 ","End":"01:19.205","Text":"but if the multiplication of the multipliers is negative then,"},{"Start":"01:19.205 ","End":"01:24.185","Text":"that changes the sign of the correlation coefficient."},{"Start":"01:24.185 ","End":"01:29.330","Text":"But otherwise, the strength of the connection stays the same."},{"Start":"01:29.330 ","End":"01:32.150","Text":"The linear transformation of 2 variables does not"},{"Start":"01:32.150 ","End":"01:35.020","Text":"change the strength of the connection between them,"},{"Start":"01:35.020 ","End":"01:39.200","Text":"it may only change the direction of the connection."},{"Start":"01:39.200 ","End":"01:41.695","Text":"Let\u0027s take a look at an example,"},{"Start":"01:41.695 ","End":"01:44.060","Text":"our example from the courses."},{"Start":"01:44.060 ","End":"01:49.310","Text":"Now, every course the student passes entitles him to 3 academic points, and we\u0027re asked,"},{"Start":"01:49.310 ","End":"01:52.130","Text":"what\u0027s the correlation coefficient between the number"},{"Start":"01:52.130 ","End":"01:55.355","Text":"of academic points earned by the student and Y?"},{"Start":"01:55.355 ","End":"01:59.915","Text":"Well, let\u0027s define w as 3 times x."},{"Start":"01:59.915 ","End":"02:02.300","Text":"That\u0027s the academic points."},{"Start":"02:02.300 ","End":"02:07.190","Text":"X would be the number of courses passed and 3 times x"},{"Start":"02:07.190 ","End":"02:12.485","Text":"would be then the academic points that he earned for each course passed."},{"Start":"02:12.485 ","End":"02:18.485","Text":"We\u0027re asked for Rho of 3x and y."},{"Start":"02:18.485 ","End":"02:27.680","Text":"Well, let\u0027s just recall what the correlation coefficient is of a linear transformation."},{"Start":"02:27.680 ","End":"02:32.969","Text":"Here we go. We see that the correlation coefficient of a linear transformation,"},{"Start":"02:32.969 ","End":"02:35.840","Text":"that equals to the correlation coefficient of"},{"Start":"02:35.840 ","End":"02:39.215","Text":"the variables before the linear transformation"},{"Start":"02:39.215 ","End":"02:45.890","Text":"if the multiplication of the multipliers of the linear transformation is greater than 0."},{"Start":"02:45.890 ","End":"02:47.555","Text":"If it\u0027s less than 0,"},{"Start":"02:47.555 ","End":"02:50.155","Text":"all we have to do is switch signs here."},{"Start":"02:50.155 ","End":"02:55.170","Text":"In our case, we have here X,"},{"Start":"02:55.170 ","End":"03:00.405","Text":"and that\u0027s transformed into 3x plus 0."},{"Start":"03:00.405 ","End":"03:04.365","Text":"Y, that remains the same."},{"Start":"03:04.365 ","End":"03:08.325","Text":"But we can write it as 1y plus 0."},{"Start":"03:08.325 ","End":"03:16.805","Text":"That means that A here would be equal to 3 and C would be equal to 1."},{"Start":"03:16.805 ","End":"03:22.130","Text":"So 8 times c equals to 3 times 1 and that equals to 3,"},{"Start":"03:22.130 ","End":"03:25.175","Text":"and that obviously is greater than 0."},{"Start":"03:25.175 ","End":"03:32.040","Text":"This means that this equals to the correlation coefficient of X and Y,"},{"Start":"03:32.040 ","End":"03:37.120","Text":"and that equals to 0.853."}],"ID":13195},{"Watched":false,"Name":"Example","Duration":"8m 17s","ChapterTopicVideoID":12725,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12725.jpeg","UploadDate":"2018-07-30T15:02:38.3700000","DurationForVideoObject":"PT8M17S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.180","Text":"I\u0027ll be using the example from the previous chapter,"},{"Start":"00:03.180 ","End":"00:05.910","Text":"so I assume that you\u0027re familiar with that."},{"Start":"00:05.910 ","End":"00:10.575","Text":"I\u0027ll be going very quickly in describing the story again."},{"Start":"00:10.575 ","End":"00:15.270","Text":"Now, a student takes semester exams 1 in economics and 1 in statistics."},{"Start":"00:15.270 ","End":"00:18.660","Text":"Assume that the chances of passing the economics exam is"},{"Start":"00:18.660 ","End":"00:23.790","Text":"0.8 and the chances of passing the statistics exam are 0.9,"},{"Start":"00:23.790 ","End":"00:28.300","Text":"the chances of passing both exams are 0.75."},{"Start":"00:28.300 ","End":"00:31.970","Text":"Now let X be the number of courses that the student passes,"},{"Start":"00:31.970 ","End":"00:35.630","Text":"and Y is an indicator variable that receives the value 1"},{"Start":"00:35.630 ","End":"00:40.330","Text":"if the student passes the economics exam and 0 if he doesn\u0027t."},{"Start":"00:40.330 ","End":"00:44.345","Text":"Here\u0027s the joint distribution function of x and y."},{"Start":"00:44.345 ","End":"00:47.630","Text":"Here again, we see the values of x at 0, 1,"},{"Start":"00:47.630 ","End":"00:49.850","Text":"and 2, and the values for y,"},{"Start":"00:49.850 ","End":"00:51.355","Text":"that\u0027s 0 and 1."},{"Start":"00:51.355 ","End":"00:53.960","Text":"We have within the table right here,"},{"Start":"00:53.960 ","End":"00:56.105","Text":"that\u0027s the joint probabilities."},{"Start":"00:56.105 ","End":"00:57.890","Text":"Now, in blue here,"},{"Start":"00:57.890 ","End":"01:00.830","Text":"that\u0027s a marginal probabilities of X."},{"Start":"01:00.830 ","End":"01:04.745","Text":"In orange, we see the marginal probabilities of Y."},{"Start":"01:04.745 ","End":"01:09.470","Text":"Now, we want to calculate the correlation coefficient of x and y."},{"Start":"01:09.470 ","End":"01:10.850","Text":"Now, what\u0027s that?"},{"Start":"01:10.850 ","End":"01:14.690","Text":"Well, that\u0027s defined as the covariance of X and Y"},{"Start":"01:14.690 ","End":"01:20.430","Text":"divided by the standard deviation of x times the standard deviation of y."},{"Start":"01:20.510 ","End":"01:24.655","Text":"In order to make this calculation easier."},{"Start":"01:24.655 ","End":"01:31.080","Text":"Then, let\u0027s first calculate the standard deviation of X and the standard deviation of Y."},{"Start":"01:31.100 ","End":"01:35.960","Text":"Here I just copied the marginal probability of X,"},{"Start":"01:35.960 ","End":"01:38.540","Text":"so that\u0027s the probability function of X."},{"Start":"01:38.540 ","End":"01:40.595","Text":"Now, what\u0027s the expectation of X?"},{"Start":"01:40.595 ","End":"01:43.965","Text":"Well, that\u0027s x times its probability."},{"Start":"01:43.965 ","End":"01:50.520","Text":"That\u0027s 0 times 0.05 plus 1 times 0.2 plus 2 times 0.75."},{"Start":"01:50.520 ","End":"01:52.275","Text":"We see the calculation here."},{"Start":"01:52.275 ","End":"01:55.415","Text":"The expectation of x is 1.7."},{"Start":"01:55.415 ","End":"01:57.260","Text":"Now what about the variance?"},{"Start":"01:57.260 ","End":"02:06.945","Text":"Well, that\u0027s the sum or the expectation of X squared minus the expectation squared of X."},{"Start":"02:06.945 ","End":"02:10.830","Text":"That will be equal to 0 squared times 0.05 plus"},{"Start":"02:10.830 ","End":"02:15.210","Text":"1 squared times 0.2 plus 2 squared times 0.75."},{"Start":"02:15.210 ","End":"02:21.220","Text":"Don\u0027t forget, we have to take away the square of the expectation."},{"Start":"02:21.220 ","End":"02:23.880","Text":"Here\u0027s the calculation right here."},{"Start":"02:23.880 ","End":"02:26.030","Text":"That comes out to 0.31."},{"Start":"02:26.030 ","End":"02:29.525","Text":"Now that\u0027s the variance and we want the standard deviation."},{"Start":"02:29.525 ","End":"02:36.790","Text":"Now the standard deviation is the square root of the variance and that equals to 0.557."},{"Start":"02:36.790 ","End":"02:44.540","Text":"Similarly, we\u0027ll calculate the expectation and variance of Y."},{"Start":"02:44.540 ","End":"02:47.300","Text":"Now here\u0027s the marginal probability of Y,"},{"Start":"02:47.300 ","End":"02:50.510","Text":"so that\u0027s the probability function of y."},{"Start":"02:50.510 ","End":"02:53.615","Text":"Now, the expectation of Y,"},{"Start":"02:53.615 ","End":"02:57.500","Text":"again, that\u0027s the value of Y times its probability."},{"Start":"02:57.500 ","End":"03:03.660","Text":"That\u0027s 0 times 0.2 plus 1 times 0.8 and that equals to 0.8."},{"Start":"03:03.660 ","End":"03:05.210","Text":"What about the variance?"},{"Start":"03:05.210 ","End":"03:10.760","Text":"Well, that\u0027s the expectation of X squared minus the expectation squared of X."},{"Start":"03:10.760 ","End":"03:16.370","Text":"That\u0027ll be equal to 0 squared times 0.2 plus 1 squared"},{"Start":"03:16.370 ","End":"03:21.945","Text":"times 0.8 minus the expectation squared."},{"Start":"03:21.945 ","End":"03:24.345","Text":"That\u0027s this squared."},{"Start":"03:24.345 ","End":"03:26.220","Text":"Here\u0027s the calculation."},{"Start":"03:26.220 ","End":"03:28.725","Text":"That turns out to be 0.16."},{"Start":"03:28.725 ","End":"03:29.880","Text":"Now that\u0027s the variance."},{"Start":"03:29.880 ","End":"03:33.200","Text":"The standard deviation is the square root of the variance,"},{"Start":"03:33.200 ","End":"03:36.220","Text":"and that turns out to be 0.4."},{"Start":"03:36.220 ","End":"03:43.670","Text":"Here we have the expectation and variance of Y. We have that."},{"Start":"03:43.670 ","End":"03:48.960","Text":"Now let\u0027s go and calculate the covariance of X and Y."},{"Start":"03:49.520 ","End":"03:54.950","Text":"The covariance of X and Y is defined as the expectation of"},{"Start":"03:54.950 ","End":"04:00.505","Text":"X times Y minus the expectation of X times the expectation of Y."},{"Start":"04:00.505 ","End":"04:05.330","Text":"That\u0027s fine. We have the expectation of X and we have the expectation of Y."},{"Start":"04:05.330 ","End":"04:06.905","Text":"We just calculated that."},{"Start":"04:06.905 ","End":"04:09.740","Text":"Now what about the expectation of X times Y?"},{"Start":"04:09.740 ","End":"04:11.585","Text":"What\u0027s this guy right here?"},{"Start":"04:11.585 ","End":"04:18.305","Text":"Well, that\u0027s defined as the sum over all combinations of x and y"},{"Start":"04:18.305 ","End":"04:27.440","Text":"of the values of X times Y times the joint probability of X and Y to values."},{"Start":"04:27.440 ","End":"04:29.285","Text":"Now, what does that mean?"},{"Start":"04:29.285 ","End":"04:32.435","Text":"That means that we go now for,"},{"Start":"04:32.435 ","End":"04:36.575","Text":"let\u0027s say x equals 0 and y equals."},{"Start":"04:36.575 ","End":"04:46.290","Text":"It\u0027ll be 0 times 0 times the joint probability of X0 and Y0, that\u0027s 0.05."},{"Start":"04:46.290 ","End":"04:54.095","Text":"It\u0027ll be 0 times 0 times 0.5 plus 0 times 1 times 0."},{"Start":"04:54.095 ","End":"05:02.465","Text":"X equals 0 times Y equaling 1 times the joint probability of x and y at 0 and 1."},{"Start":"05:02.465 ","End":"05:06.065","Text":"Now, to make things simpler,"},{"Start":"05:06.065 ","End":"05:12.215","Text":"we can get rid of this column right here and this row right here. Why is that?"},{"Start":"05:12.215 ","End":"05:18.260","Text":"We have a multiplication by 0."},{"Start":"05:18.260 ","End":"05:21.335","Text":"Here\u0027s the full calculation right here."},{"Start":"05:21.335 ","End":"05:27.660","Text":"But in essence, it\u0027ll be just 1 times 1 times 0.5."},{"Start":"05:27.660 ","End":"05:32.390","Text":"X equals 1 times Y equaling 1 times 0.05."},{"Start":"05:32.390 ","End":"05:34.250","Text":"That\u0027s the joint probability,"},{"Start":"05:34.250 ","End":"05:39.580","Text":"plus 2 times 1 times 0.75."},{"Start":"05:39.580 ","End":"05:43.715","Text":"This calculation comes out to 1.55."},{"Start":"05:43.715 ","End":"05:46.400","Text":"That\u0027s the expectation of X times Y."},{"Start":"05:46.400 ","End":"05:49.190","Text":"Again, the covariance of x and y,"},{"Start":"05:49.190 ","End":"05:52.100","Text":"well that equals to the expectation of X times Y"},{"Start":"05:52.100 ","End":"05:56.135","Text":"minus the expectation of X times the expectation of Y."},{"Start":"05:56.135 ","End":"05:57.560","Text":"Now, that equals to,"},{"Start":"05:57.560 ","End":"05:59.470","Text":"let\u0027s just plug in the numbers."},{"Start":"05:59.470 ","End":"06:03.555","Text":"That equals to 1.55 minus,"},{"Start":"06:03.555 ","End":"06:06.470","Text":"1.55 is the expectation of X,"},{"Start":"06:06.470 ","End":"06:10.280","Text":"Y minus the expectation of X times the expectation of Y."},{"Start":"06:10.280 ","End":"06:14.095","Text":"Now that comes out to 0.19."},{"Start":"06:14.095 ","End":"06:19.025","Text":"We have here the covariance of X and Y."},{"Start":"06:19.025 ","End":"06:22.355","Text":"Now what about the correlation coefficient that\u0027s rho?"},{"Start":"06:22.355 ","End":"06:26.360","Text":"Well, that\u0027s defined as the covariance of X and Y divided"},{"Start":"06:26.360 ","End":"06:30.605","Text":"by the standard deviation of x times the standard deviation of 1,"},{"Start":"06:30.605 ","End":"06:37.355","Text":"that equals to 0.19 divided by 0.557 times 0.4,"},{"Start":"06:37.355 ","End":"06:41.645","Text":"and that equals to 0.853."},{"Start":"06:41.645 ","End":"06:49.800","Text":"This then is the correlation coefficient of the variables x and y."},{"Start":"06:50.240 ","End":"06:56.945","Text":"What do we mean when we say that we have a correlation coefficient of 0.853."},{"Start":"06:56.945 ","End":"06:58.580","Text":"Well, that means that there\u0027s"},{"Start":"06:58.580 ","End":"07:05.095","Text":"a strong partial and positive linear correlation between X and Y."},{"Start":"07:05.095 ","End":"07:11.460","Text":"As X increases, we expect Y to increase and the other way round,"},{"Start":"07:11.460 ","End":"07:14.935","Text":"as Y increases, we expect X to increase."},{"Start":"07:14.935 ","End":"07:19.570","Text":"But there\u0027s no mathematical formula that can describe this connection."},{"Start":"07:19.570 ","End":"07:22.375","Text":"Now, let\u0027s just see why."},{"Start":"07:22.375 ","End":"07:25.555","Text":"Let\u0027s take a look at the definition of X."},{"Start":"07:25.555 ","End":"07:29.410","Text":"Well, X is the number of courses that a student passes."},{"Start":"07:29.410 ","End":"07:31.810","Text":"That\u0027s this thing right here. What about Y?"},{"Start":"07:31.810 ","End":"07:35.200","Text":"Where Y is the indicator variable that receives"},{"Start":"07:35.200 ","End":"07:39.700","Text":"the value one if the student passes the economics exam."},{"Start":"07:39.700 ","End":"07:43.480","Text":"As Y moves from 0-1,"},{"Start":"07:43.480 ","End":"07:48.620","Text":"as Y increases, we expect that X will increase as well."},{"Start":"07:48.620 ","End":"07:53.660","Text":"But again, there\u0027s no mathematical formula to perfectly describe this connection."},{"Start":"07:53.660 ","End":"08:00.380","Text":"There\u0027s no mathematical formula that deterministically shows this connection."},{"Start":"08:00.380 ","End":"08:04.850","Text":"On the other hand, we see that we do have a strong connection."},{"Start":"08:04.850 ","End":"08:10.260","Text":"Its measure is 0.853."},{"Start":"08:10.260 ","End":"08:17.340","Text":"That\u0027s the meaning of the correlation coefficient, being 0.853."}],"ID":13196},{"Watched":false,"Name":"Exercise 1 - Part a","Duration":"10m 19s","ChapterTopicVideoID":12726,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12726.jpeg","UploadDate":"2018-07-30T15:03:13.6200000","DurationForVideoObject":"PT10M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In this question, we\u0027re given that the chances of a student"},{"Start":"00:02.850 ","End":"00:06.930","Text":"passing the statistics exam on the first try are 0.8."},{"Start":"00:06.930 ","End":"00:08.520","Text":"If he fails on the first try,"},{"Start":"00:08.520 ","End":"00:12.810","Text":"he takes the exam again on which the chances of passing is 0.9."},{"Start":"00:12.810 ","End":"00:14.940","Text":"A student who passes on the first try,"},{"Start":"00:14.940 ","End":"00:16.680","Text":"doesn\u0027t take another exam."},{"Start":"00:16.680 ","End":"00:19.155","Text":"If the student fails on the second try,"},{"Start":"00:19.155 ","End":"00:22.560","Text":"he files a request to take the exam for the third time."},{"Start":"00:22.560 ","End":"00:26.300","Text":"The chances of the request being approved are 0.2"},{"Start":"00:26.300 ","End":"00:30.260","Text":"and the chances of actually passing it are 0.7."},{"Start":"00:30.260 ","End":"00:32.870","Text":"Now, we define the following variables."},{"Start":"00:32.870 ","End":"00:36.005","Text":"X is the number of exams taken by the student,"},{"Start":"00:36.005 ","End":"00:39.420","Text":"and Y is the number of exams on which he failed."},{"Start":"00:39.420 ","End":"00:43.580","Text":"We\u0027re asked to construct a joint probability function of X and Y,"},{"Start":"00:43.580 ","End":"00:46.970","Text":"and their marginal probability functions."},{"Start":"00:46.970 ","End":"00:50.419","Text":"The first thing that we need to do is to describe the data,"},{"Start":"00:50.419 ","End":"00:54.500","Text":"and the best way to do that is through a probability tree. Why is that?"},{"Start":"00:54.500 ","End":"01:00.635","Text":"Well, the probability tree provides us a chance to describe the data in stages,"},{"Start":"01:00.635 ","End":"01:03.230","Text":"and this is exactly what we have in the story."},{"Start":"01:03.230 ","End":"01:07.690","Text":"First of all, let\u0027s construct a probability tree."},{"Start":"01:07.690 ","End":"01:10.755","Text":"Here\u0027s a probability tree."},{"Start":"01:10.755 ","End":"01:13.970","Text":"Now for the first time that he takes the exam, well,"},{"Start":"01:13.970 ","End":"01:18.215","Text":"the probability of passing is 0.8."},{"Start":"01:18.215 ","End":"01:21.410","Text":"Now if he passes the exam, then he stops."},{"Start":"01:21.410 ","End":"01:25.205","Text":"But there\u0027s a 20 percent chance that he fails."},{"Start":"01:25.205 ","End":"01:29.740","Text":"If he fails, then he takes the exam for the second time."},{"Start":"01:29.740 ","End":"01:35.340","Text":"There\u0027s a 90 percent chance of passing on"},{"Start":"01:35.340 ","End":"01:41.025","Text":"the second try and if he passes the exam then he stops."},{"Start":"01:41.025 ","End":"01:42.450","Text":"But if he fails, well,"},{"Start":"01:42.450 ","End":"01:44.820","Text":"that\u0027s a 10 percent chance of failing,"},{"Start":"01:44.820 ","End":"01:46.430","Text":"and if he fails,"},{"Start":"01:46.430 ","End":"01:51.710","Text":"then he has to get the approval of a committee to take the exam for the third try."},{"Start":"01:51.710 ","End":"01:56.450","Text":"So the third step here is getting an approval and the probability of getting an approval,"},{"Start":"01:56.450 ","End":"02:01.155","Text":"that\u0027s 0.2, that\u0027s to get an approval."},{"Start":"02:01.155 ","End":"02:07.605","Text":"There\u0027s a 0.8 chance of not getting an approval."},{"Start":"02:07.605 ","End":"02:09.860","Text":"If he doesn\u0027t get an approval,"},{"Start":"02:09.860 ","End":"02:12.879","Text":"he stops here, but if he does get an approval,"},{"Start":"02:12.879 ","End":"02:17.379","Text":"well then he has to take the exam for the third time,"},{"Start":"02:17.379 ","End":"02:23.180","Text":"and the probability of passing on the third try that\u0027s 70 percent."},{"Start":"02:23.180 ","End":"02:27.600","Text":"So there\u0027s a 30 percent chance of failing."},{"Start":"02:28.280 ","End":"02:32.940","Text":"This then is our probability tree."},{"Start":"02:32.940 ","End":"02:36.705","Text":"Let\u0027s see about our variables."},{"Start":"02:36.705 ","End":"02:40.160","Text":"X is the number of exams taken by"},{"Start":"02:40.160 ","End":"02:44.225","Text":"the student and Y is the number of exams on which he failed."},{"Start":"02:44.225 ","End":"02:51.275","Text":"Here we\u0027ll put X and Y besides the ends of the branches."},{"Start":"02:51.275 ","End":"02:55.475","Text":"We have an end right here where he quits on the first try,"},{"Start":"02:55.475 ","End":"02:59.255","Text":"the end of this branch when he passes on the second try,"},{"Start":"02:59.255 ","End":"03:03.500","Text":"the end on this branch right here where he didn\u0027t"},{"Start":"03:03.500 ","End":"03:07.850","Text":"get an approval to go to take the exam for the third try."},{"Start":"03:07.850 ","End":"03:11.405","Text":"The end here where he passed the exam on the third try"},{"Start":"03:11.405 ","End":"03:15.250","Text":"or the end here where he failed on the third try."},{"Start":"03:15.250 ","End":"03:18.105","Text":"As we said, X is the number of exams taken."},{"Start":"03:18.105 ","End":"03:20.835","Text":"Here, X would be equal to 1,"},{"Start":"03:20.835 ","End":"03:23.450","Text":"he passed in the first try and that\u0027s it."},{"Start":"03:23.450 ","End":"03:25.600","Text":"Here, he passed in the second try,"},{"Start":"03:25.600 ","End":"03:28.260","Text":"so X equals to 2."},{"Start":"03:28.460 ","End":"03:30.990","Text":"What happens here? Well,"},{"Start":"03:30.990 ","End":"03:33.992","Text":"he took the test for the second time but"},{"Start":"03:33.992 ","End":"03:36.769","Text":"he didn\u0027t get an approval to take it for the third time,"},{"Start":"03:36.769 ","End":"03:39.565","Text":"so X here equals to 2,"},{"Start":"03:39.565 ","End":"03:43.410","Text":"and here X equals to 3,"},{"Start":"03:43.410 ","End":"03:46.470","Text":"and same here, X equals to 3."},{"Start":"03:46.470 ","End":"03:48.060","Text":"Now what about Y?"},{"Start":"03:48.060 ","End":"03:51.280","Text":"Y is the number of exams in which he failed."},{"Start":"03:51.280 ","End":"03:55.030","Text":"Now here, he didn\u0027t fail,"},{"Start":"03:55.030 ","End":"04:00.340","Text":"so Y here equals to 0. What about here?"},{"Start":"04:00.340 ","End":"04:02.470","Text":"Well, he passed on the second try,"},{"Start":"04:02.470 ","End":"04:04.450","Text":"that means that he failed on the first try,"},{"Start":"04:04.450 ","End":"04:06.880","Text":"so Y here equals to 1."},{"Start":"04:06.880 ","End":"04:09.440","Text":"What about here?"},{"Start":"04:09.830 ","End":"04:14.925","Text":"Well here, he failed on the second try,"},{"Start":"04:14.925 ","End":"04:18.494","Text":"and he didn\u0027t get approval to go to the third exam,"},{"Start":"04:18.494 ","End":"04:22.215","Text":"so Y here equals to 2."},{"Start":"04:22.215 ","End":"04:26.415","Text":"Here he passed on the third try,"},{"Start":"04:26.415 ","End":"04:28.875","Text":"so Y here equals to 2,"},{"Start":"04:28.875 ","End":"04:32.500","Text":"and here Y equals to 3."},{"Start":"04:34.130 ","End":"04:38.435","Text":"Now we know what the values of X and Y are,"},{"Start":"04:38.435 ","End":"04:40.640","Text":"what are the valid values of X and Y,"},{"Start":"04:40.640 ","End":"04:47.940","Text":"so we can start to construct our joint distribution for X and Y."},{"Start":"04:48.520 ","End":"04:53.735","Text":"Here is the empty joint probability function."},{"Start":"04:53.735 ","End":"04:55.880","Text":"That\u0027s the empty matrix right here."},{"Start":"04:55.880 ","End":"04:58.670","Text":"These are the valid values of Y 0,"},{"Start":"04:58.670 ","End":"05:00.050","Text":"1, 2, and 3."},{"Start":"05:00.050 ","End":"05:01.920","Text":"Y equals 0, Y equals 1,"},{"Start":"05:01.920 ","End":"05:04.490","Text":"2, and 3."},{"Start":"05:04.490 ","End":"05:06.590","Text":"Here are the valid values of X."},{"Start":"05:06.590 ","End":"05:10.275","Text":"That\u0027s 1, 2, 3, that\u0027s 1, 2, and 3."},{"Start":"05:10.275 ","End":"05:14.090","Text":"Here we have the marginal probability of Y,"},{"Start":"05:14.090 ","End":"05:17.260","Text":"and here we\u0027ll have the marginal probability of X."},{"Start":"05:17.260 ","End":"05:21.320","Text":"Let\u0027s just fill in the joint probabilities,"},{"Start":"05:21.320 ","End":"05:23.855","Text":"that\u0027s these white cells within the table,"},{"Start":"05:23.855 ","End":"05:28.910","Text":"and they\u0027ll be based on the probability tree right here."},{"Start":"05:30.980 ","End":"05:38.465","Text":"Where do we have probabilities where we\u0027re talking about the values of X and Y?"},{"Start":"05:38.465 ","End":"05:44.270","Text":"Well, we have values where X equals 1 and Y equals 0, that\u0027s right here."},{"Start":"05:44.270 ","End":"05:50.265","Text":"The probability of X equaling 1 and Y equaling 0,"},{"Start":"05:50.265 ","End":"05:54.540","Text":"well, that equals to 0.8."},{"Start":"05:54.540 ","End":"05:57.390","Text":"That\u0027s this branch right here."},{"Start":"05:57.390 ","End":"06:02.690","Text":"That\u0027s the only place where we have X equals 1 and Y equals 0."},{"Start":"06:02.690 ","End":"06:05.705","Text":"We\u0027ll put that here, that\u0027ll be 0.8."},{"Start":"06:05.705 ","End":"06:13.740","Text":"Now, what about Y equals 1 and X equals 2?"},{"Start":"06:13.740 ","End":"06:15.630","Text":"That\u0027s this cell right here."},{"Start":"06:15.630 ","End":"06:22.330","Text":"The probability of X equaling 2 and Y equaling 1,"},{"Start":"06:22.330 ","End":"06:24.675","Text":"well, that\u0027s this branch right here."},{"Start":"06:24.675 ","End":"06:27.630","Text":"So we have to go through 0.2 and 0.9,"},{"Start":"06:27.630 ","End":"06:30.975","Text":"so that\u0027ll be 0.2 times 0.9,"},{"Start":"06:30.975 ","End":"06:34.239","Text":"that equals to 0.18."},{"Start":"06:34.239 ","End":"06:39.510","Text":"That\u0027s what we\u0027re going to write here, 0.18."},{"Start":"06:40.000 ","End":"06:43.050","Text":"We did this and we did this."},{"Start":"06:43.050 ","End":"06:45.016","Text":"Now what about here,"},{"Start":"06:45.016 ","End":"06:52.770","Text":"the probability of X equaling 2 and Y equaling 2?"},{"Start":"06:52.770 ","End":"06:55.890","Text":"Well, that\u0027s this cell right here."},{"Start":"06:55.890 ","End":"07:02.863","Text":"Now that equals to 0.2 times 0.1 times 0.8."},{"Start":"07:02.863 ","End":"07:07.635","Text":"0.2 times 0.1 times 0.8,"},{"Start":"07:07.635 ","End":"07:12.540","Text":"that equals to 0.016."},{"Start":"07:12.540 ","End":"07:14.749","Text":"We\u0027ll write that here,"},{"Start":"07:14.749 ","End":"07:19.995","Text":"that\u0027s 0.016. We did this."},{"Start":"07:19.995 ","End":"07:22.285","Text":"Now what about X equals 3,"},{"Start":"07:22.285 ","End":"07:27.997","Text":"probability of X equaling 3 and Y equaling 2?"},{"Start":"07:27.997 ","End":"07:32.480","Text":"That\u0027ll be right here in this cell right here."},{"Start":"07:33.260 ","End":"07:37.830","Text":"X equals 3 and Y equals 2, that\u0027s right here."},{"Start":"07:37.830 ","End":"07:41.500","Text":"We\u0027re going to have to go through this branch and this branch,"},{"Start":"07:41.500 ","End":"07:44.590","Text":"and then this branch and this branch right here."},{"Start":"07:44.590 ","End":"07:48.050","Text":"So that\u0027s 0.2 times 0.1"},{"Start":"07:48.860 ","End":"07:56.070","Text":"times 0.2 times 0.7."},{"Start":"07:56.070 ","End":"08:00.550","Text":"Now that equals to 0.0028."},{"Start":"08:02.720 ","End":"08:07.000","Text":"Let\u0027s just clean that up right here."},{"Start":"08:07.000 ","End":"08:10.795","Text":"That will be 0.0028."},{"Start":"08:10.795 ","End":"08:13.180","Text":"We\u0027ll write that right here."},{"Start":"08:13.180 ","End":"08:18.400","Text":"X equals 3 and Y equals to 2, 0.0028."},{"Start":"08:18.740 ","End":"08:21.815","Text":"Now, what about this guy right here?"},{"Start":"08:21.815 ","End":"08:28.495","Text":"Well, that\u0027s the probability of X equaling 3 and Y equaling 3."},{"Start":"08:28.495 ","End":"08:30.380","Text":"Now, what does that equal to?"},{"Start":"08:30.380 ","End":"08:33.214","Text":"Well, that\u0027s this branch and this branch,"},{"Start":"08:33.214 ","End":"08:35.470","Text":"and then this and this branch."},{"Start":"08:35.470 ","End":"08:36.680","Text":"So it will be 0.2 times"},{"Start":"08:36.680 ","End":"08:44.730","Text":"0.1 times"},{"Start":"08:44.730 ","End":"08:47.453","Text":"0.2 times 0.3,"},{"Start":"08:47.453 ","End":"08:51.495","Text":"and that equals to 0.0012."},{"Start":"08:51.495 ","End":"08:56.140","Text":"We\u0027ll write that down here, that\u0027s 0.0012."},{"Start":"08:57.290 ","End":"09:02.225","Text":"Now all the other places we don\u0027t have"},{"Start":"09:02.225 ","End":"09:08.510","Text":"probabilities for the values of X and Y or the joint probabilities for X and Y."},{"Start":"09:08.510 ","End":"09:11.610","Text":"That means that they\u0027re all 0."},{"Start":"09:13.820 ","End":"09:16.295","Text":"Now having said that,"},{"Start":"09:16.295 ","End":"09:21.425","Text":"let\u0027s calculate the marginal probabilities of X and Y."},{"Start":"09:21.425 ","End":"09:24.955","Text":"Well, here that\u0027ll be 0.8."},{"Start":"09:24.955 ","End":"09:29.955","Text":"Here, that\u0027ll be 0.196."},{"Start":"09:29.955 ","End":"09:34.680","Text":"This here would be 0.004."},{"Start":"09:34.680 ","End":"09:37.530","Text":"Here that\u0027ll be 0.8."},{"Start":"09:37.530 ","End":"09:41.290","Text":"Here that\u0027ll be 0.18."},{"Start":"09:41.620 ","End":"09:49.205","Text":"Here it will be 0.0188,"},{"Start":"09:49.205 ","End":"09:56.210","Text":"and the last cell right here will be 0.0012."},{"Start":"09:56.330 ","End":"10:00.110","Text":"Last check, we have to check that the sum"},{"Start":"10:00.110 ","End":"10:03.152","Text":"of the marginal probabilities have to add up to 1,"},{"Start":"10:03.152 ","End":"10:04.640","Text":"and the sum of"},{"Start":"10:04.640 ","End":"10:10.750","Text":"all the joint probabilities they have to add up to 1 as well, and they do."},{"Start":"10:10.750 ","End":"10:14.840","Text":"Here then is the joint probability function"},{"Start":"10:14.840 ","End":"10:19.620","Text":"of X and Y and their marginal probability functions."}],"ID":13197},{"Watched":false,"Name":"Exercise 1 - Parts b-c","Duration":"4m 57s","ChapterTopicVideoID":12727,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12727.jpeg","UploadDate":"2018-07-30T15:03:26.6870000","DurationForVideoObject":"PT4M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"In this section, we\u0027re asked whether X and Y are independent variables."},{"Start":"00:04.110 ","End":"00:08.475","Text":"Well, let\u0027s just recall the definition of an independent variable."},{"Start":"00:08.475 ","End":"00:14.370","Text":"An independent variable is 1 where the probability of X and Y,"},{"Start":"00:14.370 ","End":"00:16.782","Text":"that\u0027s the joint probability of X and Y,"},{"Start":"00:16.782 ","End":"00:22.020","Text":"that equals to the probability of X times the probability of Y."},{"Start":"00:22.020 ","End":"00:24.540","Text":"The joint probability has to equal to"},{"Start":"00:24.540 ","End":"00:29.620","Text":"the marginal probability at X and a times the marginal probability at Y."},{"Start":"00:29.620 ","End":"00:33.520","Text":"Now, that\u0027s for all combinations of X and Y."},{"Start":"00:33.520 ","End":"00:37.985","Text":"If we find just 1 combination where this doesn\u0027t hold true,"},{"Start":"00:37.985 ","End":"00:40.640","Text":"then X and Y are not independent."},{"Start":"00:40.640 ","End":"00:43.625","Text":"That means that X and Y are dependent variables."},{"Start":"00:43.625 ","End":"00:54.075","Text":"Let\u0027s now take a look at the probability where X equals 1 and Y equals to 2."},{"Start":"00:54.075 ","End":"01:00.230","Text":"Now, what\u0027s the joint probability where X equals 1 and Y equals 2."},{"Start":"01:00.230 ","End":"01:04.760","Text":"X equals 1, Y equals 2, that equals to 0."},{"Start":"01:04.760 ","End":"01:13.850","Text":"Now, that has to equal to the probability of x at 1 times the probability of y at 2."},{"Start":"01:13.850 ","End":"01:17.390","Text":"Now the probability of x at 1,"},{"Start":"01:17.390 ","End":"01:19.820","Text":"the marginal probability where x equals 1,"},{"Start":"01:19.820 ","End":"01:25.775","Text":"well that\u0027s 0.8 times the marginal probability of y at 2."},{"Start":"01:25.775 ","End":"01:30.520","Text":"Well, that\u0027s 0.0188."},{"Start":"01:30.520 ","End":"01:36.920","Text":"Now, these are 2 numbers which are positive, they\u0027re positive integers."},{"Start":"01:36.920 ","End":"01:42.875","Text":"Now the multiplication of 2 positive integers,"},{"Start":"01:42.875 ","End":"01:47.170","Text":"well, the product will never be equal to 0."},{"Start":"01:47.170 ","End":"01:53.870","Text":"That just proves that there\u0027s 1 case where this does not hold true."},{"Start":"01:53.870 ","End":"02:03.310","Text":"That means that X and Y are dependent, dependent variables."},{"Start":"02:03.310 ","End":"02:08.135","Text":"In this section, it\u0027s known that the students took more than 1 exam and we\u0027re asked,"},{"Start":"02:08.135 ","End":"02:12.260","Text":"what\u0027s the probability that he failed less than 3 exams?"},{"Start":"02:12.260 ","End":"02:15.140","Text":"This is conditional probabilities."},{"Start":"02:15.140 ","End":"02:17.900","Text":"Let\u0027s just construct the probability here."},{"Start":"02:17.900 ","End":"02:19.565","Text":"That\u0027s the probability."},{"Start":"02:19.565 ","End":"02:21.410","Text":"Now, what is known?"},{"Start":"02:21.410 ","End":"02:24.410","Text":"Well, it\u0027s known that if the student took more than 1 exam,"},{"Start":"02:24.410 ","End":"02:27.815","Text":"so x, that\u0027s the number of exams that the student has taken."},{"Start":"02:27.815 ","End":"02:30.740","Text":"That has to be more than 1 and we\u0027re looking for"},{"Start":"02:30.740 ","End":"02:33.730","Text":"the probability that he failed less than 3 exams."},{"Start":"02:33.730 ","End":"02:38.840","Text":"That means that y, the number of exams that he failed has to be less than 3."},{"Start":"02:38.840 ","End":"02:44.720","Text":"Now under conditional probability in the numerator we have the intersect of these 2 guys,"},{"Start":"02:44.720 ","End":"02:46.250","Text":"and in the denominator,"},{"Start":"02:46.250 ","End":"02:48.620","Text":"we have the probability of what\u0027s known."},{"Start":"02:48.620 ","End":"02:55.515","Text":"That\u0027ll be the probability of y being less than 3 intersect,"},{"Start":"02:55.515 ","End":"02:58.120","Text":"x being greater than 1,"},{"Start":"02:58.120 ","End":"03:03.395","Text":"divided by the probability of x being greater than 1."},{"Start":"03:03.395 ","End":"03:05.585","Text":"Now that we have that,"},{"Start":"03:05.585 ","End":"03:10.850","Text":"let\u0027s take a look at our joint probability distribution."},{"Start":"03:10.850 ","End":"03:12.530","Text":"That\u0027s this table right here."},{"Start":"03:12.530 ","End":"03:16.100","Text":"Now we\u0027re looking at the probability of x being greater than 1."},{"Start":"03:16.100 ","End":"03:18.920","Text":"That means that here\u0027s x, 1, 2,"},{"Start":"03:18.920 ","End":"03:22.200","Text":"and 3, those are the values of x now x being greater than 1."},{"Start":"03:22.200 ","End":"03:26.350","Text":"We\u0027re looking at this portion of the table right here."},{"Start":"03:26.350 ","End":"03:32.340","Text":"Now, y has to be less than 3."},{"Start":"03:32.340 ","End":"03:35.795","Text":"This is a portion of the table for y."},{"Start":"03:35.795 ","End":"03:42.200","Text":"The intersect of y being less than 3 and x being greater than 1 well,"},{"Start":"03:42.200 ","End":"03:45.950","Text":"that\u0027s this part of the table."},{"Start":"03:45.950 ","End":"03:48.140","Text":"These are the cells."},{"Start":"03:48.140 ","End":"03:56.045","Text":"These are the relevant cells where x is greater than 1 and y is less than 3."},{"Start":"03:56.045 ","End":"03:59.180","Text":"Now, what\u0027s the probability of"},{"Start":"03:59.180 ","End":"04:02.630","Text":"the numerator where y is less than 3 and x is greater than 1."},{"Start":"04:02.630 ","End":"04:05.660","Text":"Well, we have to add up all these cells right here."},{"Start":"04:05.660 ","End":"04:09.145","Text":"That\u0027ll be 0 plus 0 plus 0, obviously."},{"Start":"04:09.145 ","End":"04:18.915","Text":"Then we\u0027ll have 0.18 plus 0.016 plus 0.0028."},{"Start":"04:18.915 ","End":"04:23.495","Text":"That\u0027s these 3 values right here,"},{"Start":"04:23.495 ","End":"04:28.895","Text":"divided by the probability of x being greater than 1."},{"Start":"04:28.895 ","End":"04:31.490","Text":"Now, if X equaling 1,"},{"Start":"04:31.490 ","End":"04:33.755","Text":"the marginal probability is 0.8."},{"Start":"04:33.755 ","End":"04:37.150","Text":"Well, the probability of x being greater than 1,"},{"Start":"04:37.150 ","End":"04:39.425","Text":"that\u0027s 1 minus 0.8."},{"Start":"04:39.425 ","End":"04:42.215","Text":"Now, after doing the calculation,"},{"Start":"04:42.215 ","End":"04:45.930","Text":"that comes out to 0.994."},{"Start":"04:46.010 ","End":"04:53.420","Text":"This then is the probability that the student failed lesson 3 exams,"},{"Start":"04:53.420 ","End":"04:57.060","Text":"given that he took more than 1 exam."}],"ID":13198},{"Watched":false,"Name":"Exercise 1 - Parts d-f","Duration":"10m 24s","ChapterTopicVideoID":12728,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12728.jpeg","UploadDate":"2018-07-30T15:03:58.5370000","DurationForVideoObject":"PT10M24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.950","Text":"In this section, we\u0027re asked whether x and y are fully or partially correlated,"},{"Start":"00:04.950 ","End":"00:08.385","Text":"or is it positively or negatively correlated."},{"Start":"00:08.385 ","End":"00:12.465","Text":"Now, we have to explain without doing any calculations, so let\u0027s do that."},{"Start":"00:12.465 ","End":"00:16.110","Text":"Well, first of all, let\u0027s define, again,"},{"Start":"00:16.110 ","End":"00:19.680","Text":"let\u0027s review what the definition of x. X is"},{"Start":"00:19.680 ","End":"00:25.380","Text":"the number of exams that the student has taken."},{"Start":"00:25.380 ","End":"00:31.408","Text":"What\u0027s y? Well, that\u0027s the number of failed exams."},{"Start":"00:31.408 ","End":"00:41.065","Text":"Now, it\u0027s clear that if I fail more than 1 test or 1 exam,"},{"Start":"00:41.065 ","End":"00:46.190","Text":"then I have to take more exams in order to try to pass it."},{"Start":"00:46.190 ","End":"00:50.710","Text":"That means that as x increases,"},{"Start":"00:51.320 ","End":"00:54.335","Text":"I\u0027d like to increase y as well."},{"Start":"00:54.335 ","End":"00:56.660","Text":"Or as y increases,"},{"Start":"00:56.660 ","End":"00:59.195","Text":"the number of failed exams increases,"},{"Start":"00:59.195 ","End":"01:02.960","Text":"then I have to take more exams in order to try to pass it."},{"Start":"01:02.960 ","End":"01:07.315","Text":"That means that x and y,"},{"Start":"01:07.315 ","End":"01:12.545","Text":"first of all, are positively correlated."},{"Start":"01:12.545 ","End":"01:14.929","Text":"The correlation is positive."},{"Start":"01:14.929 ","End":"01:19.595","Text":"Now, is it fully or partially correlated?"},{"Start":"01:19.595 ","End":"01:24.440","Text":"Well, x and y are partially correlated, and why is that?"},{"Start":"01:24.440 ","End":"01:26.810","Text":"Well, if I know the value of x,"},{"Start":"01:26.810 ","End":"01:31.350","Text":"I still don\u0027t know what the values of y are."},{"Start":"01:31.350 ","End":"01:35.195","Text":"That is, I don\u0027t have some mathematical formula,"},{"Start":"01:35.195 ","End":"01:39.740","Text":"a linear equation, that when I plug in the value of x,"},{"Start":"01:39.740 ","End":"01:41.810","Text":"I\u0027ll get a value for y."},{"Start":"01:41.810 ","End":"01:46.265","Text":"The example for that is that when x equals 2,"},{"Start":"01:46.265 ","End":"01:51.350","Text":"well, I don\u0027t know whether I\u0027ll be looking at y equals 1 or y equals 2."},{"Start":"01:51.350 ","End":"01:54.080","Text":"Similarly, for x equals 3,"},{"Start":"01:54.080 ","End":"01:58.265","Text":"I don\u0027t know whether y would be 2 or y would be 3,"},{"Start":"01:58.265 ","End":"02:01.335","Text":"so x and y,"},{"Start":"02:01.335 ","End":"02:06.405","Text":"we have a partial correlation."},{"Start":"02:06.405 ","End":"02:07.860","Text":"This is the answer,"},{"Start":"02:07.860 ","End":"02:10.580","Text":"x and y are correlated."},{"Start":"02:10.580 ","End":"02:15.420","Text":"They\u0027re positively correlated, but they\u0027re partially correlated."},{"Start":"02:15.580 ","End":"02:21.410","Text":"In this section, we\u0027re asked to calculate the correlation coefficient of x and y."},{"Start":"02:21.410 ","End":"02:27.150","Text":"The correlation coefficient is defined as the covariance of x and"},{"Start":"02:27.150 ","End":"02:33.770","Text":"y divided by the standard deviation of x times the standard deviation of y."},{"Start":"02:33.770 ","End":"02:36.980","Text":"In order to calculate this, first,"},{"Start":"02:36.980 ","End":"02:40.850","Text":"we need to calculate the expectation of x and y,"},{"Start":"02:40.850 ","End":"02:44.815","Text":"and then the standard deviation of x and y,"},{"Start":"02:44.815 ","End":"02:47.010","Text":"and then the covariance of x and y,"},{"Start":"02:47.010 ","End":"02:52.010","Text":"and only then can we calculate the correlation coefficient."},{"Start":"02:52.010 ","End":"02:55.100","Text":"Let\u0027s get to it. We\u0027ll start with x."},{"Start":"02:55.100 ","End":"02:58.860","Text":"Now, the expectation of x, well,"},{"Start":"02:58.860 ","End":"03:02.595","Text":"that equals to the sum of x times the probability."},{"Start":"03:02.595 ","End":"03:08.385","Text":"That means that we\u0027re looking at 1 times 0.8 plus"},{"Start":"03:08.385 ","End":"03:16.590","Text":"2 times 0.196 plus 3 times 0.004,"},{"Start":"03:16.590 ","End":"03:22.185","Text":"and that equals to 1.204a]Now,"},{"Start":"03:22.185 ","End":"03:25.400","Text":"what about the variance of x?"},{"Start":"03:25.400 ","End":"03:27.205","Text":"Well, the variance of x,"},{"Start":"03:27.205 ","End":"03:31.030","Text":"that equals to the sum of x squared times"},{"Start":"03:31.030 ","End":"03:36.160","Text":"the probability minus the expectation squared of x."},{"Start":"03:36.160 ","End":"03:43.035","Text":"Well, that\u0027ll be 1 squared times 0.8 plus 2 squared times 0.196"},{"Start":"03:43.035 ","End":"03:50.925","Text":"plus 3 squared times 0.004 minus 1.204 squared."},{"Start":"03:50.925 ","End":"03:57.371","Text":"Now, that equals to 0.1704."},{"Start":"03:57.371 ","End":"04:03.790","Text":"That means that the standard deviation of x, well,"},{"Start":"04:03.790 ","End":"04:08.005","Text":"that\u0027s the square root of 0.1704,"},{"Start":"04:08.005 ","End":"04:14.150","Text":"and that equals to 0.4126."},{"Start":"04:14.840 ","End":"04:20.345","Text":"Now, let\u0027s take a look at the expectation of y."},{"Start":"04:20.345 ","End":"04:23.210","Text":"Well, again, the expectation of y,"},{"Start":"04:23.210 ","End":"04:27.605","Text":"that\u0027s the sum of the value of y times its probability,"},{"Start":"04:27.605 ","End":"04:37.450","Text":"so that equals to 0 times 0.8 plus 1 times this probability plus 1 times"},{"Start":"04:37.450 ","End":"04:48.695","Text":"0.18 plus 2 times 0.0188 plus 3 times 0.0012."},{"Start":"04:48.695 ","End":"04:55.264","Text":"Now, that equals to 0.2212."},{"Start":"04:55.264 ","End":"05:00.630","Text":"That\u0027s a 0 right here."},{"Start":"05:00.910 ","End":"05:06.375","Text":"Now, what about the variance of y?"},{"Start":"05:06.375 ","End":"05:11.450","Text":"Well, that equals to the sum of y squared"},{"Start":"05:11.450 ","End":"05:16.740","Text":"times its probability minus the expectation squared."},{"Start":"05:16.740 ","End":"05:25.775","Text":"That\u0027ll be 0 squared times 0.8 plus 1 squared times 0.18 plus 2 squared times"},{"Start":"05:25.775 ","End":"05:30.480","Text":"0.0188 plus 3 squared times"},{"Start":"05:30.480 ","End":"05:38.100","Text":"0.0012 minus now 0.2212 squared,"},{"Start":"05:38.100 ","End":"05:44.200","Text":"and that equals to 0.217."},{"Start":"05:44.200 ","End":"05:48.370","Text":"The standard deviation of y,"},{"Start":"05:49.030 ","End":"05:54.380","Text":"that equals to the square root of 0.217,"},{"Start":"05:54.380 ","End":"06:02.060","Text":"and that equals to 0.4659."},{"Start":"06:02.060 ","End":"06:06.675","Text":"Excellent. Now that we\u0027ve calculated"},{"Start":"06:06.675 ","End":"06:11.767","Text":"the standard deviation of x and standard deviation of y,"},{"Start":"06:11.767 ","End":"06:15.035","Text":"let\u0027s take a look at the covariance of x and y."},{"Start":"06:15.035 ","End":"06:20.840","Text":"Well, the covariance of x and y, well,"},{"Start":"06:20.840 ","End":"06:24.410","Text":"that equals to the expectation of x times"},{"Start":"06:24.410 ","End":"06:31.510","Text":"y minus the expectation of x times the expectation of y."},{"Start":"06:31.510 ","End":"06:35.070","Text":"Now, we have the expectation of x and we have the expectation of y,"},{"Start":"06:35.070 ","End":"06:39.140","Text":"so let\u0027s calculate the expectation of x times y."},{"Start":"06:39.140 ","End":"06:44.860","Text":"Well, that\u0027s the sum of the value of x times the value of y"},{"Start":"06:44.860 ","End":"06:50.900","Text":"times the joint probability of x and y for all values of x and y."},{"Start":"06:50.900 ","End":"06:53.780","Text":"Let\u0027s just take a look at our table."},{"Start":"06:53.780 ","End":"06:56.765","Text":"We\u0027re looking now to calculate this,"},{"Start":"06:56.765 ","End":"07:00.415","Text":"the expectation of x times y."},{"Start":"07:00.415 ","End":"07:02.610","Text":"Here\u0027s our table,"},{"Start":"07:02.610 ","End":"07:07.220","Text":"and we can see that here we can get"},{"Start":"07:07.220 ","End":"07:12.715","Text":"rid of this row right here because the value of y here is 0."},{"Start":"07:12.715 ","End":"07:18.860","Text":"Then here we can get rid of this row because the remaining joint probabilities here,"},{"Start":"07:18.860 ","End":"07:21.388","Text":"well, that equals to 0 as well."},{"Start":"07:21.388 ","End":"07:24.800","Text":"Any place that we have a joint probability of 0,"},{"Start":"07:24.800 ","End":"07:27.140","Text":"well, then I can just disregard that,"},{"Start":"07:27.140 ","End":"07:30.470","Text":"so basically I have 4 values that I"},{"Start":"07:30.470 ","End":"07:34.400","Text":"have to deal with in order to calculate the expectation of x times y."},{"Start":"07:34.400 ","End":"07:41.689","Text":"That\u0027s 2 times 1 times this probability plus 2 times 2 times this probability,"},{"Start":"07:41.689 ","End":"07:43.025","Text":"and so on and so forth."},{"Start":"07:43.025 ","End":"07:45.560","Text":"Let\u0027s just calculate that."},{"Start":"07:45.560 ","End":"07:51.840","Text":"That\u0027ll be equal to 2 times 1 times"},{"Start":"07:51.840 ","End":"08:02.025","Text":"0.18 plus 3 times 2 times 0.0028,"},{"Start":"08:02.025 ","End":"08:04.260","Text":"and so on and so forth,"},{"Start":"08:04.260 ","End":"08:10.473","Text":"and that should equal 0.4516."},{"Start":"08:10.473 ","End":"08:15.800","Text":"The covariance of x and y, well,"},{"Start":"08:15.800 ","End":"08:18.650","Text":"that equals to this expression right here,"},{"Start":"08:18.650 ","End":"08:23.675","Text":"the expectation of x times y minus the expectation of x times the expectation of y."},{"Start":"08:23.675 ","End":"08:25.910","Text":"Well, the expectation of x times y,"},{"Start":"08:25.910 ","End":"08:26.982","Text":"that\u0027s this thing right here,"},{"Start":"08:26.982 ","End":"08:30.740","Text":"that\u0027ll be 0.4516,"},{"Start":"08:30.740 ","End":"08:34.725","Text":"minus the expectation of x,"},{"Start":"08:34.725 ","End":"08:41.240","Text":"that\u0027s 1.204, times the expectation of y,"},{"Start":"08:41.240 ","End":"08:48.556","Text":"that\u0027s 0.2212, and that equals to 0.1853."},{"Start":"08:48.556 ","End":"08:53.505","Text":"That\u0027s the covariance of x and y."},{"Start":"08:53.505 ","End":"09:00.755","Text":"The correlation coefficient, that equals to the covariance"},{"Start":"09:00.755 ","End":"09:08.000","Text":"of x and y divided by the standard deviation of x times the standard deviation of y."},{"Start":"09:08.000 ","End":"09:10.260","Text":"Now, that\u0027s equal to,"},{"Start":"09:10.260 ","End":"09:12.525","Text":"well, the covariance of x and y,"},{"Start":"09:12.525 ","End":"09:13.708","Text":"that\u0027s this guy right here,"},{"Start":"09:13.708 ","End":"09:17.683","Text":"that\u0027ll be 0.1853,"},{"Start":"09:17.683 ","End":"09:22.640","Text":"divided by 0.4126,"},{"Start":"09:22.640 ","End":"09:24.605","Text":"that\u0027s the standard deviation of x,"},{"Start":"09:24.605 ","End":"09:31.160","Text":"times the standard deviation of y, that\u0027s 0.4659."},{"Start":"09:31.160 ","End":"09:37.075","Text":"Now, that equals to 0.963."},{"Start":"09:37.075 ","End":"09:39.860","Text":"Here, as we said,"},{"Start":"09:39.860 ","End":"09:42.320","Text":"that\u0027s the correlation coefficient of x and y."},{"Start":"09:42.320 ","End":"09:44.180","Text":"We see that\u0027s very close to 1,"},{"Start":"09:44.180 ","End":"09:49.340","Text":"so it\u0027s a strong correlation and it\u0027s positive."},{"Start":"09:49.340 ","End":"09:53.605","Text":"This is the answer for this section."},{"Start":"09:53.605 ","End":"09:58.070","Text":"In this section, we\u0027re asked whether the variables are uncorrelated."},{"Start":"09:58.070 ","End":"09:59.630","Text":"Well, we\u0027ve calculated,"},{"Start":"09:59.630 ","End":"10:01.352","Text":"in the previous section,"},{"Start":"10:01.352 ","End":"10:06.500","Text":"that the correlation coefficient equals to 0.963."},{"Start":"10:06.500 ","End":"10:08.780","Text":"Now, we\u0027ve also learned that"},{"Start":"10:08.780 ","End":"10:13.076","Text":"the correlation coefficient for uncorrelated variables, well,"},{"Start":"10:13.076 ","End":"10:14.375","Text":"that equals to 0,"},{"Start":"10:14.375 ","End":"10:16.415","Text":"and this isn\u0027t the case here,"},{"Start":"10:16.415 ","End":"10:21.360","Text":"so that means that x and y are correlated."}],"ID":13199},{"Watched":false,"Name":"Exercise 2 - Parts a-b","Duration":"5m 59s","ChapterTopicVideoID":12729,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12729.jpeg","UploadDate":"2018-07-30T15:04:20.3230000","DurationForVideoObject":"PT5M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.525","Text":"In this question, we\u0027re given that a coin is tossed 3 times, now,"},{"Start":"00:03.525 ","End":"00:07.890","Text":"we defined X as the number of heads obtained on the first 2 tosses,"},{"Start":"00:07.890 ","End":"00:11.610","Text":"and Y as the number of heads obtained in the last 2 tosses."},{"Start":"00:11.610 ","End":"00:15.930","Text":"We\u0027re asked to construct a joint probability function of X and Y,"},{"Start":"00:15.930 ","End":"00:18.840","Text":"and their marginal probability functions."},{"Start":"00:18.840 ","End":"00:23.840","Text":"The first thing that we want to do is to describe the data."},{"Start":"00:23.840 ","End":"00:29.490","Text":"The best way to do that is to just write out all the combinations that we have of"},{"Start":"00:29.490 ","End":"00:38.440","Text":"the 3 tosses and then count the number of tosses for X, and for Y."},{"Start":"00:39.470 ","End":"00:45.500","Text":"Here are all the combinations of throwing or tossing a coin 3 times."},{"Start":"00:45.500 ","End":"00:47.060","Text":"This is the first task,"},{"Start":"00:47.060 ","End":"00:48.560","Text":"this is the second task,"},{"Start":"00:48.560 ","End":"00:50.255","Text":"and this is the third task."},{"Start":"00:50.255 ","End":"00:53.488","Text":"We have here 8 combinations."},{"Start":"00:53.488 ","End":"00:55.060","Text":"Now what\u0027s X?"},{"Start":"00:55.060 ","End":"00:59.750","Text":"X is the number of heads on the first 2 tosses."},{"Start":"00:59.750 ","End":"01:01.565","Text":"Well, here we have 2."},{"Start":"01:01.565 ","End":"01:03.755","Text":"Here we have 2."},{"Start":"01:03.755 ","End":"01:07.125","Text":"Here we have 1, again 1,"},{"Start":"01:07.125 ","End":"01:09.240","Text":"and here we have 1,"},{"Start":"01:09.240 ","End":"01:13.995","Text":"here 1, and here we have 0 and 0."},{"Start":"01:13.995 ","End":"01:19.805","Text":"Now, y is the number of heads on the last 2 tosses."},{"Start":"01:19.805 ","End":"01:21.500","Text":"Well, here we have 2."},{"Start":"01:21.500 ","End":"01:25.425","Text":"Here, we have 1, here we have 1,"},{"Start":"01:25.425 ","End":"01:32.985","Text":"0, 2, 1, 1, and 0."},{"Start":"01:32.985 ","End":"01:40.580","Text":"This then is the data that we\u0027ve received from the story."},{"Start":"01:40.580 ","End":"01:46.475","Text":"Now, X can have a valid value of 0,1 or 2,"},{"Start":"01:46.475 ","End":"01:51.175","Text":"and same as Y. Y can have a valid value of 0,1 and 2."},{"Start":"01:51.175 ","End":"01:56.800","Text":"Let\u0027s now construct the joint probability function of X and Y."},{"Start":"01:56.800 ","End":"02:02.630","Text":"Here\u0027s the empty table in which we\u0027ll construct a joint probability function."},{"Start":"02:02.630 ","End":"02:06.560","Text":"First, let\u0027s just fill in the joint probabilities."},{"Start":"02:06.560 ","End":"02:11.760","Text":"Now, what\u0027s the probability of X equaling 0 and Y equaling 0?"},{"Start":"02:11.760 ","End":"02:14.301","Text":"That\u0027s this cell right here."},{"Start":"02:14.301 ","End":"02:18.750","Text":"We only have 1 combination of 0 and 0 here."},{"Start":"02:18.750 ","End":"02:23.000","Text":"That means that the probability is 1 divided by a."},{"Start":"02:23.000 ","End":"02:29.325","Text":"Now, what about X equaling 0 and Y equaling 1?"},{"Start":"02:29.325 ","End":"02:31.875","Text":"X equaling 0 and Y equaling 1."},{"Start":"02:31.875 ","End":"02:33.570","Text":"Again, there\u0027s only 1 of these,"},{"Start":"02:33.570 ","End":"02:35.370","Text":"so that\u0027ll be 1/8."},{"Start":"02:35.370 ","End":"02:39.145","Text":"What about X equals 0 and Y equals 2?"},{"Start":"02:39.145 ","End":"02:42.440","Text":"Well, there\u0027s no such combination here,"},{"Start":"02:42.440 ","End":"02:44.435","Text":"so the probability is 0."},{"Start":"02:44.435 ","End":"02:51.570","Text":"Now, what about X equaling 1,"},{"Start":"02:51.570 ","End":"02:53.970","Text":"and Y equaling 0? That\u0027s right here."},{"Start":"02:53.970 ","End":"02:57.000","Text":"This is 1/8. Now,"},{"Start":"02:57.000 ","End":"03:00.330","Text":"what about X equaling 1 and Y equaling 1?"},{"Start":"03:00.330 ","End":"03:06.255","Text":"Well, we have 1 combination here and 1 combination here,"},{"Start":"03:06.255 ","End":"03:08.700","Text":"so that\u0027s 2 divided by 8."},{"Start":"03:08.700 ","End":"03:11.925","Text":"What about X equaling 1 and Y equaling 2?"},{"Start":"03:11.925 ","End":"03:15.360","Text":"Well, that\u0027s here. There will be 1 divided by 8."},{"Start":"03:15.360 ","End":"03:19.635","Text":"Now, what about X equaling 2, Y 0."},{"Start":"03:19.635 ","End":"03:24.095","Text":"We don\u0027t have a combination like that, so that\u0027s 0."},{"Start":"03:24.095 ","End":"03:29.090","Text":"What about X equaling 2 and Y equaling 1."},{"Start":"03:29.090 ","End":"03:31.010","Text":"That\u0027s here."},{"Start":"03:31.010 ","End":"03:33.460","Text":"We have 1/8."},{"Start":"03:33.460 ","End":"03:37.260","Text":"What about X equaling 2 and Y equaling 2 again,"},{"Start":"03:37.260 ","End":"03:40.665","Text":"we have 1 combination like that, so that\u0027s 1/8."},{"Start":"03:40.665 ","End":"03:44.375","Text":"Here we have the joint probabilities of X and Y."},{"Start":"03:44.375 ","End":"03:46.805","Text":"What about the marginal probabilities?"},{"Start":"03:46.805 ","End":"03:50.645","Text":"Well, let\u0027s first take a look at the marginal probability of Y."},{"Start":"03:50.645 ","End":"03:52.715","Text":"Well, when Y equals 0,"},{"Start":"03:52.715 ","End":"03:55.390","Text":"that\u0027ll be 2/8 here."},{"Start":"03:55.390 ","End":"03:58.425","Text":"That\u0027ll be 4/8 here,"},{"Start":"03:58.425 ","End":"04:00.755","Text":"and that\u0027ll be 2/8 here."},{"Start":"04:00.755 ","End":"04:02.405","Text":"Now, same thing here,"},{"Start":"04:02.405 ","End":"04:03.805","Text":"that\u0027ll be 2/8,"},{"Start":"04:03.805 ","End":"04:08.340","Text":"4/8, and 2/8."},{"Start":"04:08.340 ","End":"04:11.900","Text":"Here then we\u0027ve constructed the joint probability function of"},{"Start":"04:11.900 ","End":"04:16.355","Text":"X and Y and also the marginal probability functions."},{"Start":"04:16.355 ","End":"04:20.825","Text":"In this section, we\u0027re asked whether X and Y are independent variables."},{"Start":"04:20.825 ","End":"04:25.130","Text":"Well, the definition of independent variables is this."},{"Start":"04:25.130 ","End":"04:28.670","Text":"That\u0027s the joint probability where X equals to"},{"Start":"04:28.670 ","End":"04:32.660","Text":"some value K and Y equaling some value L. Well,"},{"Start":"04:32.660 ","End":"04:40.340","Text":"that has to be equal to the multiplication of the marginal probabilities of X equaling"},{"Start":"04:40.340 ","End":"04:44.060","Text":"K times Y equaling L for"},{"Start":"04:44.060 ","End":"04:48.470","Text":"all values of K and L. That means for all combinations of X and Y."},{"Start":"04:48.470 ","End":"04:55.385","Text":"Then if I can find at least 1 case where this equation doesn\u0027t hold true,"},{"Start":"04:55.385 ","End":"05:01.319","Text":"then X and Y are not independent but dependent variables,"},{"Start":"05:01.319 ","End":"05:11.210","Text":"so let\u0027s just take a look at the probability of X equaling 1 and Y equaling 2."},{"Start":"05:11.210 ","End":"05:17.060","Text":"Well, we\u0027re looking at the probability of X equaling"},{"Start":"05:17.060 ","End":"05:23.210","Text":"1 times the probability of Y being equal to 2."},{"Start":"05:23.210 ","End":"05:27.205","Text":"Well, what\u0027s the marginal probability of X equaling 1?"},{"Start":"05:27.205 ","End":"05:29.470","Text":"That\u0027s 4/8."},{"Start":"05:29.470 ","End":"05:32.420","Text":"What\u0027s the marginal probability of Y equaling 2?"},{"Start":"05:32.420 ","End":"05:34.100","Text":"That\u0027s 2/8."},{"Start":"05:34.100 ","End":"05:40.685","Text":"Now, what\u0027s the probability or the joint probability of X equaling 1 and Y equaling 2?"},{"Start":"05:40.685 ","End":"05:44.195","Text":"Well, that\u0027s right here, that\u0027s 1/8."},{"Start":"05:44.195 ","End":"05:48.065","Text":"Clearly, we can see that the multiplication of"},{"Start":"05:48.065 ","End":"05:53.105","Text":"these values right here do not equal to this value right here."},{"Start":"05:53.105 ","End":"05:59.260","Text":"That means that X and Y then are dependent."}],"ID":13200},{"Watched":false,"Name":"Exercise 2 - Parts c-e","Duration":"9m 27s","ChapterTopicVideoID":12730,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12730.jpeg","UploadDate":"2018-07-30T15:04:48.6800000","DurationForVideoObject":"PT9M27S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this section, we\u0027re asked what\u0027s the correlation coefficient of X and"},{"Start":"00:03.570 ","End":"00:08.085","Text":"Y and whether the variables are correlated or not."},{"Start":"00:08.085 ","End":"00:13.740","Text":"The definition of the correlation coefficient is the covariance of X and"},{"Start":"00:13.740 ","End":"00:20.235","Text":"Y divided by the standard deviation of x times the standard deviation of Y."},{"Start":"00:20.235 ","End":"00:23.280","Text":"Now, in order to calculate that,"},{"Start":"00:23.280 ","End":"00:27.510","Text":"I first need to calculate the expectation of X,"},{"Start":"00:27.510 ","End":"00:29.070","Text":"the expectation of Y,"},{"Start":"00:29.070 ","End":"00:30.930","Text":"the standard deviation of X,"},{"Start":"00:30.930 ","End":"00:32.625","Text":"the standard deviation of Y,"},{"Start":"00:32.625 ","End":"00:35.190","Text":"the covariance of X and Y,"},{"Start":"00:35.190 ","End":"00:40.160","Text":"and only then can I calculate the correlation coefficient."},{"Start":"00:40.160 ","End":"00:42.570","Text":"Let\u0027s get to work. Now,"},{"Start":"00:42.570 ","End":"00:44.645","Text":"the expectation of X,"},{"Start":"00:44.645 ","End":"00:49.640","Text":"we\u0027ll start with X that equals to the sum of X times its probability,"},{"Start":"00:49.640 ","End":"00:54.620","Text":"so that will be equal to 0 times 2/8"},{"Start":"00:54.620 ","End":"01:01.675","Text":"plus 1 times 4/8 plus 2 times 2/8,"},{"Start":"01:01.675 ","End":"01:04.690","Text":"and that equals to 1."},{"Start":"01:04.690 ","End":"01:07.640","Text":"Now, what about the variance of X?"},{"Start":"01:07.640 ","End":"01:11.900","Text":"Well, that equals to the sum of X squared"},{"Start":"01:11.900 ","End":"01:16.310","Text":"times its probability minus the expectation squared."},{"Start":"01:16.310 ","End":"01:22.595","Text":"Then be 0 squared times 2/8 plus 1 squared times"},{"Start":"01:22.595 ","End":"01:30.045","Text":"4/8 plus 2 squared times 2/8 minus 1 squared."},{"Start":"01:30.045 ","End":"01:33.420","Text":"Now, that equals to 1.5."},{"Start":"01:33.420 ","End":"01:38.240","Text":"That means that the standard deviation of X,"},{"Start":"01:38.240 ","End":"01:42.820","Text":"well that equals to the square root of a 1/2."},{"Start":"01:42.820 ","End":"01:47.705","Text":"Now, what about the expectation and standard deviation of Y?"},{"Start":"01:47.705 ","End":"01:50.795","Text":"Well, if we take a look at the table here,"},{"Start":"01:50.795 ","End":"01:57.290","Text":"we see that the marginal probabilities of X and Y are equal,"},{"Start":"01:57.290 ","End":"02:00.155","Text":"they have the same distribution."},{"Start":"02:00.155 ","End":"02:05.435","Text":"Not only that, but the distribution is symmetrical around 1."},{"Start":"02:05.435 ","End":"02:10.700","Text":"We can say that the expectation of Y now,"},{"Start":"02:10.700 ","End":"02:13.025","Text":"well that equals to 1,"},{"Start":"02:13.025 ","End":"02:19.205","Text":"because we\u0027ve calculated the expectation for X and the distributions are the same,"},{"Start":"02:19.205 ","End":"02:25.085","Text":"so the expectation of Y equals to 1 and the standard deviation of Y,"},{"Start":"02:25.085 ","End":"02:29.710","Text":"well that equals to the square root of a 1/2."},{"Start":"02:29.710 ","End":"02:32.765","Text":"Now, what about the covariance of X and Y?"},{"Start":"02:32.765 ","End":"02:36.065","Text":"Well, the covariance of X and Y,"},{"Start":"02:36.065 ","End":"02:40.310","Text":"well that equals to the expectation of X times Y"},{"Start":"02:40.310 ","End":"02:45.885","Text":"minus the expectation of X times the expectation of Y,"},{"Start":"02:45.885 ","End":"02:48.350","Text":"so we have the expectation of X,"},{"Start":"02:48.350 ","End":"02:50.480","Text":"we have the expectation of Y,"},{"Start":"02:50.480 ","End":"02:54.335","Text":"and we need to calculate the expectation of X times Y."},{"Start":"02:54.335 ","End":"02:57.744","Text":"Well, the expectation of X times Y,"},{"Start":"02:57.744 ","End":"02:59.870","Text":"well, that equals to what?"},{"Start":"02:59.870 ","End":"03:04.130","Text":"Well, that\u0027s the multiplication of the values"},{"Start":"03:04.130 ","End":"03:08.915","Text":"of X by Y times the joint probabilities of X and Y."},{"Start":"03:08.915 ","End":"03:11.795","Text":"But if we look at the table again,"},{"Start":"03:11.795 ","End":"03:14.390","Text":"we see that we can get rid of this row right"},{"Start":"03:14.390 ","End":"03:17.780","Text":"here and this column right here, why is that?"},{"Start":"03:17.780 ","End":"03:23.200","Text":"Well, because X has a value of 0 or Y has a value of 0,"},{"Start":"03:23.200 ","End":"03:28.070","Text":"so we\u0027re interested then in only these 4 cells right here,"},{"Start":"03:28.070 ","End":"03:35.855","Text":"so that would be 1 times 1 times 2/8,"},{"Start":"03:35.855 ","End":"03:40.580","Text":"plus 1 times 2 times 1/8,"},{"Start":"03:40.580 ","End":"03:45.730","Text":"plus 2 times 1 times 1/8,"},{"Start":"03:45.730 ","End":"03:54.750","Text":"plus 2 times 2 times 1/8 and that equals to 1.25."},{"Start":"03:54.750 ","End":"04:01.915","Text":"Excellent. Now the covariance of X and Y, well,"},{"Start":"04:01.915 ","End":"04:04.760","Text":"that equals to the expectation of X times Y,"},{"Start":"04:04.760 ","End":"04:08.720","Text":"that\u0027s 1.25 minus the expectation of X,"},{"Start":"04:08.720 ","End":"04:11.120","Text":"that\u0027s 1 times the expectation of Y,"},{"Start":"04:11.120 ","End":"04:12.425","Text":"that\u0027s 1 as well,"},{"Start":"04:12.425 ","End":"04:15.950","Text":"so that equals to 0.25."},{"Start":"04:15.950 ","End":"04:18.465","Text":"That\u0027s the covariance of X and Y."},{"Start":"04:18.465 ","End":"04:23.165","Text":"Excellent. Now that I have all the data,"},{"Start":"04:23.165 ","End":"04:27.425","Text":"now it can calculate the correlation coefficient,"},{"Start":"04:27.425 ","End":"04:31.535","Text":"so the correlation coefficient that\u0027s row,"},{"Start":"04:31.535 ","End":"04:34.010","Text":"that equals to the covariance of X and Y"},{"Start":"04:34.010 ","End":"04:38.090","Text":"divided by the standard deviation of X times the standard deviation of Y."},{"Start":"04:38.090 ","End":"04:40.370","Text":"Well, the covariance of X and Y,"},{"Start":"04:40.370 ","End":"04:46.640","Text":"that\u0027s 0.25 divided by the standard deviation of X,"},{"Start":"04:46.640 ","End":"04:52.160","Text":"that\u0027s the square root of a 1/2 times the square root of a 1/2,"},{"Start":"04:52.160 ","End":"04:54.155","Text":"that\u0027s the standard deviation of Y."},{"Start":"04:54.155 ","End":"04:58.055","Text":"Now, that equals to 1.5."},{"Start":"04:58.055 ","End":"05:06.575","Text":"Excellent. Now we can answer the question whether the variables are correlated or not."},{"Start":"05:06.575 ","End":"05:11.705","Text":"Well, when our variables are not correlated or uncorrelated."},{"Start":"05:11.705 ","End":"05:15.530","Text":"Well, when the correlation coefficient equals to 0,"},{"Start":"05:15.530 ","End":"05:18.065","Text":"and that\u0027s not the case here."},{"Start":"05:18.065 ","End":"05:23.695","Text":"That means that X and Y are correlated."},{"Start":"05:23.695 ","End":"05:26.950","Text":"In this section, we assume that exactly 1"},{"Start":"05:26.950 ","End":"05:29.600","Text":"of the first 2 tosses was heads and we\u0027re asked,"},{"Start":"05:29.600 ","End":"05:33.710","Text":"what\u0027s the probability that the last 2 tosses were heads?"},{"Start":"05:33.710 ","End":"05:37.610","Text":"This is a conditional probability and we\u0027re asked him,"},{"Start":"05:37.610 ","End":"05:40.085","Text":"what\u0027s the probability of what?"},{"Start":"05:40.085 ","End":"05:42.200","Text":"Now, since this is a conditional probability,"},{"Start":"05:42.200 ","End":"05:43.670","Text":"what\u0027s given to us?"},{"Start":"05:43.670 ","End":"05:45.950","Text":"Well, we\u0027re given that X,"},{"Start":"05:45.950 ","End":"05:48.410","Text":"the number of heads in the first 2 tosses,"},{"Start":"05:48.410 ","End":"05:53.480","Text":"that has to equal to 1 and we\u0027re asked about the probability that the last 2 tosses,"},{"Start":"05:53.480 ","End":"05:58.805","Text":"that\u0027s Y that has to equal to 2."},{"Start":"05:58.805 ","End":"06:01.820","Text":"Now, how do we solve this?"},{"Start":"06:01.820 ","End":"06:06.110","Text":"Well, in the denominator we have the probability what\u0027s given,"},{"Start":"06:06.110 ","End":"06:08.390","Text":"that\u0027s X equals to 1."},{"Start":"06:08.390 ","End":"06:11.480","Text":"In the numerator we have the intersection,"},{"Start":"06:11.480 ","End":"06:17.860","Text":"that\u0027s the probability of Y equaling 2 and X equaling 1."},{"Start":"06:19.070 ","End":"06:21.470","Text":"That equals to what?"},{"Start":"06:21.470 ","End":"06:24.870","Text":"Well, the probability of X equaling 1, well,"},{"Start":"06:24.870 ","End":"06:27.780","Text":"that\u0027s 4/8, that\u0027s right here,"},{"Start":"06:27.780 ","End":"06:31.025","Text":"so, that would be 4/8 in the denominator."},{"Start":"06:31.025 ","End":"06:34.730","Text":"Now, what\u0027s the probability of Y equaling 2 and X equaling 1?"},{"Start":"06:34.730 ","End":"06:41.380","Text":"Well, here\u0027s Y equal 2 and x equaling 1, so that\u0027s 1/8."},{"Start":"06:41.380 ","End":"06:48.015","Text":"That means that this then equals to 1/4,"},{"Start":"06:48.015 ","End":"06:53.075","Text":"that\u0027s the probability that the last 2 tosses were heads,"},{"Start":"06:53.075 ","End":"07:00.430","Text":"given that in the first 2 tosses there was exactly 1 heads."},{"Start":"07:00.430 ","End":"07:06.005","Text":"In this section, we\u0027re asked if there was at least 1 heads and the last 2 tosses,"},{"Start":"07:06.005 ","End":"07:10.190","Text":"what\u0027s the probability of exactly 1 heads and the first 2 tosses?"},{"Start":"07:10.190 ","End":"07:14.600","Text":"This is a conditional probability we\u0027re asked for the probability then,"},{"Start":"07:14.600 ","End":"07:15.965","Text":"and now what\u0027s given?"},{"Start":"07:15.965 ","End":"07:20.010","Text":"If there was at least 1 heads on the last 2 tosses,"},{"Start":"07:20.010 ","End":"07:22.505","Text":"that means that Y, the last 2 tosses,"},{"Start":"07:22.505 ","End":"07:24.485","Text":"is greater or equal to 1,"},{"Start":"07:24.485 ","End":"07:26.180","Text":"at least 1 heads,"},{"Start":"07:26.180 ","End":"07:31.800","Text":"and we\u0027re asked what\u0027s the probability of exactly 1 heads and the first 2 tosses,"},{"Start":"07:31.800 ","End":"07:36.545","Text":"that means that X, the first 2 tosses that has to be equal to 1."},{"Start":"07:36.545 ","End":"07:39.200","Text":"Now, what\u0027s this probability?"},{"Start":"07:39.200 ","End":"07:42.985","Text":"Well, that equals 2 in the denominator,"},{"Start":"07:42.985 ","End":"07:45.680","Text":"we have the probability of what\u0027s given,"},{"Start":"07:45.680 ","End":"07:47.915","Text":"Y being greater or equal to 1,"},{"Start":"07:47.915 ","End":"07:49.700","Text":"and in the numerator, well,"},{"Start":"07:49.700 ","End":"07:50.720","Text":"we have the intersect,"},{"Start":"07:50.720 ","End":"07:56.195","Text":"that\u0027s the probability of X equaling 1 and Y being greater or equal to 1."},{"Start":"07:56.195 ","End":"08:03.305","Text":"Now let\u0027s take a look at the table right here and see what the probabilities are."},{"Start":"08:03.305 ","End":"08:07.430","Text":"Well, let\u0027s take a look at the intersection that"},{"Start":"08:07.430 ","End":"08:12.305","Text":"where X equals 1 and y is greater or equals to 1."},{"Start":"08:12.305 ","End":"08:15.590","Text":"Well, when Y is greater or equal to 1,"},{"Start":"08:15.590 ","End":"08:20.790","Text":"we\u0027re looking at these cells right here. Now, why is that?"},{"Start":"08:20.790 ","End":"08:23.205","Text":"Well, Y is either 1 or 2,"},{"Start":"08:23.205 ","End":"08:28.055","Text":"so these are the cells that we\u0027re interested in."},{"Start":"08:28.055 ","End":"08:31.180","Text":"Now what about when x equals to 1?"},{"Start":"08:31.180 ","End":"08:33.675","Text":"Well, when X equals to 1,"},{"Start":"08:33.675 ","End":"08:37.690","Text":"then we\u0027re looking at these cells right here,"},{"Start":"08:37.690 ","End":"08:40.850","Text":"so what\u0027s the intersection?"},{"Start":"08:40.850 ","End":"08:45.080","Text":"Well, the intersection or these 2 cells right here,"},{"Start":"08:45.080 ","End":"08:49.585","Text":"that\u0027s 2/8 plus 1/8."},{"Start":"08:49.585 ","End":"08:53.180","Text":"These are the cells that are the intersection"},{"Start":"08:53.180 ","End":"08:57.880","Text":"between X equals 1 and Y being greater or equal to 1."},{"Start":"08:57.880 ","End":"09:05.480","Text":"That\u0027s these 2 cells divided by the probability of Y being greater than or equal to 1."},{"Start":"09:05.480 ","End":"09:08.675","Text":"What does these marginal probabilities right here?"},{"Start":"09:08.675 ","End":"09:14.615","Text":"That would be 4/8 plus 2/8, 4/8 plus 2/8."},{"Start":"09:14.615 ","End":"09:20.770","Text":"Now, that equals to 3/8 divided by 6/8,"},{"Start":"09:20.770 ","End":"09:23.250","Text":"and that equals to 1.5,"},{"Start":"09:23.250 ","End":"09:28.290","Text":"so that\u0027s the answer for this conditional probability."}],"ID":13201},{"Watched":false,"Name":"Exercise 3 - Part a","Duration":"6m 47s","ChapterTopicVideoID":12731,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12731.jpeg","UploadDate":"2018-07-30T15:05:09.4800000","DurationForVideoObject":"PT6M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.174","Text":"In this question, we have 3 different balls that are distributed among 3 different boxes."},{"Start":"00:05.174 ","End":"00:07.410","Text":"We define the following variables."},{"Start":"00:07.410 ","End":"00:10.350","Text":"X is the number of balls in the first box,"},{"Start":"00:10.350 ","End":"00:12.870","Text":"and Y is the number of balls in the second box."},{"Start":"00:12.870 ","End":"00:16.770","Text":"We\u0027re asked to construct the joint probability function of X and Y."},{"Start":"00:16.770 ","End":"00:18.285","Text":"Well, first of all,"},{"Start":"00:18.285 ","End":"00:20.235","Text":"let\u0027s look at our sample space."},{"Start":"00:20.235 ","End":"00:24.060","Text":"Well, we have 3 different balls and for each ball,"},{"Start":"00:24.060 ","End":"00:27.525","Text":"we can put it in 1 of 3 different boxes."},{"Start":"00:27.525 ","End":"00:32.355","Text":"The first ball has 3 boxes that it could go into."},{"Start":"00:32.355 ","End":"00:36.630","Text":"The second ball can also have 3 different boxes that it can go to."},{"Start":"00:36.630 ","End":"00:39.150","Text":"The third ball, the same."},{"Start":"00:39.150 ","End":"00:44.210","Text":"In all, we have 27 different combinations where"},{"Start":"00:44.210 ","End":"00:49.365","Text":"the 3 balls can be placed into the 3 boxes."},{"Start":"00:49.365 ","End":"00:51.920","Text":"This is the total sample space."},{"Start":"00:51.920 ","End":"00:54.860","Text":"Now I can write out all the possible combinations,"},{"Start":"00:54.860 ","End":"00:57.260","Text":"but it\u0027ll be pretty tiresome."},{"Start":"00:57.260 ","End":"01:00.320","Text":"It\u0027s a fairly large sample space."},{"Start":"01:00.320 ","End":"01:02.885","Text":"I\u0027ll be using combinatorics here."},{"Start":"01:02.885 ","End":"01:11.420","Text":"The other thing that I want to find out is the valid values of X and Y. X,"},{"Start":"01:11.420 ","End":"01:14.330","Text":"as we said, is the number of balls in the first box."},{"Start":"01:14.330 ","End":"01:17.660","Text":"Well, that number can be either 0,"},{"Start":"01:17.660 ","End":"01:20.680","Text":"1, 2 or 3."},{"Start":"01:20.680 ","End":"01:25.460","Text":"We can have 0 balls in the first box or 1 or 2 or 3 balls."},{"Start":"01:25.460 ","End":"01:27.575","Text":"The same goes for Y,"},{"Start":"01:27.575 ","End":"01:30.020","Text":"where Y is the number of balls in the second box."},{"Start":"01:30.020 ","End":"01:31.520","Text":"That also equals 0,"},{"Start":"01:31.520 ","End":"01:33.920","Text":"1, 2, and 3."},{"Start":"01:33.920 ","End":"01:41.690","Text":"Having said that, let\u0027s just write out the probability function of X and Y,"},{"Start":"01:41.690 ","End":"01:44.360","Text":"or at least the empty matrix,"},{"Start":"01:44.360 ","End":"01:47.725","Text":"and then we\u0027ll fit it up with our probabilities."},{"Start":"01:47.725 ","End":"01:53.390","Text":"This is our empty matrix and what we have to do is to complete the matrix by"},{"Start":"01:53.390 ","End":"01:58.955","Text":"filling in the probabilities in the white cells right here, the joint probabilities."},{"Start":"01:58.955 ","End":"02:04.114","Text":"Let\u0027s first look at some impossible combinations."},{"Start":"02:04.114 ","End":"02:07.100","Text":"Well, if X equals 3,"},{"Start":"02:07.100 ","End":"02:10.400","Text":"that means that there are 3 balls placed in the first box,"},{"Start":"02:10.400 ","End":"02:12.605","Text":"then Y equals 1,"},{"Start":"02:12.605 ","End":"02:15.830","Text":"where 1 ball is placed in the second box."},{"Start":"02:15.830 ","End":"02:21.340","Text":"Well, that\u0027s impossible because we only have 3 balls and they\u0027re place in the first box."},{"Start":"02:21.340 ","End":"02:24.050","Text":"That\u0027s 0 here."},{"Start":"02:24.050 ","End":"02:27.245","Text":"Likewise, that\u0027s 0 here and 0 here."},{"Start":"02:27.245 ","End":"02:28.860","Text":"What about Y?"},{"Start":"02:28.860 ","End":"02:31.950","Text":"When Y equals 3 and X equals 1,"},{"Start":"02:31.950 ","End":"02:35.010","Text":"if 3 balls are placed in the second box,"},{"Start":"02:35.010 ","End":"02:39.720","Text":"then no balls can be placed in the first box."},{"Start":"02:39.720 ","End":"02:42.930","Text":"The probabilities here are 0 and 0."},{"Start":"02:42.930 ","End":"02:46.545","Text":"What about here,"},{"Start":"02:46.545 ","End":"02:48.765","Text":"where X equals 2 and Y equals 2?"},{"Start":"02:48.765 ","End":"02:51.710","Text":"Again, there are 4 balls that have to be distributed,"},{"Start":"02:51.710 ","End":"02:52.760","Text":"but there are only 3 balls,"},{"Start":"02:52.760 ","End":"02:56.020","Text":"that\u0027s 0 as well."},{"Start":"02:56.020 ","End":"03:01.310","Text":"Let\u0027s take a look at the probability."},{"Start":"03:01.310 ","End":"03:04.865","Text":"We\u0027ll take this cell for example right here,"},{"Start":"03:04.865 ","End":"03:10.965","Text":"where X equals to 3 and Y equals to 0."},{"Start":"03:10.965 ","End":"03:15.665","Text":"Well, that means the 3 balls are placed in the first box."},{"Start":"03:15.665 ","End":"03:20.285","Text":"Well, that equals to 1 divided by 27. Why is that?"},{"Start":"03:20.285 ","End":"03:23.825","Text":"We only have 1 combination where"},{"Start":"03:23.825 ","End":"03:30.565","Text":"all the 3 balls are in the first box and we have 27 possible combinations."},{"Start":"03:30.565 ","End":"03:33.119","Text":"This is 1 divided by 27."},{"Start":"03:33.119 ","End":"03:35.885","Text":"We\u0027ll put that here, 1 divided by 27."},{"Start":"03:35.885 ","End":"03:43.280","Text":"We can see that the distribution of X or Y are symmetrical."},{"Start":"03:43.280 ","End":"03:45.560","Text":"Why is that?"},{"Start":"03:45.560 ","End":"03:48.230","Text":"Well, the distribution of X,"},{"Start":"03:48.230 ","End":"03:50.510","Text":"where X equals 0, 1, 2, or 3,"},{"Start":"03:50.510 ","End":"03:53.840","Text":"that means the number of balls in the first box,"},{"Start":"03:53.840 ","End":"03:56.840","Text":"that distribution is symmetrical when talking"},{"Start":"03:56.840 ","End":"04:00.575","Text":"about the second box and the number of balls in the second box."},{"Start":"04:00.575 ","End":"04:05.840","Text":"Here, where X equals 3 and Y equals 0,"},{"Start":"04:05.840 ","End":"04:11.570","Text":"that equals to the probability of where Y equals 3 and X equals 0."},{"Start":"04:11.570 ","End":"04:15.140","Text":"That here will also be 1 divided by 27."},{"Start":"04:15.140 ","End":"04:23.240","Text":"Let\u0027s take a look at the probability of X equaling 2,"},{"Start":"04:23.240 ","End":"04:27.630","Text":"and y equaling 0."},{"Start":"04:27.630 ","End":"04:33.335","Text":"What does that mean? The 2 balls are placed in the first box,"},{"Start":"04:33.335 ","End":"04:37.030","Text":"none are placed in the second box."},{"Start":"04:37.030 ","End":"04:38.945","Text":"What do we have here?"},{"Start":"04:38.945 ","End":"04:42.020","Text":"Out of the 3 balls,"},{"Start":"04:42.020 ","End":"04:44.300","Text":"we want to choose 2 of them,"},{"Start":"04:44.300 ","End":"04:48.575","Text":"and we have a total of 27."},{"Start":"04:48.575 ","End":"04:53.255","Text":"That means that that equals to 3 divided by 27."},{"Start":"04:53.255 ","End":"04:57.330","Text":"Where X equals 2 and Y equals 0,"},{"Start":"04:57.330 ","End":"05:00.345","Text":"that\u0027s 3 divided by 27."},{"Start":"05:00.345 ","End":"05:06.270","Text":"Again, we have here a symmetrical joint probability."},{"Start":"05:06.270 ","End":"05:10.070","Text":"Here, where Y equals 2 and X equals 0,"},{"Start":"05:10.070 ","End":"05:13.405","Text":"that also would be 3 divided by 27."},{"Start":"05:13.405 ","End":"05:19.270","Text":"Let\u0027s take a look at the probability of X equaling 1,"},{"Start":"05:19.270 ","End":"05:22.940","Text":"and Y equaling 0."},{"Start":"05:22.940 ","End":"05:27.640","Text":"Well, now we have to choose 1 out of the 3 balls,"},{"Start":"05:27.640 ","End":"05:32.735","Text":"and we have a total of 27 possible combinations."},{"Start":"05:32.735 ","End":"05:36.710","Text":"That equals also to 3 divided by 27."},{"Start":"05:36.710 ","End":"05:40.550","Text":"When X equals 1 and Y equals 0,"},{"Start":"05:40.550 ","End":"05:43.500","Text":"that\u0027s 3 divided by 27,"},{"Start":"05:43.500 ","End":"05:48.875","Text":"and symmetrically, that also is 3 divided by 27."},{"Start":"05:48.875 ","End":"05:53.570","Text":"What about when the probability of x equaling 0,"},{"Start":"05:53.570 ","End":"05:55.820","Text":"and y equaling 0?"},{"Start":"05:55.820 ","End":"06:01.770","Text":"That means that no balls are in the first box and no balls are in the second box."},{"Start":"06:01.770 ","End":"06:04.110","Text":"That means that all 3 balls are in the third box."},{"Start":"06:04.110 ","End":"06:09.955","Text":"That means that we have one possible combination that that\u0027s 1 divided by 27."},{"Start":"06:09.955 ","End":"06:18.639","Text":"I\u0027ll let you guys figure out the other combination,"},{"Start":"06:18.639 ","End":"06:20.970","Text":"the other joint probabilities here,"},{"Start":"06:20.970 ","End":"06:22.850","Text":"but I\u0027ll give you the answer right here."},{"Start":"06:22.850 ","End":"06:25.775","Text":"That\u0027ll be 6 divided by 27,"},{"Start":"06:25.775 ","End":"06:29.455","Text":"that\u0027ll be 3 divided by 27,"},{"Start":"06:29.455 ","End":"06:33.515","Text":"and that will also be 3 divided by 27."},{"Start":"06:33.515 ","End":"06:35.175","Text":"These 3 cells,"},{"Start":"06:35.175 ","End":"06:36.380","Text":"I\u0027ve given you the answers,"},{"Start":"06:36.380 ","End":"06:38.660","Text":"but you have to figure it out by yourselves."},{"Start":"06:38.660 ","End":"06:47.550","Text":"Why is that? This then is the joint probability function of X and Y."}],"ID":13202},{"Watched":false,"Name":"Exercise 3 - Part b","Duration":"4m 10s","ChapterTopicVideoID":12732,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12732.jpeg","UploadDate":"2018-07-30T15:05:26.3170000","DurationForVideoObject":"PT4M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.835","Text":"In this section, we\u0027re asked whether the variables are uncorrelated."},{"Start":"00:03.835 ","End":"00:08.885","Text":"In order to see whether they\u0027re uncorrelated or not we need to go to the covariance."},{"Start":"00:08.885 ","End":"00:14.040","Text":"The covariance of x and y. What does that equals?"},{"Start":"00:14.040 ","End":"00:17.880","Text":"It\u0027s defined as the expectation of x times y"},{"Start":"00:17.880 ","End":"00:23.085","Text":"minus the expectation of x times the expectation of y."},{"Start":"00:23.085 ","End":"00:26.235","Text":"In order for the variables to be uncorrelated,"},{"Start":"00:26.235 ","End":"00:28.635","Text":"that means that the covariance,"},{"Start":"00:28.635 ","End":"00:30.960","Text":"it has to be equal to 0."},{"Start":"00:30.960 ","End":"00:36.495","Text":"That means that the expectation of x times y,"},{"Start":"00:36.495 ","End":"00:43.030","Text":"that has to be equal to the expectation of x times the expectation of y."},{"Start":"00:43.310 ","End":"00:48.045","Text":"In order to see if this is true or not,"},{"Start":"00:48.045 ","End":"00:52.000","Text":"let\u0027s take a look at the expectation of x and the expectation of y."},{"Start":"00:52.000 ","End":"00:53.780","Text":"In order to calculate that,"},{"Start":"00:53.780 ","End":"00:57.530","Text":"we need to calculate the marginal probabilities of x and y."},{"Start":"00:57.530 ","End":"01:00.770","Text":"Let\u0027s take a look at the marginal probabilities of x."},{"Start":"01:00.770 ","End":"01:03.560","Text":"That\u0027s 1 plus 3 plus 3 plus 1."},{"Start":"01:03.560 ","End":"01:06.535","Text":"That\u0027s 8 divided by 27,"},{"Start":"01:06.535 ","End":"01:10.820","Text":"here, that\u0027ll be 12 divided by 27."},{"Start":"01:10.820 ","End":"01:13.370","Text":"That\u0027s 6 divided by 27,"},{"Start":"01:13.370 ","End":"01:15.425","Text":"and that\u0027s 1 divided by 27."},{"Start":"01:15.425 ","End":"01:21.617","Text":"Since the distribution of x and y is identical,"},{"Start":"01:21.617 ","End":"01:24.290","Text":"then the marginal distribution is also identical."},{"Start":"01:24.290 ","End":"01:27.365","Text":"That means that here that will be 8 divided by 27,"},{"Start":"01:27.365 ","End":"01:30.215","Text":"that\u0027s 12 divided by 27."},{"Start":"01:30.215 ","End":"01:33.080","Text":"Here, that\u0027ll be 6 divided by 27,"},{"Start":"01:33.080 ","End":"01:35.180","Text":"and that\u0027s 1 divided by 27."},{"Start":"01:35.180 ","End":"01:39.065","Text":"Let\u0027s just make sure that everything adds up to 1."},{"Start":"01:39.065 ","End":"01:42.620","Text":"8 plus 12 plus 6 plus 1, that\u0027s 27."},{"Start":"01:42.620 ","End":"01:44.990","Text":"27 divided by 27 equals 1."},{"Start":"01:44.990 ","End":"01:52.340","Text":"Same thing here and the joint probabilities also add up to 1. We\u0027re fine here."},{"Start":"01:52.340 ","End":"01:56.870","Text":"Now let\u0027s calculate the expectation of x."},{"Start":"01:56.870 ","End":"01:59.005","Text":"The expectation of x,"},{"Start":"01:59.005 ","End":"02:03.110","Text":"that equals to the sum of x times its probabilities."},{"Start":"02:03.110 ","End":"02:07.880","Text":"Where x equals 0, we don\u0027t need to do some hard work here."},{"Start":"02:07.880 ","End":"02:09.890","Text":"Here where y equals 0,"},{"Start":"02:09.890 ","End":"02:11.990","Text":"we don\u0027t need to work hard here as well,"},{"Start":"02:11.990 ","End":"02:16.115","Text":"so we can delete these columns. What do we have there?"},{"Start":"02:16.115 ","End":"02:17.495","Text":"Expectation of x,"},{"Start":"02:17.495 ","End":"02:20.270","Text":"that\u0027s 1 times its probabilities."},{"Start":"02:20.270 ","End":"02:29.930","Text":"So it\u0027s 1 times 12 divided by 27 plus 2 times 6 divided by 27 plus so on and so forth."},{"Start":"02:29.930 ","End":"02:35.300","Text":"That equals to 1. Since the distribution of x and y is the same,"},{"Start":"02:35.300 ","End":"02:38.420","Text":"that means that the expectation of x and y are the same."},{"Start":"02:38.420 ","End":"02:44.635","Text":"That means that the expectation of x equals to 1 and that equals to the expectation of y."},{"Start":"02:44.635 ","End":"02:47.970","Text":"That also equals to 1."},{"Start":"02:47.970 ","End":"02:53.205","Text":"What about the expectation of x times y?"},{"Start":"02:53.205 ","End":"03:00.785","Text":"Let\u0027s take a look, that\u0027s 1 times 1 times 6 divided by 27."},{"Start":"03:00.785 ","End":"03:04.370","Text":"X equaling 1 times y equals 1 times the joint probability."},{"Start":"03:04.370 ","End":"03:06.170","Text":"That\u0027s 6 divided by 27."},{"Start":"03:06.170 ","End":"03:12.135","Text":"That\u0027s 1 times 1 times 6 divided by 27 plus,"},{"Start":"03:12.135 ","End":"03:16.860","Text":"now it\u0027ll be 2 times 1 times 3 divided by 27."},{"Start":"03:16.860 ","End":"03:20.960","Text":"It would be 2 times 1 times 3 divided by 27,"},{"Start":"03:20.960 ","End":"03:22.850","Text":"and so on and so forth."},{"Start":"03:22.850 ","End":"03:27.085","Text":"That equals to 2/3."},{"Start":"03:27.085 ","End":"03:29.835","Text":"We have the expectation of x,"},{"Start":"03:29.835 ","End":"03:31.490","Text":"we have the expectation of y,"},{"Start":"03:31.490 ","End":"03:34.025","Text":"and we have the expectation of x times y."},{"Start":"03:34.025 ","End":"03:37.385","Text":"If we have this equality right here,"},{"Start":"03:37.385 ","End":"03:40.130","Text":"that means that the variables are uncorrelated because"},{"Start":"03:40.130 ","End":"03:45.050","Text":"this equality means that the covariance equals to 0."},{"Start":"03:45.050 ","End":"03:48.230","Text":"What\u0027s the expectation of x times y?"},{"Start":"03:48.230 ","End":"03:50.830","Text":"That\u0027s 2/3."},{"Start":"03:50.830 ","End":"03:53.535","Text":"What\u0027s the expectation of x?"},{"Start":"03:53.535 ","End":"03:57.600","Text":"That\u0027s 1 and the expectation of y that\u0027s also 1."},{"Start":"03:57.600 ","End":"04:03.210","Text":"We see that 2/3 does not equal to 1 times 1."},{"Start":"04:03.210 ","End":"04:09.450","Text":"That means that x and y, they\u0027re correlated."}],"ID":13203},{"Watched":false,"Name":"Exercise 4 - Part a","Duration":"6m 9s","ChapterTopicVideoID":12733,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12733.jpeg","UploadDate":"2018-07-30T15:06:01.2430000","DurationForVideoObject":"PT6M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.910","Text":"In this question a balanced die is thrown twice."},{"Start":"00:02.910 ","End":"00:05.550","Text":"We define X is the larger number of"},{"Start":"00:05.550 ","End":"00:10.650","Text":"the 2 throws and Y is the number of even numbers received after the 2 throws,"},{"Start":"00:10.650 ","End":"00:14.790","Text":"and we\u0027re asked to find the joint probability function of X and Y."},{"Start":"00:14.790 ","End":"00:20.010","Text":"Well, the first thing that we want to know is to find out what is our sample space."},{"Start":"00:20.010 ","End":"00:22.260","Text":"Well, on the first throw,"},{"Start":"00:22.260 ","End":"00:24.750","Text":"we have 6 possible results,"},{"Start":"00:24.750 ","End":"00:28.350","Text":"and on the second throw we also have 6 possible results,"},{"Start":"00:28.350 ","End":"00:33.325","Text":"so we have basically 36 possible combinations."},{"Start":"00:33.325 ","End":"00:37.286","Text":"This is the sample space that we\u0027re dealing with."},{"Start":"00:37.286 ","End":"00:44.120","Text":"Let\u0027s write down the values of X and Y on the first and second throw."},{"Start":"00:44.390 ","End":"00:51.248","Text":"Here\u0027s a table that has the possible values on the first throw and the second throw."},{"Start":"00:51.248 ","End":"00:53.730","Text":"Here we see the values of the first throw from 1 to"},{"Start":"00:53.730 ","End":"00:56.970","Text":"6 and the values on the second throw from 1 to 6,"},{"Start":"00:56.970 ","End":"00:59.615","Text":"and within the table in the white cells,"},{"Start":"00:59.615 ","End":"01:04.940","Text":"we want to write the values of the variables for X and Y,"},{"Start":"01:04.940 ","End":"01:06.950","Text":"where X is the largest number of"},{"Start":"01:06.950 ","End":"01:12.065","Text":"the 2 throws and Y is the number of even numbers received after the 2 throws."},{"Start":"01:12.065 ","End":"01:14.897","Text":"Let\u0027s write down the values of X,"},{"Start":"01:14.897 ","End":"01:19.035","Text":"so if we have on the first throw 1 and then the second throw 1,"},{"Start":"01:19.035 ","End":"01:21.165","Text":"well, the larger number is 1."},{"Start":"01:21.165 ","End":"01:26.805","Text":"What about on the first throw we have 2 and on the second throw we have 1."},{"Start":"01:26.805 ","End":"01:28.920","Text":"Well, the larger number is 2."},{"Start":"01:28.920 ","End":"01:30.765","Text":"In here that\u0027ll be 3,"},{"Start":"01:30.765 ","End":"01:33.690","Text":"4, 5, and 6."},{"Start":"01:33.690 ","End":"01:35.310","Text":"What about here?"},{"Start":"01:35.310 ","End":"01:38.640","Text":"The first throw is 1 and the second throw is 2."},{"Start":"01:38.640 ","End":"01:41.355","Text":"Well, 2 is the larger number,"},{"Start":"01:41.355 ","End":"01:45.564","Text":"and the first throw 2 and the second throw 2, that\u0027ll be 2,"},{"Start":"01:45.564 ","End":"01:50.760","Text":"that\u0027ll be 3, 4, 5, and 6."},{"Start":"01:50.760 ","End":"01:57.040","Text":"Here\u0027s a table with all of the values of X on the first and second throw."},{"Start":"01:57.040 ","End":"01:58.675","Text":"Now, what about Y?"},{"Start":"01:58.675 ","End":"02:02.590","Text":"The number of even numbers received after 2 throws."},{"Start":"02:02.590 ","End":"02:07.680","Text":"Well, if we have here where the first throw we got 1,"},{"Start":"02:07.680 ","End":"02:09.010","Text":"the second throw we got 1, well,"},{"Start":"02:09.010 ","End":"02:10.810","Text":"how many even numbers do we have?"},{"Start":"02:10.810 ","End":"02:12.555","Text":"Well, we have 0."},{"Start":"02:12.555 ","End":"02:15.405","Text":"On the first throw we had a 2,"},{"Start":"02:15.405 ","End":"02:16.860","Text":"and then the second throw we had a 1,"},{"Start":"02:16.860 ","End":"02:18.655","Text":"well, how many even numbers there?"},{"Start":"02:18.655 ","End":"02:20.650","Text":"Well, that\u0027s 1 even number,"},{"Start":"02:20.650 ","End":"02:23.180","Text":"and here that\u0027ll be 0."},{"Start":"02:23.180 ","End":"02:24.930","Text":"Here, that\u0027ll be 1, here,"},{"Start":"02:24.930 ","End":"02:27.245","Text":"that\u0027ll be 0, and here that\u0027ll be 1."},{"Start":"02:27.245 ","End":"02:28.900","Text":"Now what about here?"},{"Start":"02:28.900 ","End":"02:31.030","Text":"Well, on the first throw that\u0027s a 1,"},{"Start":"02:31.030 ","End":"02:32.980","Text":"second throw that\u0027s a 2,"},{"Start":"02:32.980 ","End":"02:35.510","Text":"so we have 1 even number."},{"Start":"02:35.540 ","End":"02:39.540","Text":"On the first throw we have a 2,"},{"Start":"02:39.540 ","End":"02:42.375","Text":"on the second throw we have a 2, well, that\u0027s 2."},{"Start":"02:42.375 ","End":"02:44.430","Text":"We have 1, 2,"},{"Start":"02:44.430 ","End":"02:46.334","Text":"and a 1 and a 2,"},{"Start":"02:46.334 ","End":"02:48.905","Text":"so here\u0027s a table."},{"Start":"02:48.905 ","End":"02:52.670","Text":"After we\u0027ve written down the value of"},{"Start":"02:52.670 ","End":"02:57.005","Text":"Y which was the even numbers received after 2 throws."},{"Start":"02:57.005 ","End":"03:03.835","Text":"Now, we have all the possible values for X and Y in our sample space."},{"Start":"03:03.835 ","End":"03:09.210","Text":"Let\u0027s write down then the joint probability function of X and Y."},{"Start":"03:09.500 ","End":"03:14.500","Text":"Here\u0027s the joint probability template for X and Y,"},{"Start":"03:14.500 ","End":"03:18.770","Text":"and what we want to do is we want to write down the joint probabilities right"},{"Start":"03:18.770 ","End":"03:24.215","Text":"here in the white cells for all possible values of X and Y."},{"Start":"03:24.215 ","End":"03:30.960","Text":"Now, what happens when X equals 1 and Y equals 0?"},{"Start":"03:30.960 ","End":"03:35.380","Text":"Well, X equals 1 and Y equals 0."},{"Start":"03:36.590 ","End":"03:39.440","Text":"This value right here."},{"Start":"03:39.440 ","End":"03:45.340","Text":"Now do we have any other values where X equals 1 and Y equals 0? Well, no."},{"Start":"03:45.340 ","End":"03:48.230","Text":"How many possible values do we have here?"},{"Start":"03:48.230 ","End":"03:50.660","Text":"Well, we have 36 possible values,"},{"Start":"03:50.660 ","End":"03:53.007","Text":"so we only have 1 of them,"},{"Start":"03:53.007 ","End":"03:57.320","Text":"so here, that\u0027ll be 1 divided by 36."},{"Start":"03:57.320 ","End":"04:02.760","Text":"Now, what about where X equals 2 and Y equals 0?"},{"Start":"04:02.760 ","End":"04:06.195","Text":"Well, when X equals 2 and Y is 0,"},{"Start":"04:06.195 ","End":"04:08.510","Text":"I don\u0027t see anything here,"},{"Start":"04:08.510 ","End":"04:10.895","Text":"so that\u0027ll be 0 here."},{"Start":"04:10.895 ","End":"04:16.245","Text":"What about when X equals 3 and Y equals 0?"},{"Start":"04:16.245 ","End":"04:18.360","Text":"Well, when X equals 3 and Y equals 0,"},{"Start":"04:18.360 ","End":"04:21.750","Text":"that\u0027s right here, that\u0027s 1."},{"Start":"04:21.750 ","End":"04:26.490","Text":"Right here, that\u0027s 2 and right here that\u0027s 3."},{"Start":"04:26.490 ","End":"04:29.865","Text":"We have here 3 divided by 36."},{"Start":"04:29.865 ","End":"04:31.800","Text":"What about X equals 4,"},{"Start":"04:31.800 ","End":"04:34.570","Text":"Y equals 0? That\u0027s 0."},{"Start":"04:35.300 ","End":"04:38.760","Text":"X equals 5 and Y equals 0,"},{"Start":"04:38.760 ","End":"04:42.090","Text":"well, that\u0027s right here, that\u0027s 1,"},{"Start":"04:42.090 ","End":"04:46.095","Text":"2, 3,"},{"Start":"04:46.095 ","End":"04:50.010","Text":"4, and 5."},{"Start":"04:50.010 ","End":"04:54.540","Text":"That\u0027s 5 divided by 36."},{"Start":"04:54.540 ","End":"04:58.430","Text":"What about when X equals 6 and Y equals 0?"},{"Start":"04:58.430 ","End":"05:02.340","Text":"Well, we don\u0027t have that, that\u0027s 0 right here."},{"Start":"05:02.480 ","End":"05:08.000","Text":"Here we have the joint probability function all filled out."},{"Start":"05:08.000 ","End":"05:11.200","Text":"The joint probabilities here are filled out."},{"Start":"05:11.200 ","End":"05:14.055","Text":"Now, what about the marginal probabilities?"},{"Start":"05:14.055 ","End":"05:17.520","Text":"Well, for Y, that\u0027ll be 5 plus 3 plus 1,"},{"Start":"05:17.520 ","End":"05:21.830","Text":"that\u0027ll be 9 divided by 36."},{"Start":"05:21.830 ","End":"05:25.528","Text":"Here that\u0027ll be 2 plus 2 plus 4 plus 4 plus 6,"},{"Start":"05:25.528 ","End":"05:30.125","Text":"that\u0027ll be 18 divided by 36,"},{"Start":"05:30.125 ","End":"05:33.020","Text":"and that\u0027s 9 divided by 36."},{"Start":"05:33.020 ","End":"05:37.195","Text":"Here, that\u0027ll be 1 divided by 36,"},{"Start":"05:37.195 ","End":"05:43.035","Text":"3 divided by 36,5 divided by 36,"},{"Start":"05:43.035 ","End":"05:45.600","Text":"7 divided by 36,"},{"Start":"05:45.600 ","End":"05:47.970","Text":"9 divided by 36,"},{"Start":"05:47.970 ","End":"05:50.505","Text":"and 11 divided by 36."},{"Start":"05:50.505 ","End":"05:56.135","Text":"Let\u0027s just check that the marginal probabilities add up to 1 and they do,"},{"Start":"05:56.135 ","End":"06:01.490","Text":"and so do all the probabilities within the table,"},{"Start":"06:01.490 ","End":"06:05.028","Text":"the joint probabilities,"},{"Start":"06:05.028 ","End":"06:10.680","Text":"so this then is the joint probability function of X and Y."}],"ID":13204},{"Watched":false,"Name":"Exercise 4 - Part b","Duration":"5m 15s","ChapterTopicVideoID":12734,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12734.jpeg","UploadDate":"2018-07-30T15:06:52.0070000","DurationForVideoObject":"PT5M15S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.005","Text":"In this section, we\u0027re asked to calculate the correlation coefficient of X and Y."},{"Start":"00:04.005 ","End":"00:07.245","Text":"The correlation coefficient of X and Y,"},{"Start":"00:07.245 ","End":"00:11.535","Text":"that equals to the covariance of X and Y"},{"Start":"00:11.535 ","End":"00:17.100","Text":"divided by the standard deviation of X times the standard deviation of Y."},{"Start":"00:17.100 ","End":"00:19.170","Text":"In order to calculate that,"},{"Start":"00:19.170 ","End":"00:21.990","Text":"we need to calculate the standard deviation of X,"},{"Start":"00:21.990 ","End":"00:23.445","Text":"the standard deviation of Y,"},{"Start":"00:23.445 ","End":"00:26.160","Text":"and the covariance of X and Y."},{"Start":"00:26.160 ","End":"00:29.220","Text":"The covariance of X and Y,"},{"Start":"00:29.220 ","End":"00:32.950","Text":"that equals to the expectation of X times Y"},{"Start":"00:32.950 ","End":"00:40.165","Text":"minus the expectation of X times the expectation of Y."},{"Start":"00:40.165 ","End":"00:44.930","Text":"Here we need to calculate the expectation of X, the expectation of Y,"},{"Start":"00:44.930 ","End":"00:47.296","Text":"and the expectation of X times Y,"},{"Start":"00:47.296 ","End":"00:50.370","Text":"so let\u0027s get to work."},{"Start":"00:51.020 ","End":"00:54.345","Text":"The expectation of X,"},{"Start":"00:54.345 ","End":"00:59.030","Text":"that equals to the sum of the values of X times its probability."},{"Start":"00:59.030 ","End":"01:06.320","Text":"That\u0027s 1 times 1 divided by 36 plus 2 times 3 divided by 36, and so on and so forth."},{"Start":"01:06.320 ","End":"01:07.805","Text":"Let\u0027s just write that down."},{"Start":"01:07.805 ","End":"01:14.930","Text":"1 times 1 divided by 36 plus 2 times 3 divided by 36,"},{"Start":"01:14.930 ","End":"01:16.565","Text":"and so on and so forth,"},{"Start":"01:16.565 ","End":"01:22.335","Text":"and that is equal to 4.472."},{"Start":"01:22.335 ","End":"01:25.015","Text":"Now, what about the variance of X?"},{"Start":"01:25.015 ","End":"01:29.390","Text":"The variance of X is the sum of the values"},{"Start":"01:29.390 ","End":"01:33.965","Text":"of X squared times the probability minus the expectation squared."},{"Start":"01:33.965 ","End":"01:42.830","Text":"That\u0027ll be 1 squared times 1 divided by 36 plus 2 squared times 3 divided by 36 plus,"},{"Start":"01:42.830 ","End":"01:47.465","Text":"and so on and so forth, minus 4.472 squared."},{"Start":"01:47.465 ","End":"01:51.872","Text":"That equals to 1.973,"},{"Start":"01:51.872 ","End":"01:57.390","Text":"so what about the expectation of Y?"},{"Start":"01:57.390 ","End":"02:02.515","Text":"Similarly, we can calculate the expectation of Y,"},{"Start":"02:02.515 ","End":"02:05.650","Text":"and that will be equal to 1."},{"Start":"02:05.650 ","End":"02:08.920","Text":"What about the variance of Y?"},{"Start":"02:08.920 ","End":"02:12.770","Text":"Similarly to how we calculate the variance for X,"},{"Start":"02:12.770 ","End":"02:15.750","Text":"that will equal to 0.5."},{"Start":"02:15.750 ","End":"02:18.520","Text":"I\u0027m sparing you all the gory details,"},{"Start":"02:18.520 ","End":"02:20.485","Text":"the gory calculations here."},{"Start":"02:20.485 ","End":"02:25.285","Text":"What about the covariance of X and Y?"},{"Start":"02:25.285 ","End":"02:30.070","Text":"As we said, that equals to the expectation of X times"},{"Start":"02:30.070 ","End":"02:36.440","Text":"Y minus the expectation of X times the expectation of Y,"},{"Start":"02:36.440 ","End":"02:39.730","Text":"now we have the expectation of X, that\u0027s here,"},{"Start":"02:39.730 ","End":"02:42.110","Text":"and we have the expectation of Y, that\u0027s here,"},{"Start":"02:42.110 ","End":"02:45.350","Text":"so what\u0027s the expectation of X times Y?"},{"Start":"02:45.350 ","End":"02:48.810","Text":"The expectation of X times Y,"},{"Start":"02:48.810 ","End":"02:50.630","Text":"that equals to the following."},{"Start":"02:50.630 ","End":"02:58.490","Text":"Let\u0027s get rid of all the cells that have 0 or all values of Y that have 0,"},{"Start":"02:58.490 ","End":"03:00.320","Text":"and we get this."},{"Start":"03:00.320 ","End":"03:06.620","Text":"We have 2 times 1 times 2 divided by 36. What\u0027s that?"},{"Start":"03:06.620 ","End":"03:09.135","Text":"That\u0027s X equaling 2,"},{"Start":"03:09.135 ","End":"03:13.190","Text":"Y equaling 1, so it\u0027s 2 times 1 times the joint probability,"},{"Start":"03:13.190 ","End":"03:17.540","Text":"and that equals to 2 divided by 36,"},{"Start":"03:17.540 ","End":"03:24.480","Text":"plus, now we have 3 times 1 times 2 divided by 36."},{"Start":"03:24.480 ","End":"03:29.160","Text":"It\u0027ll be 3 times 1 times 2 divided by 36."},{"Start":"03:29.160 ","End":"03:32.115","Text":"X equals 3, Y equals 1,"},{"Start":"03:32.115 ","End":"03:33.720","Text":"and the joint probabilities,"},{"Start":"03:33.720 ","End":"03:35.250","Text":"2 divided by 36,"},{"Start":"03:35.250 ","End":"03:37.560","Text":"and so on and so forth."},{"Start":"03:37.560 ","End":"03:46.650","Text":"That equals to 4.722."},{"Start":"03:46.650 ","End":"03:48.860","Text":"Now let\u0027s get back to our covariance."},{"Start":"03:48.860 ","End":"03:54.020","Text":"That\u0027s the expectation of X times Y minus the expectation of X times expectation of Y."},{"Start":"03:54.020 ","End":"03:56.630","Text":"The expectation of X times Y,"},{"Start":"03:56.630 ","End":"04:01.085","Text":"that\u0027s 4.722, here we just calculated that,"},{"Start":"04:01.085 ","End":"04:03.560","Text":"minus the expectation of X,"},{"Start":"04:03.560 ","End":"04:12.675","Text":"that\u0027s 4.472 times the expectation of Y, that equals to 1."},{"Start":"04:12.675 ","End":"04:16.710","Text":"All that comes out to 0.25."},{"Start":"04:16.710 ","End":"04:20.325","Text":"That\u0027s the covariance of X and Y."},{"Start":"04:20.325 ","End":"04:26.530","Text":"Excellent. We can go ahead and calculate our correlation coefficient."},{"Start":"04:26.530 ","End":"04:29.855","Text":"The correlation coefficient, as we said,"},{"Start":"04:29.855 ","End":"04:32.330","Text":"equals to the covariance of X and Y divided by"},{"Start":"04:32.330 ","End":"04:36.040","Text":"the standard deviation of X times the standard deviation of Y."},{"Start":"04:36.040 ","End":"04:39.345","Text":"The correlation coefficient of X and Y,"},{"Start":"04:39.345 ","End":"04:41.915","Text":"that equals to the covariance of X and Y."},{"Start":"04:41.915 ","End":"04:46.235","Text":"That\u0027s 0.25 divided by,"},{"Start":"04:46.235 ","End":"04:47.893","Text":"what\u0027s the standard deviation of X?"},{"Start":"04:47.893 ","End":"04:52.310","Text":"That\u0027s the square root of 1.973."},{"Start":"04:52.310 ","End":"04:56.980","Text":"Here\u0027s the variance of X and we have to take the square root of that times,"},{"Start":"04:56.980 ","End":"04:58.350","Text":"now, what\u0027s the variance of Y?"},{"Start":"04:58.350 ","End":"05:04.295","Text":"That\u0027s 0.5, so the standard deviation is the square root of 0.5."},{"Start":"05:04.295 ","End":"05:09.120","Text":"This ratio comes out to 0.252."},{"Start":"05:09.980 ","End":"05:16.110","Text":"This then is the correlation coefficient of X and Y."}],"ID":13205},{"Watched":false,"Name":"Exercise 4 - Part c","Duration":"3m 29s","ChapterTopicVideoID":12735,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12735.jpeg","UploadDate":"2018-07-30T15:07:14.2800000","DurationForVideoObject":"PT3M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.135","Text":"This section we\u0027re asked to find the probability distribution of Y given that X equals 2."},{"Start":"00:06.135 ","End":"00:15.615","Text":"Let\u0027s find, or let\u0027s calculate the probability of y for all values of y. Y equals 0,"},{"Start":"00:15.615 ","End":"00:17.370","Text":"y equals 1 and y equals 2,"},{"Start":"00:17.370 ","End":"00:23.940","Text":"given that x equals 2 and then we\u0027ll write the probability distribution."},{"Start":"00:23.940 ","End":"00:32.805","Text":"We have here the probability of Y equals 0 given that X equals to 2."},{"Start":"00:32.805 ","End":"00:35.610","Text":"Well, in the conditional probability,"},{"Start":"00:35.610 ","End":"00:39.870","Text":"in the denominator, we have the probability of what\u0027s given, what\u0027s known."},{"Start":"00:39.870 ","End":"00:42.075","Text":"That\u0027s x equals 2."},{"Start":"00:42.075 ","End":"00:44.720","Text":"In the numerator, we have the intersect,"},{"Start":"00:44.720 ","End":"00:50.940","Text":"that\u0027s the probability of y equaling 0 and x equals to 2."},{"Start":"00:51.200 ","End":"00:53.930","Text":"What do we have in the denominator?"},{"Start":"00:53.930 ","End":"00:57.900","Text":"Well, when x equals 2 the probability of x equals 2,"},{"Start":"00:57.900 ","End":"01:01.175","Text":"that\u0027s 3 divided by 36, that\u0027s right here."},{"Start":"01:01.175 ","End":"01:03.095","Text":"What about the intercept?"},{"Start":"01:03.095 ","End":"01:11.820","Text":"Y equals 0 and x equals 2 so that\u0027s the probability 0 so we have 0."},{"Start":"01:11.820 ","End":"01:18.170","Text":"What about the probability of Y equals 1 given that x equals 2?"},{"Start":"01:18.170 ","End":"01:23.795","Text":"Well, again, in the denominator we have the probability of X equaling 2."},{"Start":"01:23.795 ","End":"01:26.600","Text":"In the numerator we have the intersects,"},{"Start":"01:26.600 ","End":"01:31.155","Text":"that\u0027s the probability of y equaling 1 and X equaling 2."},{"Start":"01:31.155 ","End":"01:34.940","Text":"Now, that equals 2 in the denominator."},{"Start":"01:34.940 ","End":"01:41.390","Text":"Again, there will be 3 divided by 36 and the probability of the intersection,"},{"Start":"01:41.390 ","End":"01:47.000","Text":"basically the joint distribution of X equaling 2 and y equaling 1."},{"Start":"01:47.000 ","End":"01:49.085","Text":"Well, that\u0027s right here."},{"Start":"01:49.085 ","End":"01:52.910","Text":"That\u0027s still divided by 36."},{"Start":"01:52.910 ","End":"01:57.515","Text":"The probability here comes out to 2/3."},{"Start":"01:57.515 ","End":"02:02.380","Text":"What about the probability of Y now equaling to 2?"},{"Start":"02:02.380 ","End":"02:05.130","Text":"Given that X equals to 2."},{"Start":"02:05.130 ","End":"02:07.420","Text":"Again, in the denominator,"},{"Start":"02:07.420 ","End":"02:11.509","Text":"we have the probability of X equaling 2 and in the numerator"},{"Start":"02:11.509 ","End":"02:16.110","Text":"we have the intersect X equals 2."},{"Start":"02:16.110 ","End":"02:18.060","Text":"Now that equals 2,"},{"Start":"02:18.060 ","End":"02:20.230","Text":"3 divided by 36,"},{"Start":"02:20.230 ","End":"02:23.060","Text":"that\u0027s the probability of X equaling 2"},{"Start":"02:23.060 ","End":"02:28.550","Text":"and the joint probability of X equaling 2 and y equaling 2."},{"Start":"02:28.550 ","End":"02:30.500","Text":"Well, that\u0027s right here,"},{"Start":"02:30.500 ","End":"02:33.155","Text":"that\u0027s 1 divided by 36."},{"Start":"02:33.155 ","End":"02:35.845","Text":"We have here 1/3."},{"Start":"02:35.845 ","End":"02:43.230","Text":"let\u0027s now write the probability distribution of Y given that X equals to 2."},{"Start":"02:43.580 ","End":"02:48.930","Text":"Here we have the values of Y given X equals to"},{"Start":"02:48.930 ","End":"02:56.150","Text":"2 and here we\u0027ll have the probability of Y given X equaling 2."},{"Start":"02:56.150 ","End":"03:00.630","Text":"Now, the values of y given X equals 2,"},{"Start":"03:00.630 ","End":"03:03.940","Text":"well, that\u0027s either 0,1 or 2."},{"Start":"03:03.940 ","End":"03:06.830","Text":"Now for 0, the probability is 0,"},{"Start":"03:06.830 ","End":"03:08.465","Text":"so we won\u0027t write that down."},{"Start":"03:08.465 ","End":"03:13.790","Text":"It\u0027ll be just y equaling 1 and y equal 2 and the probability for that,"},{"Start":"03:13.790 ","End":"03:18.105","Text":"well, that\u0027s 2/3 here and 1 third here."},{"Start":"03:18.105 ","End":"03:22.895","Text":"Now, let\u0027s just make sure the sum of the probabilities adds up to 1 so we\u0027re okay,"},{"Start":"03:22.895 ","End":"03:30.120","Text":"so this then is the probability distribution of Y given that X equals to 2."}],"ID":13206},{"Watched":false,"Name":"Exercise 5 - Part a","Duration":"4m 42s","ChapterTopicVideoID":12736,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12736.jpeg","UploadDate":"2018-07-30T15:07:48.0670000","DurationForVideoObject":"PT4M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.345","Text":"In This question, we\u0027re given a building with 5 apartments."},{"Start":"00:03.345 ","End":"00:08.040","Text":"Now, Apartment 1 and Apartment 3 have been renovated while the other haven\u0027t."},{"Start":"00:08.040 ","End":"00:10.380","Text":"2 apartments are randomly selected."},{"Start":"00:10.380 ","End":"00:12.690","Text":"We define the following variables."},{"Start":"00:12.690 ","End":"00:16.290","Text":"X is the number of renovated apartments that are selected,"},{"Start":"00:16.290 ","End":"00:19.920","Text":"and Y is the number of odd-numbered apartments selected."},{"Start":"00:19.920 ","End":"00:21.570","Text":"We\u0027re asked to construct"},{"Start":"00:21.570 ","End":"00:27.090","Text":"the joint probability function and the marginal probability function of X and Y."},{"Start":"00:27.090 ","End":"00:32.565","Text":"The first thing that we want to know is we want to know what our sample sizes."},{"Start":"00:32.565 ","End":"00:39.040","Text":"Well, we have 5 apartments and we sample 2 of them,"},{"Start":"00:39.040 ","End":"00:42.925","Text":"so we have 10 different possibilities."},{"Start":"00:42.925 ","End":"00:45.290","Text":"Because this isn\u0027t a large number,"},{"Start":"00:45.290 ","End":"00:51.996","Text":"let\u0027s just write down all the possible combinations of the apartments sampled,"},{"Start":"00:51.996 ","End":"00:57.020","Text":"so here then are all the possible combinations of the partners that we can sample."},{"Start":"00:57.020 ","End":"00:59.720","Text":"We can sample Apartment 1 and Apartment 2,"},{"Start":"00:59.720 ","End":"01:01.265","Text":"Apartment 1 and 3,"},{"Start":"01:01.265 ","End":"01:02.750","Text":"Apartment 2 and 4,"},{"Start":"01:02.750 ","End":"01:04.130","Text":"Apartment 4 and 5,"},{"Start":"01:04.130 ","End":"01:05.615","Text":"and so on and so forth."},{"Start":"01:05.615 ","End":"01:09.635","Text":"Now, let\u0027s see what are the values of"},{"Start":"01:09.635 ","End":"01:14.009","Text":"X and Y for each 1 of the combination of the apartment sampled."},{"Start":"01:14.009 ","End":"01:23.715","Text":"Now, X then is the renovated apartments and Y are the odd number apartments."},{"Start":"01:23.715 ","End":"01:27.050","Text":"How many renovated apartments do we have here?"},{"Start":"01:27.050 ","End":"01:31.550","Text":"Well, don\u0027t forget, Apartment 1 and Apartment 3 were renovated."},{"Start":"01:31.550 ","End":"01:34.800","Text":"Here we\u0027ve selected apartment number 1,"},{"Start":"01:34.800 ","End":"01:37.670","Text":"so we have 1 renovated apartment here."},{"Start":"01:37.670 ","End":"01:39.440","Text":"Now, how many do we have here?"},{"Start":"01:39.440 ","End":"01:42.605","Text":"We have 2, Apartments 1 and 3 are renovated."},{"Start":"01:42.605 ","End":"01:44.240","Text":"How many do we have here?"},{"Start":"01:44.240 ","End":"01:46.310","Text":"1 and 1."},{"Start":"01:46.310 ","End":"01:47.960","Text":"What about 2 and 3?"},{"Start":"01:47.960 ","End":"01:50.330","Text":"3 is renovated and 2 isn\u0027t."},{"Start":"01:50.330 ","End":"01:52.300","Text":"We have 1 here,"},{"Start":"01:52.300 ","End":"01:54.900","Text":"2 and 4 that\u0027s 0 apartments,"},{"Start":"01:54.900 ","End":"01:56.325","Text":"2 and 5 is 0,"},{"Start":"01:56.325 ","End":"01:57.970","Text":"3 and 4 is 1,"},{"Start":"01:57.970 ","End":"01:59.250","Text":"3 is renovated,"},{"Start":"01:59.250 ","End":"02:03.645","Text":"3 and 5 are 1 and 4 and 5 that\u0027s 0."},{"Start":"02:03.645 ","End":"02:08.300","Text":"These are the valid values for X. What about Y?"},{"Start":"02:08.300 ","End":"02:12.110","Text":"As we said, Y are the odd-numbered apartments,"},{"Start":"02:12.110 ","End":"02:14.285","Text":"the number of odd numbered apartments."},{"Start":"02:14.285 ","End":"02:16.370","Text":"How many do we have here?"},{"Start":"02:16.370 ","End":"02:19.430","Text":"Well, we have 1 odd number apartment here, that\u0027s 1."},{"Start":"02:19.430 ","End":"02:21.140","Text":"Here, we have 2."},{"Start":"02:21.140 ","End":"02:22.925","Text":"Here, we have 1."},{"Start":"02:22.925 ","End":"02:25.010","Text":"Here, we have 2."},{"Start":"02:25.010 ","End":"02:27.125","Text":"Here, we have 1."},{"Start":"02:27.125 ","End":"02:31.890","Text":"That\u0027s 0 here, 1 here,"},{"Start":"02:32.900 ","End":"02:37.210","Text":"2, and 1."},{"Start":"02:37.400 ","End":"02:42.750","Text":"These are the valid values for Y."},{"Start":"02:43.300 ","End":"02:47.135","Text":"Now that we have the valid values for X and Y,"},{"Start":"02:47.135 ","End":"02:50.075","Text":"let\u0027s construct the joint probability function"},{"Start":"02:50.075 ","End":"02:54.060","Text":"and the marginal probability functions of X and Y."},{"Start":"02:54.650 ","End":"02:58.790","Text":"This is a table that will contain the joint probability function,"},{"Start":"02:58.790 ","End":"03:02.005","Text":"the marginal probability function of X and Y."},{"Start":"03:02.005 ","End":"03:05.405","Text":"Here in the white cells right here,"},{"Start":"03:05.405 ","End":"03:09.200","Text":"we\u0027ll write down the joint probabilities of X and Y."},{"Start":"03:09.200 ","End":"03:10.730","Text":"Now let\u0027s take a look at this."},{"Start":"03:10.730 ","End":"03:13.615","Text":"When X equals 0 and Y equals 0,"},{"Start":"03:13.615 ","End":"03:21.120","Text":"we have only 1 case of that right here out of 10 possibilities here."},{"Start":"03:21.120 ","End":"03:25.100","Text":"That\u0027ll be 1 divided by 10 or 0.1."},{"Start":"03:25.100 ","End":"03:31.444","Text":"Now, what about when X equals 1 and Y equals 0?"},{"Start":"03:31.444 ","End":"03:34.010","Text":"We don\u0027t have anything,"},{"Start":"03:34.010 ","End":"03:35.615","Text":"so that\u0027s 0 here."},{"Start":"03:35.615 ","End":"03:40.375","Text":"The same thing for X equals 2 and Y equals 0,"},{"Start":"03:40.375 ","End":"03:42.470","Text":"that\u0027ll be 0 here."},{"Start":"03:42.470 ","End":"03:46.540","Text":"Let\u0027s just fill in the rest of the table quickly."},{"Start":"03:46.540 ","End":"03:52.250","Text":"Here we have the joint probabilities, already written down."},{"Start":"03:52.250 ","End":"03:59.310","Text":"As we said, when X equals 0 and Y equals 1 for example,"},{"Start":"03:59.310 ","End":"04:01.410","Text":"we have 2 cases of that right here,"},{"Start":"04:01.410 ","End":"04:03.630","Text":"right here, and right here."},{"Start":"04:03.630 ","End":"04:08.895","Text":"That means we have 2 divided by 10 or 0.2 and so on and so forth."},{"Start":"04:08.895 ","End":"04:13.430","Text":"Now let\u0027s write down the marginal probabilities."},{"Start":"04:13.430 ","End":"04:15.950","Text":"All we have to do is add up the rows and columns."},{"Start":"04:15.950 ","End":"04:22.280","Text":"Now for X equals 0, that\u0027s 0.1 plus 0.2 and that\u0027ll be 0.3,"},{"Start":"04:22.280 ","End":"04:26.170","Text":"and here that\u0027ll be 0.6, that\u0027ll be 0.1."},{"Start":"04:26.170 ","End":"04:29.865","Text":"For Y, that\u0027ll be 0.1,"},{"Start":"04:29.865 ","End":"04:35.280","Text":"that\u0027ll be 0.6 and that will be 0.3."},{"Start":"04:35.280 ","End":"04:38.810","Text":"Here we are. That\u0027s the joint probability function"},{"Start":"04:38.810 ","End":"04:42.540","Text":"and the marginal probability function of X and Y."}],"ID":13207},{"Watched":false,"Name":"Exercise 5 - Parts b-c","Duration":"3m 43s","ChapterTopicVideoID":12739,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12739.jpeg","UploadDate":"2018-07-30T15:08:58.8030000","DurationForVideoObject":"PT3M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.380","Text":"In this section, we\u0027re asked whether the variables X and Y are correlated."},{"Start":"00:04.380 ","End":"00:06.570","Text":"Well, in order to answer that question,"},{"Start":"00:06.570 ","End":"00:10.395","Text":"we need to look at the covariance to see if it equals to 0 or not."},{"Start":"00:10.395 ","End":"00:12.345","Text":"If it equals to 0,"},{"Start":"00:12.345 ","End":"00:14.775","Text":"then the variables are not correlated."},{"Start":"00:14.775 ","End":"00:16.500","Text":"If it doesn\u0027t equal to 0,"},{"Start":"00:16.500 ","End":"00:19.418","Text":"then the variables are correlated,"},{"Start":"00:19.418 ","End":"00:23.055","Text":"so again, the covariance of X and Y,"},{"Start":"00:23.055 ","End":"00:24.510","Text":"that\u0027s what we\u0027re looking at."},{"Start":"00:24.510 ","End":"00:28.650","Text":"That\u0027s defined as the expectation of X times Y"},{"Start":"00:28.650 ","End":"00:34.800","Text":"minus the expectation of X times the expectation of Y."},{"Start":"00:34.800 ","End":"00:39.950","Text":"All we need to do is we need to calculate these expectations right here."},{"Start":"00:39.950 ","End":"00:42.665","Text":"The expectation of X,"},{"Start":"00:42.665 ","End":"00:46.880","Text":"well that equals to the sum of X times its probabilities,"},{"Start":"00:46.880 ","End":"00:48.710","Text":"that\u0027s 0 times 0.3,"},{"Start":"00:48.710 ","End":"00:50.390","Text":"plus 1 times 0.6,"},{"Start":"00:50.390 ","End":"00:55.805","Text":"plus 2 times 0.1, and that equals to 0.8."},{"Start":"00:55.805 ","End":"01:02.150","Text":"Similarly, the expectation of Y is the sum of Y times its probabilities,"},{"Start":"01:02.150 ","End":"01:05.315","Text":"and that comes out to 1.2."},{"Start":"01:05.315 ","End":"01:08.735","Text":"Now, what about the expectation of X times Y?"},{"Start":"01:08.735 ","End":"01:13.280","Text":"Well, that equals to the value of X times the value of"},{"Start":"01:13.280 ","End":"01:18.880","Text":"Y times the joint probability for all values of X and Y."},{"Start":"01:18.880 ","End":"01:26.509","Text":"That\u0027ll be 0 times 0 times 0.1 plus 0 times 1 times 0,"},{"Start":"01:26.509 ","End":"01:28.610","Text":"plus 0 times 2 times 0,"},{"Start":"01:28.610 ","End":"01:30.025","Text":"and so on and so forth."},{"Start":"01:30.025 ","End":"01:34.245","Text":"Now that equals to 1.2 as well."},{"Start":"01:34.245 ","End":"01:39.740","Text":"The covariance of X and Y,"},{"Start":"01:39.740 ","End":"01:42.170","Text":"that equals to this expression right here."},{"Start":"01:42.170 ","End":"01:44.210","Text":"Well, that\u0027s the expectation of X times Y."},{"Start":"01:44.210 ","End":"01:49.280","Text":"That\u0027s equals to 1.2 minus the expectation of X,"},{"Start":"01:49.280 ","End":"01:54.375","Text":"that\u0027s 0.8, times the expectation of Y, that\u0027s 1.2."},{"Start":"01:54.375 ","End":"01:58.725","Text":"That equals to 0.24."},{"Start":"01:58.725 ","End":"02:01.560","Text":"Now, that does not equal to 0,"},{"Start":"02:01.560 ","End":"02:06.040","Text":"so that means that the variables are correlated."},{"Start":"02:06.040 ","End":"02:08.170","Text":"In this section, we\u0027re asked,"},{"Start":"02:08.170 ","End":"02:10.865","Text":"what\u0027s the correlation coefficient of X and Y?"},{"Start":"02:10.865 ","End":"02:15.320","Text":"Well, the correlation coefficient rho of X and Y,"},{"Start":"02:15.320 ","End":"02:19.570","Text":"that equals to the covariance of X and Y"},{"Start":"02:19.570 ","End":"02:25.860","Text":"divided by the standard deviation of X times the standard deviation of Y."},{"Start":"02:26.350 ","End":"02:35.730","Text":"Let\u0027s just calculate the variance of X here."},{"Start":"02:35.730 ","End":"02:40.160","Text":"Well, that equals to the sum of X squared times"},{"Start":"02:40.160 ","End":"02:45.915","Text":"its probability minus the expectation squared of X."},{"Start":"02:45.915 ","End":"02:50.585","Text":"Now, that comes out to 0.36."},{"Start":"02:50.585 ","End":"02:55.210","Text":"Similarly, the variance of Y,"},{"Start":"02:55.210 ","End":"02:59.390","Text":"well, that equals to 0.36 as well."},{"Start":"02:59.390 ","End":"03:02.075","Text":"Now what about the covariance of X and Y?"},{"Start":"03:02.075 ","End":"03:07.045","Text":"Well, we\u0027ve calculated the covariance of X and Y,"},{"Start":"03:07.045 ","End":"03:09.590","Text":"we\u0027ve calculated that in the previous section,"},{"Start":"03:09.590 ","End":"03:12.000","Text":"that equals to 0.24."},{"Start":"03:13.430 ","End":"03:18.285","Text":"The correlation coefficient of X and Y,"},{"Start":"03:18.285 ","End":"03:20.840","Text":"that equals to covariance of X and Y."},{"Start":"03:20.840 ","End":"03:26.870","Text":"That\u0027s 0.24 divided by the square root of 0.36."},{"Start":"03:26.870 ","End":"03:31.125","Text":"That\u0027s 0.6 times 0.6."},{"Start":"03:31.125 ","End":"03:34.310","Text":"That\u0027s the square root of 0.36."},{"Start":"03:34.310 ","End":"03:39.155","Text":"Now, that then equals to 2/3."},{"Start":"03:39.155 ","End":"03:43.320","Text":"This then is a correlation coefficient of X and Y."}],"ID":13208},{"Watched":false,"Name":"Exercise 5 - Part d","Duration":"5m 12s","ChapterTopicVideoID":12737,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12737.jpeg","UploadDate":"2018-07-30T15:08:24.0530000","DurationForVideoObject":"PT5M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.140","Text":"In this section, we\u0027re asked what\u0027s the correlation coefficient between the number of"},{"Start":"00:04.140 ","End":"00:08.475","Text":"renovated apartments and the number of even numbered apartments that were sampled."},{"Start":"00:08.475 ","End":"00:10.875","Text":"Let\u0027s just remind ourselves."},{"Start":"00:10.875 ","End":"00:18.195","Text":"X is the number of renovated apartments,"},{"Start":"00:18.195 ","End":"00:28.005","Text":"and Y is the number of odd numbered apartments."},{"Start":"00:28.005 ","End":"00:31.920","Text":"We\u0027re looking for the correlation coefficient between x,"},{"Start":"00:31.920 ","End":"00:35.175","Text":"the number of renovated apartments and the number of"},{"Start":"00:35.175 ","End":"00:39.088","Text":"even numbered apartments not the number of odd numbered apartments,"},{"Start":"00:39.088 ","End":"00:44.910","Text":"so we\u0027re going to have to transform y from odd number to even number apartments."},{"Start":"00:44.910 ","End":"00:47.310","Text":"Let\u0027s define a new random variable,"},{"Start":"00:47.310 ","End":"00:53.850","Text":"let\u0027s call that W and that equals to 2 now minus y. Why is that?"},{"Start":"00:53.850 ","End":"00:58.020","Text":"Well, 2 is a number of apartments that we sampled,"},{"Start":"00:58.020 ","End":"01:01.220","Text":"y is the number of odd numbered apartments."},{"Start":"01:01.220 ","End":"01:06.321","Text":"2 minus y would be the number of even numbered apartments,"},{"Start":"01:06.321 ","End":"01:12.290","Text":"so We\u0027re looking for the correlation coefficient between x and w. Now,"},{"Start":"01:12.290 ","End":"01:14.480","Text":"let\u0027s just write this like this."},{"Start":"01:14.480 ","End":"01:19.805","Text":"That\u0027s the correlation coefficient of x and 2 minus y."},{"Start":"01:19.805 ","End":"01:23.670","Text":"Now, that can also be written as the correlation coefficient between"},{"Start":"01:23.670 ","End":"01:27.935","Text":"x and minus 1y plus 2."},{"Start":"01:27.935 ","End":"01:30.140","Text":"Now, once we write it like this,"},{"Start":"01:30.140 ","End":"01:34.440","Text":"we see that we have a linear transformation on y."},{"Start":"01:34.660 ","End":"01:41.155","Text":"How does this linear transformation influence the correlation coefficient?"},{"Start":"01:41.155 ","End":"01:43.440","Text":"Well, we know this,"},{"Start":"01:43.440 ","End":"01:53.565","Text":"we know that the correlation coefficient of ax plus b and cy plus d,"},{"Start":"01:53.565 ","End":"01:57.710","Text":"that\u0027s a linear transformation on x and the linear transformation y,"},{"Start":"01:57.710 ","End":"01:59.300","Text":"well, that equals to what?"},{"Start":"01:59.300 ","End":"02:03.925","Text":"That equals to the correlation coefficient of x and y,"},{"Start":"02:03.925 ","End":"02:14.315","Text":"where 8 times c is greater than 0 and minus the correlation coefficient of x and y,"},{"Start":"02:14.315 ","End":"02:17.255","Text":"where a times c is less than 0."},{"Start":"02:17.255 ","End":"02:18.650","Text":"Now, what does that mean?"},{"Start":"02:18.650 ","End":"02:21.950","Text":"That means that a linear transformation does not"},{"Start":"02:21.950 ","End":"02:25.170","Text":"influence the strength of the connection between x and y,"},{"Start":"02:25.170 ","End":"02:29.135","Text":"all it does is it may change the direction."},{"Start":"02:29.135 ","End":"02:33.155","Text":"Now, getting back to our problem here,"},{"Start":"02:33.155 ","End":"02:39.200","Text":"a equals to 1 and c that equals to minus 1,"},{"Start":"02:39.200 ","End":"02:40.960","Text":"so 8 times c,"},{"Start":"02:40.960 ","End":"02:44.565","Text":"that equals to minus 1 which is less than 0."},{"Start":"02:44.565 ","End":"02:52.655","Text":"That means that this equals to minus the correlation coefficient of x and y."},{"Start":"02:52.655 ","End":"02:59.300","Text":"Now we\u0027ve calculated this correlation coefficient in the last section that equal to 2/3."},{"Start":"02:59.300 ","End":"03:02.940","Text":"Our answer here is minus 2/3."},{"Start":"03:03.400 ","End":"03:07.865","Text":"In this section, we\u0027re asked about the correlation coefficient between the number of"},{"Start":"03:07.865 ","End":"03:12.700","Text":"even number apartments and the number of odd numbered apartments in the sample."},{"Start":"03:12.700 ","End":"03:15.975","Text":"Why then was the number of"},{"Start":"03:15.975 ","End":"03:21.215","Text":"odd numbered apartments and w was the number of even numbered apartments."},{"Start":"03:21.215 ","End":"03:27.260","Text":"We\u0027re looking for the correlation coefficient between y and"},{"Start":"03:27.260 ","End":"03:34.460","Text":"w. Now that equals to the correlation coefficient between y and 2 minus y."},{"Start":"03:34.460 ","End":"03:37.940","Text":"Again, we can write this as between"},{"Start":"03:37.940 ","End":"03:43.375","Text":"the correlation coefficient between y and minus 1y plus 2."},{"Start":"03:43.375 ","End":"03:46.670","Text":"Now, this is really a no brainer and why is that?"},{"Start":"03:46.670 ","End":"03:49.880","Text":"Because the correlation coefficient Rho"},{"Start":"03:49.880 ","End":"03:54.110","Text":"provides a measure of the linear dependency between the 2 variables."},{"Start":"03:54.110 ","End":"04:01.600","Text":"Now here, there\u0027s a perfect dependency between y and w and it\u0027s linear."},{"Start":"04:01.600 ","End":"04:06.635","Text":"That means that if I know the value of y,"},{"Start":"04:06.635 ","End":"04:11.780","Text":"then I obviously know the value of w. Because of this dependency,"},{"Start":"04:11.780 ","End":"04:13.850","Text":"it\u0027s a perfect dependency."},{"Start":"04:13.850 ","End":"04:16.235","Text":"That equals to 1."},{"Start":"04:16.235 ","End":"04:18.815","Text":"Now, the question then becomes,"},{"Start":"04:18.815 ","End":"04:22.999","Text":"what\u0027s the direction of this correlation coefficient?"},{"Start":"04:22.999 ","End":"04:24.919","Text":"Is it positive or negative?"},{"Start":"04:24.919 ","End":"04:27.980","Text":"Well, it\u0027s negative because first of all,"},{"Start":"04:27.980 ","End":"04:31.100","Text":"let\u0027s look at the multipliers."},{"Start":"04:31.100 ","End":"04:38.535","Text":"The multiplier of y is 1 and the multiplier of w is minus 1."},{"Start":"04:38.535 ","End":"04:41.685","Text":"A times c is less than 0,"},{"Start":"04:41.685 ","End":"04:44.070","Text":"so that means that this is minus 1."},{"Start":"04:44.070 ","End":"04:47.660","Text":"This then is the correlation coefficient between"},{"Start":"04:47.660 ","End":"04:53.260","Text":"the number of odd numbered departments and the number of even numbered departments."},{"Start":"04:53.260 ","End":"04:59.510","Text":"It makes sense because if you know the number of odd numbered apartments in the sample,"},{"Start":"04:59.510 ","End":"05:06.965","Text":"then you can easily calculate the number of even numbered apartments in your sample."},{"Start":"05:06.965 ","End":"05:12.270","Text":"This is the meaning of a perfect linear dependency."}],"ID":13209},{"Watched":false,"Name":"Exercise 5 - Part e","Duration":"3m 41s","ChapterTopicVideoID":12738,"CourseChapterTopicPlaylistID":245019,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12738.jpeg","UploadDate":"2018-07-30T15:08:47.5770000","DurationForVideoObject":"PT3M41S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.414","Text":"In this section, we\u0027re given that every renovated apartment costs 2 million dollars and"},{"Start":"00:04.414 ","End":"00:09.000","Text":"very apartment that\u0027s not renovated costs 1.5 million dollars."},{"Start":"00:09.000 ","End":"00:11.940","Text":"We\u0027re asked about the correlation between the cost of"},{"Start":"00:11.940 ","End":"00:17.190","Text":"the sampled departments and the number of even-numbered apartments."},{"Start":"00:17.190 ","End":"00:19.880","Text":"Let\u0027s just define our variables."},{"Start":"00:19.880 ","End":"00:25.070","Text":"Let\u0027s define C as the cost of the sampled apartments."},{"Start":"00:25.070 ","End":"00:30.375","Text":"How much do the apartments that we sampled cost?"},{"Start":"00:30.375 ","End":"00:34.970","Text":"Well, they cost this, it\u0027s 2 million dollars times x,"},{"Start":"00:34.970 ","End":"00:38.420","Text":"which is the number of renovated apartments,"},{"Start":"00:38.420 ","End":"00:44.790","Text":"plus 1.5 million dollars times 2 minus x."},{"Start":"00:44.790 ","End":"00:48.620","Text":"That\u0027s the number of apartments that have not been renovated."},{"Start":"00:48.620 ","End":"00:51.440","Text":"Now, after doing a little bit of manipulation here,"},{"Start":"00:51.440 ","End":"00:58.000","Text":"that comes out to 0.5 times x plus 3."},{"Start":"00:58.100 ","End":"01:01.200","Text":"What about the other variable?"},{"Start":"01:01.200 ","End":"01:02.220","Text":"Well, the other variables,"},{"Start":"01:02.220 ","End":"01:04.430","Text":"the number of even-numbered apartments."},{"Start":"01:04.430 ","End":"01:06.560","Text":"Well, we defined that in the last section,"},{"Start":"01:06.560 ","End":"01:10.400","Text":"that\u0027s W. That equals to 2 minus y."},{"Start":"01:10.400 ","End":"01:13.280","Text":"Now, what are we looking for?"},{"Start":"01:13.280 ","End":"01:14.860","Text":"The correlation."},{"Start":"01:14.860 ","End":"01:21.950","Text":"The correlation coefficient between C and W. Let\u0027s just write that out."},{"Start":"01:21.950 ","End":"01:29.660","Text":"That\u0027s a correlation coefficient of 0.5 x plus 3 and 2 minus y."},{"Start":"01:29.660 ","End":"01:31.565","Text":"Well, let\u0027s write it out again."},{"Start":"01:31.565 ","End":"01:36.935","Text":"That\u0027ll be a correlation coefficient of 0.5 x plus 3."},{"Start":"01:36.935 ","End":"01:43.940","Text":"We\u0027ll write this out as minus 1 times y plus 2. Here we are."},{"Start":"01:43.940 ","End":"01:50.870","Text":"We have the correlation coefficient of a linear transformation on both variables."},{"Start":"01:50.870 ","End":"01:56.845","Text":"Now, how does a linear transformation affect a correlation coefficient?"},{"Start":"01:56.845 ","End":"02:03.810","Text":"We know that a linear transformation ax plus"},{"Start":"02:03.810 ","End":"02:11.495","Text":"b and cy plus d. That affects our correlation coefficient in this way."},{"Start":"02:11.495 ","End":"02:15.650","Text":"That\u0027ll be the correlation coefficient of x and y,"},{"Start":"02:15.650 ","End":"02:19.310","Text":"where a times c is greater than 0 and"},{"Start":"02:19.310 ","End":"02:25.250","Text":"minus the correlation coefficient of x and y where a times c is less than 0."},{"Start":"02:25.250 ","End":"02:29.210","Text":"All that means is that a linear transformation does not"},{"Start":"02:29.210 ","End":"02:34.040","Text":"affect the strength of the connection between x and y."},{"Start":"02:34.040 ","End":"02:37.790","Text":"All it may affect is the direction."},{"Start":"02:37.790 ","End":"02:40.430","Text":"Now, in our case,"},{"Start":"02:40.430 ","End":"02:44.930","Text":"a here, the multiplier for x, that equals to 0.5,"},{"Start":"02:44.930 ","End":"02:46.490","Text":"and the multiplier for C,"},{"Start":"02:46.490 ","End":"02:47.900","Text":"that equals to minus 1,"},{"Start":"02:47.900 ","End":"02:51.680","Text":"so a times c that equals to minus 0.5,"},{"Start":"02:51.680 ","End":"02:54.505","Text":"and obviously that\u0027s less than 0."},{"Start":"02:54.505 ","End":"03:01.670","Text":"The correlation coefficient of this linear transformation on both x and y, well,"},{"Start":"03:01.670 ","End":"03:10.355","Text":"that equals to minus the correlation coefficient of x and y."},{"Start":"03:10.355 ","End":"03:15.545","Text":"Now, that equals to minus 2/3s."},{"Start":"03:15.545 ","End":"03:18.500","Text":"2/3s was a correlation coefficient of x and y."},{"Start":"03:18.500 ","End":"03:21.535","Text":"We\u0027ve calculated that in the last section."},{"Start":"03:21.535 ","End":"03:25.230","Text":"Only here the direction is changed."},{"Start":"03:25.230 ","End":"03:27.405","Text":"All that means, again,"},{"Start":"03:27.405 ","End":"03:34.340","Text":"is that there\u0027s a negative correlation of 2/3s between the cost of"},{"Start":"03:34.340 ","End":"03:41.790","Text":"the apartment sample and the number of even numbered apartments."}],"ID":13210}],"Thumbnail":null,"ID":245019},{"Name":"Linear Combinations","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Tutorial 1","Duration":"2m 22s","ChapterTopicVideoID":12714,"CourseChapterTopicPlaylistID":245020,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12714.jpeg","UploadDate":"2018-07-30T15:05:33.4930000","DurationForVideoObject":"PT2M22S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"In this chapter we\u0027ll be talking about"},{"Start":"00:02.310 ","End":"00:06.750","Text":"the 2-dimensional random variables, specifically linear combinations."},{"Start":"00:06.750 ","End":"00:09.765","Text":"Now, the expectation variance of a sum of variables."},{"Start":"00:09.765 ","End":"00:11.910","Text":"Now, if we have 2 variables X and Y,"},{"Start":"00:11.910 ","End":"00:16.170","Text":"and we want to know the expectation of X plus Y, well,"},{"Start":"00:16.170 ","End":"00:18.570","Text":"that equals to the sum of the expectations,"},{"Start":"00:18.570 ","End":"00:21.870","Text":"the expectation of X plus the expectation of Y."},{"Start":"00:21.870 ","End":"00:26.490","Text":"If we\u0027re looking for the variance of the sum of the 2 variables, well,"},{"Start":"00:26.490 ","End":"00:29.640","Text":"that equals to the variance of X plus the variance of Y"},{"Start":"00:29.640 ","End":"00:33.720","Text":"plus 2 times the covariance of X and Y."},{"Start":"00:33.720 ","End":"00:36.870","Text":"Let\u0027s just take a look at an example."},{"Start":"00:37.550 ","End":"00:40.560","Text":"Assume for 2 random variables,"},{"Start":"00:40.560 ","End":"00:42.080","Text":"we have the expectation of X,"},{"Start":"00:42.080 ","End":"00:45.080","Text":"that\u0027s 80, and the expectation of Y, that\u0027s 70."},{"Start":"00:45.080 ","End":"00:47.690","Text":"Now, the standard deviation of X is 15,"},{"Start":"00:47.690 ","End":"00:49.790","Text":"the standard deviation of Y, that\u0027s 20,"},{"Start":"00:49.790 ","End":"00:53.240","Text":"the covariance of X and Y equals to 200."},{"Start":"00:53.240 ","End":"01:00.270","Text":"We\u0027re asked to find the expectation and variance of the sum of the variables. Here we go."},{"Start":"01:00.270 ","End":"01:03.305","Text":"The expectation of X plus Y, well,"},{"Start":"01:03.305 ","End":"01:07.505","Text":"that equals to the expectation of X plus the expectation of Y."},{"Start":"01:07.505 ","End":"01:09.950","Text":"In our case, the expectation of X,"},{"Start":"01:09.950 ","End":"01:15.695","Text":"that\u0027s 80, plus the expectation of Y, that\u0027s 70."},{"Start":"01:15.695 ","End":"01:20.755","Text":"That will be equal to 150."},{"Start":"01:20.755 ","End":"01:23.285","Text":"Now, what about the variance of X and Y?"},{"Start":"01:23.285 ","End":"01:25.700","Text":"Well, that equals to the variance of X plus the variance of"},{"Start":"01:25.700 ","End":"01:28.430","Text":"Y plus 2 times the covariance of X and Y."},{"Start":"01:28.430 ","End":"01:32.875","Text":"Now, here, we have standard deviations, not variances."},{"Start":"01:32.875 ","End":"01:39.120","Text":"So that has to equal to 15 squared plus the variance of Y,"},{"Start":"01:39.120 ","End":"01:41.614","Text":"well that\u0027s 20 squared,"},{"Start":"01:41.614 ","End":"01:43.640","Text":"plus 2 times,"},{"Start":"01:43.640 ","End":"01:46.810","Text":"now the covariance of X and Y, that\u0027s 200."},{"Start":"01:46.810 ","End":"01:55.230","Text":"Now, that equals to 225 plus 400 plus 400."},{"Start":"01:55.230 ","End":"01:59.300","Text":"So that\u0027s 1,025. Now,"},{"Start":"01:59.300 ","End":"02:05.375","Text":"a thing that you have to take into consideration is that if X and Y are uncorrelated,"},{"Start":"02:05.375 ","End":"02:09.380","Text":"then the variance of the sum becomes the sum of"},{"Start":"02:09.380 ","End":"02:13.790","Text":"the variances because the covariance equals 0,"},{"Start":"02:13.790 ","End":"02:15.665","Text":"2 times 0 equals 0."},{"Start":"02:15.665 ","End":"02:22.440","Text":"So the variance of X plus Y equals to the variance of X plus the variance of Y."}],"ID":13211},{"Watched":false,"Name":"Tutorial 2","Duration":"1m 36s","ChapterTopicVideoID":12715,"CourseChapterTopicPlaylistID":245020,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12715.jpeg","UploadDate":"2018-07-30T15:05:39.9030000","DurationForVideoObject":"PT1M36S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.705","Text":"When dealing with a difference of 2 random variables, X and Y,"},{"Start":"00:03.705 ","End":"00:09.300","Text":"the expectation of X minus Y that equals to the difference in the expectations,"},{"Start":"00:09.300 ","End":"00:12.825","Text":"the expectation of X minus the expectation of Y,"},{"Start":"00:12.825 ","End":"00:17.055","Text":"and the variance of X minus Y."},{"Start":"00:17.055 ","End":"00:21.150","Text":"That equals to the variance of X plus the variance of"},{"Start":"00:21.150 ","End":"00:25.500","Text":"Y minus 2 times the covariance of X and Y."},{"Start":"00:25.500 ","End":"00:30.330","Text":"Again, let\u0027s just take a look at an example. Here we go."},{"Start":"00:30.330 ","End":"00:32.730","Text":"This is our example and we\u0027re asked here to find"},{"Start":"00:32.730 ","End":"00:35.745","Text":"the expectation and variance of Y minus X."},{"Start":"00:35.745 ","End":"00:37.860","Text":"Now here we have X minus Y,"},{"Start":"00:37.860 ","End":"00:38.970","Text":"but it really doesn\u0027t matter."},{"Start":"00:38.970 ","End":"00:42.220","Text":"We\u0027re looking at the expectation of Y minus X,"},{"Start":"00:42.220 ","End":"00:48.555","Text":"that equals to the expectation of Y minus the expectation of X."},{"Start":"00:48.555 ","End":"00:50.670","Text":"The expectation of Y,"},{"Start":"00:50.670 ","End":"00:55.500","Text":"that\u0027s 70, and the expectation of X that\u0027s 80."},{"Start":"00:55.500 ","End":"00:57.530","Text":"The expectation of the difference,"},{"Start":"00:57.530 ","End":"01:01.370","Text":"that\u0027s the difference in the expectations and that\u0027ll be minus 10."},{"Start":"01:01.370 ","End":"01:02.780","Text":"Now what about the variance?"},{"Start":"01:02.780 ","End":"01:07.800","Text":"The variance of X minus Y equals to the variance of X."},{"Start":"01:07.800 ","End":"01:12.980","Text":"That\u0027ll be 15 squared plus the variance of Y,"},{"Start":"01:12.980 ","End":"01:18.860","Text":"that\u0027ll be 20 squared minus now 2 times the covariance of X and Y,"},{"Start":"01:18.860 ","End":"01:25.460","Text":"that\u0027ll be 2 times 200 and that equals to 225 plus 400,"},{"Start":"01:25.460 ","End":"01:30.815","Text":"that\u0027s 625 minus 400, that\u0027s 225."},{"Start":"01:30.815 ","End":"01:35.850","Text":"This is then the variance of X minus Y."}],"ID":13212},{"Watched":false,"Name":"Tutorial 3","Duration":"3m 54s","ChapterTopicVideoID":12716,"CourseChapterTopicPlaylistID":245020,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12716.jpeg","UploadDate":"2018-07-30T15:06:06.8870000","DurationForVideoObject":"PT3M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"Let\u0027s talk now about linear combinations."},{"Start":"00:03.210 ","End":"00:08.340","Text":"We perform linear combinations on our random variables X and Y."},{"Start":"00:08.340 ","End":"00:12.360","Text":"The template for linear combination is this 1."},{"Start":"00:12.360 ","End":"00:16.890","Text":"You have aX plus b plus cY plus d,"},{"Start":"00:16.890 ","End":"00:21.780","Text":"and we define that expression as a new variable W. Now,"},{"Start":"00:21.780 ","End":"00:24.420","Text":"how do we perform this linear combination."},{"Start":"00:24.420 ","End":"00:29.190","Text":"We take constants a and b and c and d are parameters?"},{"Start":"00:29.190 ","End":"00:30.390","Text":"They\u0027re constants."},{"Start":"00:30.390 ","End":"00:34.965","Text":"Where a is the multiplier for X and c is the multiplier for Y."},{"Start":"00:34.965 ","End":"00:37.050","Text":"Now, having said that,"},{"Start":"00:37.050 ","End":"00:39.015","Text":"what\u0027s the expectation of W?"},{"Start":"00:39.015 ","End":"00:43.400","Text":"Well, that equals to a times the expectation of X plus b plus"},{"Start":"00:43.400 ","End":"00:47.935","Text":"c times the expectation of Y plus d. The variance of W, well,"},{"Start":"00:47.935 ","End":"00:52.040","Text":"that equals to a squared times the variance of X plus c squared times"},{"Start":"00:52.040 ","End":"00:57.395","Text":"the variance of Y plus 2 times a times c times the covariance of X and Y."},{"Start":"00:57.395 ","End":"00:59.840","Text":"Now, apart from that,"},{"Start":"00:59.840 ","End":"01:05.495","Text":"we want to see how this linear combination affects the covariance."},{"Start":"01:05.495 ","End":"01:09.620","Text":"Well, the covariance of this linear transformation of X and"},{"Start":"01:09.620 ","End":"01:15.845","Text":"Y is equal to a times c times the covariance of X and Y."},{"Start":"01:15.845 ","End":"01:20.600","Text":"Let\u0027s just take a look at an example and make things a little bit clearer."},{"Start":"01:20.600 ","End":"01:25.400","Text":"In our example, we\u0027re asked to find the expectation variance of the variable W,"},{"Start":"01:25.400 ","End":"01:29.285","Text":"which is equal to 2x plus 3y."},{"Start":"01:29.285 ","End":"01:35.015","Text":"W here, that equals to 2x plus 3y."},{"Start":"01:35.015 ","End":"01:41.225","Text":"Now, let\u0027s take a look at our template for linear combination that equals to W,"},{"Start":"01:41.225 ","End":"01:46.980","Text":"that equals to aX plus b plus cY"},{"Start":"01:46.980 ","End":"01:53.040","Text":"plus d. Let\u0027s bring this into this form and that\u0027ll be equal to,"},{"Start":"01:53.040 ","End":"02:00.075","Text":"W will then be equal to 2x plus 0 plus 3y plus 0."},{"Start":"02:00.075 ","End":"02:05.630","Text":"That means that a will be equal to 2, b equals 0,"},{"Start":"02:05.630 ","End":"02:07.969","Text":"c equals to 3,"},{"Start":"02:07.969 ","End":"02:12.090","Text":"and d equals to 0."},{"Start":"02:12.090 ","End":"02:16.005","Text":"What\u0027s the expectation then of W?"},{"Start":"02:16.005 ","End":"02:22.835","Text":"Well, that equals to a times the expectation of X plus b"},{"Start":"02:22.835 ","End":"02:30.270","Text":"plus c times the expectation of Y plus d. What is that equal to?"},{"Start":"02:30.270 ","End":"02:32.040","Text":"Well, a here equals to 2."},{"Start":"02:32.040 ","End":"02:33.795","Text":"The expectation of X,"},{"Start":"02:33.795 ","End":"02:43.250","Text":"well that equal to 80 plus b equals 0 plus c is equal to 3 times the expectation of Y,"},{"Start":"02:43.250 ","End":"02:44.870","Text":"that equals to 70,"},{"Start":"02:44.870 ","End":"02:46.250","Text":"plus d, well,"},{"Start":"02:46.250 ","End":"02:47.735","Text":"that equals to 0."},{"Start":"02:47.735 ","End":"02:51.590","Text":"Then all of this equals to 370."},{"Start":"02:51.590 ","End":"02:55.415","Text":"Now, what about the variance of W?"},{"Start":"02:55.415 ","End":"03:03.725","Text":"Well, that equals to a squared times the variance of X plus c squared"},{"Start":"03:03.725 ","End":"03:13.490","Text":"times the variance of Y plus 2 times a times c times the covariance of X and Y."},{"Start":"03:13.490 ","End":"03:15.830","Text":"Now, in our case,"},{"Start":"03:15.830 ","End":"03:17.315","Text":"let\u0027s just plug in the numbers."},{"Start":"03:17.315 ","End":"03:18.440","Text":"A equals 2,"},{"Start":"03:18.440 ","End":"03:22.550","Text":"so that\u0027ll be 2 squared times the variance of X."},{"Start":"03:22.550 ","End":"03:26.330","Text":"Well that\u0027s 15 squared plus c squared,"},{"Start":"03:26.330 ","End":"03:29.450","Text":"that will be 3 squared times the variance of Y,"},{"Start":"03:29.450 ","End":"03:34.980","Text":"that\u0027ll be 20 squared plus 2 times 2."},{"Start":"03:34.980 ","End":"03:36.120","Text":"2 times a,"},{"Start":"03:36.120 ","End":"03:37.980","Text":"a is 2, times c,"},{"Start":"03:37.980 ","End":"03:42.335","Text":"that\u0027s 3, times the covariance of X and Y, that\u0027s 200."},{"Start":"03:42.335 ","End":"03:47.535","Text":"Now, all that equals to 6,900."},{"Start":"03:47.535 ","End":"03:50.300","Text":"This then is the variance of W,"},{"Start":"03:50.300 ","End":"03:54.720","Text":"and this is the expectation of W."}],"ID":13213},{"Watched":false,"Name":"Exercise 1 - Parts a-c","Duration":"5m 32s","ChapterTopicVideoID":12717,"CourseChapterTopicPlaylistID":245020,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12717.jpeg","UploadDate":"2018-07-30T15:06:36.4730000","DurationForVideoObject":"PT5M32S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.070","Text":"In this question, we\u0027re given the following joint probability function, we have x."},{"Start":"00:05.070 ","End":"00:07.140","Text":"The values of x are 1, 2, and 3."},{"Start":"00:07.140 ","End":"00:08.220","Text":"We have y."},{"Start":"00:08.220 ","End":"00:10.490","Text":"The values of y are 2 and 3."},{"Start":"00:10.490 ","End":"00:14.580","Text":"We have the joint probabilities here and some of the marginal probabilities,"},{"Start":"00:14.580 ","End":"00:17.700","Text":"and we\u0027re asked to fill in the missing probabilities."},{"Start":"00:17.700 ","End":"00:21.930","Text":"Since we know that all the probabilities have to add up to 1,"},{"Start":"00:21.930 ","End":"00:24.075","Text":"the joint probabilities have to add up to 1,"},{"Start":"00:24.075 ","End":"00:27.850","Text":"the marginal probabilities have to add up as 1 as well."},{"Start":"00:27.850 ","End":"00:31.515","Text":"So that\u0027s 0.6 plus 0.4,"},{"Start":"00:31.515 ","End":"00:33.285","Text":"that\u0027ll be equal to 1."},{"Start":"00:33.285 ","End":"00:36.330","Text":"Now 0.2 plus 0.1,"},{"Start":"00:36.330 ","End":"00:40.781","Text":"that\u0027s 0.3, and that means that this is 0.1."},{"Start":"00:40.781 ","End":"00:43.454","Text":"Here we have 0.4,"},{"Start":"00:43.454 ","End":"00:45.880","Text":"so this must be 0.2."},{"Start":"00:45.880 ","End":"00:47.670","Text":"So that\u0027s 0.4,"},{"Start":"00:47.670 ","End":"00:51.575","Text":"that\u0027s 0.2, and that\u0027s 0.4."},{"Start":"00:51.575 ","End":"00:57.050","Text":"This then is the joint probability function of x and y,"},{"Start":"00:57.050 ","End":"01:00.335","Text":"including the marginal probabilities."},{"Start":"01:00.335 ","End":"01:04.205","Text":"In this section, we\u0027re asked whether the variables x and y are dependent."},{"Start":"01:04.205 ","End":"01:07.865","Text":"We know that if they\u0027re independent,"},{"Start":"01:07.865 ","End":"01:13.130","Text":"then the probability of x equaling k and y equaling l,"},{"Start":"01:13.130 ","End":"01:19.145","Text":"that should equal to the probability of x equal to k times the probability of y"},{"Start":"01:19.145 ","End":"01:26.030","Text":"equaling l for all values of k and l. What are we looking for?"},{"Start":"01:26.030 ","End":"01:31.760","Text":"Let\u0027s take a look now at the cells right here at the joint probabilities."},{"Start":"01:31.760 ","End":"01:37.730","Text":"Now if we find at least 1 case where this equation doesn\u0027t hold true,"},{"Start":"01:37.730 ","End":"01:40.405","Text":"then we know that the variables are dependent."},{"Start":"01:40.405 ","End":"01:48.315","Text":"Let\u0027s take a look at the cell where x equals 2 and y equals to 3."},{"Start":"01:48.315 ","End":"01:54.240","Text":"The probability of x equaling 2 and y equaling 3,"},{"Start":"01:54.240 ","End":"02:02.765","Text":"does that equal to the probability of x equaling 2 times the probability of y equaling 3?"},{"Start":"02:02.765 ","End":"02:05.945","Text":"That\u0027s what we have to look at right here."},{"Start":"02:05.945 ","End":"02:12.245","Text":"The joint probability, x equals 2 and y equals 3, that\u0027s 0.1."},{"Start":"02:12.245 ","End":"02:15.935","Text":"Now the probability of x equaling 2, that\u0027s 0.2."},{"Start":"02:15.935 ","End":"02:17.591","Text":"It\u0027s right here."},{"Start":"02:17.591 ","End":"02:19.910","Text":"The probability of y equaling 3,"},{"Start":"02:19.910 ","End":"02:22.520","Text":"that\u0027s right here, that\u0027s 0.4."},{"Start":"02:22.520 ","End":"02:28.190","Text":"We see that 0.2 times 0.4 does not equal to 0.1."},{"Start":"02:28.190 ","End":"02:33.930","Text":"Therefore, x and y are dependent."},{"Start":"02:34.360 ","End":"02:38.855","Text":"In this section, we\u0027re asked whether the variables are uncorrelated."},{"Start":"02:38.855 ","End":"02:42.170","Text":"In order for x and y to be uncorrelated,"},{"Start":"02:42.170 ","End":"02:45.165","Text":"then the covariance of x and y,"},{"Start":"02:45.165 ","End":"02:47.035","Text":"that has to be equal to 0."},{"Start":"02:47.035 ","End":"02:49.315","Text":"Now what\u0027s the definition of the covariance?"},{"Start":"02:49.315 ","End":"02:55.060","Text":"Covariance of x and y that equals to the expectation of x times"},{"Start":"02:55.060 ","End":"03:01.585","Text":"y minus the expectation of x times the expectation of y."},{"Start":"03:01.585 ","End":"03:05.335","Text":"Now in order for that to be equal to 0, well,"},{"Start":"03:05.335 ","End":"03:09.070","Text":"that means that the expectation of x times y,"},{"Start":"03:09.070 ","End":"03:15.615","Text":"that has to be equal to the expectation of x times the expectation of y."},{"Start":"03:15.615 ","End":"03:19.510","Text":"What we need to do here is to calculate the expectations,"},{"Start":"03:19.510 ","End":"03:20.620","Text":"the expectation of x,"},{"Start":"03:20.620 ","End":"03:24.205","Text":"the expectation of y, and the expectation of x times y,"},{"Start":"03:24.205 ","End":"03:27.950","Text":"in order to see whether this equation holds true."},{"Start":"03:27.950 ","End":"03:31.685","Text":"So let\u0027s get to work. The expectation of x,"},{"Start":"03:31.685 ","End":"03:36.770","Text":"that equals to the sum of x times the probability of x."},{"Start":"03:36.770 ","End":"03:40.010","Text":"Now that equals to 1 times"},{"Start":"03:40.010 ","End":"03:48.705","Text":"0.4 plus 2 times 0.2 plus 3 times 0.4."},{"Start":"03:48.705 ","End":"03:51.794","Text":"Now that equals to 2."},{"Start":"03:51.794 ","End":"03:54.650","Text":"What about the expectation of y?"},{"Start":"03:54.650 ","End":"04:00.500","Text":"Similarly, that\u0027s the sum of y times the probability of y."},{"Start":"04:00.500 ","End":"04:08.740","Text":"That equals to 2 times 0.6 plus 3 times 0.4."},{"Start":"04:08.740 ","End":"04:12.570","Text":"Now that equals to 2.4."},{"Start":"04:12.570 ","End":"04:17.645","Text":"Now what about the expectation of x times y?"},{"Start":"04:17.645 ","End":"04:27.440","Text":"That equals to the value of x times the value of y times the joint probability."},{"Start":"04:27.440 ","End":"04:34.850","Text":"So that will be 1 times 2 times 0.2 plus, now,"},{"Start":"04:34.850 ","End":"04:40.805","Text":"2 times 2 times 0.1 plus"},{"Start":"04:40.805 ","End":"04:47.090","Text":"3 times 2 times 0.3 plus,"},{"Start":"04:47.090 ","End":"04:53.135","Text":"and so on and so forth for all combinations of x and y and their joint probabilities,"},{"Start":"04:53.135 ","End":"04:56.840","Text":"and that equals to 4.7."},{"Start":"04:56.840 ","End":"04:59.330","Text":"Now what we need to know,"},{"Start":"04:59.330 ","End":"05:01.490","Text":"the expectation of x times y,"},{"Start":"05:01.490 ","End":"05:05.300","Text":"does that equal to the expectation of x times the expectation of y?"},{"Start":"05:05.300 ","End":"05:09.389","Text":"The expectation of x times y, that\u0027s 4.7."},{"Start":"05:09.389 ","End":"05:12.080","Text":"What about the expectation of x?"},{"Start":"05:12.080 ","End":"05:13.775","Text":"That equals to 2,"},{"Start":"05:13.775 ","End":"05:15.450","Text":"and the expectation of y,"},{"Start":"05:15.450 ","End":"05:17.565","Text":"that equals to 2.4."},{"Start":"05:17.565 ","End":"05:22.511","Text":"Now 2 times 2.4 does not equal to 4.7."},{"Start":"05:22.511 ","End":"05:30.240","Text":"That means that x and y are correlated."}],"ID":13214},{"Watched":false,"Name":"Exercise 1 - Parts d-f","Duration":"4m 52s","ChapterTopicVideoID":12718,"CourseChapterTopicPlaylistID":245020,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12718.jpeg","UploadDate":"2018-07-30T15:06:57.8530000","DurationForVideoObject":"PT4M52S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this section, we\u0027re asked to calculate the covariance of x and y."},{"Start":"00:03.750 ","End":"00:07.335","Text":"Well, if you recall from the last section,"},{"Start":"00:07.335 ","End":"00:11.085","Text":"the covariance of x and y, well,"},{"Start":"00:11.085 ","End":"00:14.895","Text":"that equal to the expectation of x times y"},{"Start":"00:14.895 ","End":"00:19.815","Text":"minus the expectation of y times the expectation of y."},{"Start":"00:19.815 ","End":"00:22.515","Text":"Now, in the last section,"},{"Start":"00:22.515 ","End":"00:26.100","Text":"the expectation of x times y,"},{"Start":"00:26.100 ","End":"00:27.315","Text":"we\u0027ve calculated that,"},{"Start":"00:27.315 ","End":"00:32.640","Text":"that was 4.7 and the expectation of x that equal to 2,"},{"Start":"00:32.640 ","End":"00:34.725","Text":"and the expectation of y,"},{"Start":"00:34.725 ","End":"00:37.830","Text":"well that equal to 2.4."},{"Start":"00:37.830 ","End":"00:49.190","Text":"The covariance of x and y that equal to 4.7 minus 2 times 2.4,"},{"Start":"00:49.190 ","End":"00:53.630","Text":"that equals to 4.7 minus 4.8,"},{"Start":"00:53.630 ","End":"00:56.705","Text":"that equals to minus 0.1."},{"Start":"00:56.705 ","End":"01:01.010","Text":"This then is the covariance of x and y."},{"Start":"01:01.010 ","End":"01:04.100","Text":"In this section, we\u0027re asked to calculate the expectation of"},{"Start":"01:04.100 ","End":"01:07.252","Text":"variance of the sum of the 2 variables."},{"Start":"01:07.252 ","End":"01:10.850","Text":"Now if we recall the expectation of x plus y,"},{"Start":"01:10.850 ","End":"01:14.150","Text":"that\u0027s the sum of the 2 variables that equals to the expectation"},{"Start":"01:14.150 ","End":"01:17.840","Text":"of x plus the expectation of y."},{"Start":"01:17.840 ","End":"01:20.660","Text":"Now what about the variance of the sum?"},{"Start":"01:20.660 ","End":"01:25.090","Text":"Well, that equals to the variance of x plus the variance of"},{"Start":"01:25.090 ","End":"01:31.490","Text":"y plus 2 times the covariance of x and y."},{"Start":"01:31.490 ","End":"01:34.370","Text":"Let\u0027s get to work."},{"Start":"01:34.370 ","End":"01:38.210","Text":"The expectation of x plus y."},{"Start":"01:38.210 ","End":"01:41.360","Text":"Well, that equals to the expectation of x plus the expectation of y."},{"Start":"01:41.360 ","End":"01:47.270","Text":"Now, the expectation of x is 2 and the expectation of y is 2.4,"},{"Start":"01:47.270 ","End":"01:51.125","Text":"so that equals to 4.4."},{"Start":"01:51.125 ","End":"01:53.630","Text":"Now what about the variance of x plus y?"},{"Start":"01:53.630 ","End":"01:55.655","Text":"Well, that equals to this expression,"},{"Start":"01:55.655 ","End":"01:59.290","Text":"but we don\u0027t have the variance of x and variance of y."},{"Start":"01:59.290 ","End":"02:01.070","Text":"Let\u0027s calculate that now."},{"Start":"02:01.070 ","End":"02:02.960","Text":"The variance of x,"},{"Start":"02:02.960 ","End":"02:07.430","Text":"that equals to the sum of x squared times the probability of"},{"Start":"02:07.430 ","End":"02:13.760","Text":"x minus the expectation squared of x."},{"Start":"02:13.760 ","End":"02:19.610","Text":"Now that equals to 1 squared times 0.4"},{"Start":"02:19.610 ","End":"02:26.780","Text":"plus 2 squared times 0.2 plus 3 squared times 0.4 minus."},{"Start":"02:26.780 ","End":"02:28.760","Text":"Now the expectation squared,"},{"Start":"02:28.760 ","End":"02:34.200","Text":"now that\u0027ll be 2 squared and that equals to 0.8."},{"Start":"02:34.200 ","End":"02:36.485","Text":"What about the variance of y?"},{"Start":"02:36.485 ","End":"02:40.250","Text":"Well, similarly, that\u0027s the sum of y squared times"},{"Start":"02:40.250 ","End":"02:45.120","Text":"the probability of y minus the expectation squared of y."},{"Start":"02:45.120 ","End":"02:50.680","Text":"Now that equals to 2 squared times 0.6 plus 3"},{"Start":"02:50.680 ","End":"02:57.175","Text":"squared times 0.4 minus 2.4 squared."},{"Start":"02:57.175 ","End":"03:01.510","Text":"Now that equals to 0.24."},{"Start":"03:01.510 ","End":"03:07.930","Text":"Now we can calculate the variance of x plus y."},{"Start":"03:07.930 ","End":"03:10.060","Text":"Well that equals to the variance of x,"},{"Start":"03:10.060 ","End":"03:13.345","Text":"that\u0027s 0.8, plus the variance of y,"},{"Start":"03:13.345 ","End":"03:18.300","Text":"that\u0027s 0.24, plus 2 times,"},{"Start":"03:18.300 ","End":"03:21.465","Text":"well the covariance of x and y."},{"Start":"03:21.465 ","End":"03:25.180","Text":"That equals to minus 0.1."},{"Start":"03:26.600 ","End":"03:33.710","Text":"That comes out to 0.84."},{"Start":"03:33.710 ","End":"03:35.330","Text":"In this section, we\u0027re asked to calculate"},{"Start":"03:35.330 ","End":"03:38.479","Text":"the expectation variance of the difference of the 2 variables."},{"Start":"03:38.479 ","End":"03:42.320","Text":"Now, let\u0027s recall the expectation of x minus y."},{"Start":"03:42.320 ","End":"03:47.695","Text":"Well that equals to the expectation of x minus the expectation of y."},{"Start":"03:47.695 ","End":"03:50.890","Text":"The variance of X minus y, well,"},{"Start":"03:50.890 ","End":"03:55.400","Text":"that equals to the variance of x plus the variance of y"},{"Start":"03:55.400 ","End":"04:00.765","Text":"minus 2 times the covariance of x and y."},{"Start":"04:00.765 ","End":"04:05.075","Text":"In our case, the expectation of x minus y,"},{"Start":"04:05.075 ","End":"04:11.525","Text":"well that equals to the expectation of x that was 2 and the expectation of y,"},{"Start":"04:11.525 ","End":"04:13.130","Text":"that was 2.4,"},{"Start":"04:13.130 ","End":"04:15.530","Text":"so the expectation of x minus y,"},{"Start":"04:15.530 ","End":"04:18.260","Text":"that equals to minus 0.4."},{"Start":"04:18.260 ","End":"04:21.900","Text":"Now, what about the variance of x minus y?"},{"Start":"04:21.900 ","End":"04:25.940","Text":"Well, we\u0027ve calculated the variances in the last section."},{"Start":"04:25.940 ","End":"04:31.220","Text":"The variance of x was 0.8 and the variance of y was"},{"Start":"04:31.220 ","End":"04:37.535","Text":"0.24 now to minus 2 times the covariance of x and y."},{"Start":"04:37.535 ","End":"04:40.715","Text":"Well, that equals to minus 1."},{"Start":"04:40.715 ","End":"04:44.345","Text":"All this comes out to 1.24."},{"Start":"04:44.345 ","End":"04:47.660","Text":"There we go. Here\u0027s the expectation of the difference of"},{"Start":"04:47.660 ","End":"04:52.830","Text":"the variables and here\u0027s the variance of the difference of the variables."}],"ID":13215},{"Watched":false,"Name":"Exercise 2","Duration":"8m 47s","ChapterTopicVideoID":12719,"CourseChapterTopicPlaylistID":245020,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12719.jpeg","UploadDate":"2018-07-30T15:07:47.9270000","DurationForVideoObject":"PT8M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.100","Text":"In this question, we\u0027re given an exam that\u0027s"},{"Start":"00:02.100 ","End":"00:05.280","Text":"composed of a quantitative and this verbal section."},{"Start":"00:05.280 ","End":"00:07.170","Text":"The expectation of the grades in"},{"Start":"00:07.170 ","End":"00:11.055","Text":"the quantitative section is a 100 with a standard deviation of 20,"},{"Start":"00:11.055 ","End":"00:12.840","Text":"and the expectation of the grades in"},{"Start":"00:12.840 ","End":"00:16.350","Text":"the verbal section is 90 with a standard deviation of 15."},{"Start":"00:16.350 ","End":"00:19.080","Text":"The correlation coefficient of the grades on"},{"Start":"00:19.080 ","End":"00:22.708","Text":"the quantitative and verbal sections is 0.8,"},{"Start":"00:22.708 ","End":"00:24.719","Text":"and we\u0027re asked to calculate the covariance"},{"Start":"00:24.719 ","End":"00:27.525","Text":"of the quantitative grading, the verbal grade."},{"Start":"00:27.525 ","End":"00:31.665","Text":"The first thing that we need to do is to define our variables."},{"Start":"00:31.665 ","End":"00:36.720","Text":"We\u0027ll define X as the quantitative grade,"},{"Start":"00:36.720 ","End":"00:47.710","Text":"and y would be the verbal grade."},{"Start":"00:47.720 ","End":"00:53.610","Text":"Now, the expectation of X that equals to 100,"},{"Start":"00:53.610 ","End":"00:55.050","Text":"it\u0027s given right here,"},{"Start":"00:55.050 ","End":"00:58.795","Text":"and the standard deviation of x, well that\u0027s 20."},{"Start":"00:58.795 ","End":"01:01.960","Text":"What about the expectation of y?"},{"Start":"01:01.960 ","End":"01:04.435","Text":"That\u0027s the expectation as a verbal grade."},{"Start":"01:04.435 ","End":"01:06.768","Text":"Well, that equals to 90,"},{"Start":"01:06.768 ","End":"01:08.605","Text":"and the standard deviation of y,"},{"Start":"01:08.605 ","End":"01:11.303","Text":"well that equals to 15,"},{"Start":"01:11.303 ","End":"01:16.840","Text":"and the correlation coefficient between x and y, that equals to 0.8."},{"Start":"01:16.840 ","End":"01:19.330","Text":"Now, we\u0027re asked to calculate the covariance"},{"Start":"01:19.330 ","End":"01:21.875","Text":"of the quantitative grade and the verbal grade."},{"Start":"01:21.875 ","End":"01:23.620","Text":"Let\u0027s get to work."},{"Start":"01:23.620 ","End":"01:28.370","Text":"We know that the correlation coefficient between x and y,"},{"Start":"01:28.370 ","End":"01:31.700","Text":"well that equals to the covariance of x and"},{"Start":"01:31.700 ","End":"01:36.890","Text":"y divided by the standard deviation of x times the standard deviation of y."},{"Start":"01:36.890 ","End":"01:41.675","Text":"Now, we have the covariance of x and y, that\u0027s 0.8."},{"Start":"01:41.675 ","End":"01:46.280","Text":"Now, that equals to the covariance of x and y."},{"Start":"01:46.280 ","End":"01:48.350","Text":"That\u0027s what we\u0027re trying to find out right here."},{"Start":"01:48.350 ","End":"01:52.700","Text":"The covariance divided by the standard deviation of x."},{"Start":"01:52.700 ","End":"01:54.695","Text":"That equals to 20,"},{"Start":"01:54.695 ","End":"01:56.720","Text":"times the standard deviation of y,"},{"Start":"01:56.720 ","End":"01:58.730","Text":"that equals to 15."},{"Start":"01:58.730 ","End":"02:02.900","Text":"That means that the covariance of x and y,"},{"Start":"02:02.900 ","End":"02:07.940","Text":"that equals to 240."},{"Start":"02:07.940 ","End":"02:10.970","Text":"In this section, we\u0027re asked to calculate the expectation and the variance of"},{"Start":"02:10.970 ","End":"02:15.025","Text":"the sum of the grades and the quantitative section, and the verbal section."},{"Start":"02:15.025 ","End":"02:19.550","Text":"We know that the expectation of the sum of 2 variables,"},{"Start":"02:19.550 ","End":"02:25.865","Text":"that equals to the expectation of x plus the expectation of y,"},{"Start":"02:25.865 ","End":"02:28.895","Text":"and the variance of x plus y."},{"Start":"02:28.895 ","End":"02:33.745","Text":"That equals to the variance of x plus the variance of y,"},{"Start":"02:33.745 ","End":"02:40.290","Text":"plus 2 times the covariance of x and y."},{"Start":"02:40.370 ","End":"02:44.945","Text":"Let\u0027s calculate the expectation of x plus y."},{"Start":"02:44.945 ","End":"02:48.245","Text":"The expectation of x is a 100,"},{"Start":"02:48.245 ","End":"02:50.355","Text":"and the expectation of y,"},{"Start":"02:50.355 ","End":"02:54.930","Text":"that\u0027s 90, so that equals to 190."},{"Start":"02:54.930 ","End":"02:58.455","Text":"What about the variance of x plus y?"},{"Start":"02:58.455 ","End":"03:01.640","Text":"Well, that is the variance of x."},{"Start":"03:01.640 ","End":"03:03.620","Text":"Now, we have the standard deviation."},{"Start":"03:03.620 ","End":"03:05.495","Text":"The standard deviation is 20,"},{"Start":"03:05.495 ","End":"03:08.555","Text":"so the variance of x is 20 squared,"},{"Start":"03:08.555 ","End":"03:14.690","Text":"and the variance of y then is 15 squared plus 2 times the covariance of x and y."},{"Start":"03:14.690 ","End":"03:17.668","Text":"Now we\u0027ve calculated the covariance in the last section,"},{"Start":"03:17.668 ","End":"03:20.470","Text":"and that equal to 240."},{"Start":"03:20.470 ","End":"03:27.915","Text":"After calculation, that comes out to 1,105."},{"Start":"03:27.915 ","End":"03:33.010","Text":"This then is the expectation of x plus y,"},{"Start":"03:33.010 ","End":"03:36.650","Text":"and this then is the variance of x plus y."},{"Start":"03:36.650 ","End":"03:39.650","Text":"In this section, we\u0027re asked to calculate the expectation of variance of"},{"Start":"03:39.650 ","End":"03:44.530","Text":"the difference between the grades and the quantitative section of the verbal section."},{"Start":"03:44.530 ","End":"03:48.570","Text":"The expectation of x minus y,"},{"Start":"03:48.570 ","End":"03:53.825","Text":"that equals to the expectation of x minus the expectation of y,"},{"Start":"03:53.825 ","End":"03:56.875","Text":"and the variance of x minus y,"},{"Start":"03:56.875 ","End":"04:01.415","Text":"that equals to the variance of x plus the variance of y"},{"Start":"04:01.415 ","End":"04:07.165","Text":"minus 2 times the covariance of x and y."},{"Start":"04:07.165 ","End":"04:10.365","Text":"The expectation of x minus y,"},{"Start":"04:10.365 ","End":"04:12.530","Text":"that equals to the expectation of x,"},{"Start":"04:12.530 ","End":"04:17.700","Text":"and that\u0027s 100 minus the expectation of y, that\u0027s 90."},{"Start":"04:17.700 ","End":"04:20.130","Text":"The expectation of x minus y,"},{"Start":"04:20.130 ","End":"04:21.900","Text":"that equals to 10."},{"Start":"04:21.900 ","End":"04:25.575","Text":"What about the variance of x minus y?"},{"Start":"04:25.575 ","End":"04:29.840","Text":"The variance of x is 20 squared,"},{"Start":"04:29.840 ","End":"04:32.620","Text":"and the variance of y, that\u0027s 15 squared."},{"Start":"04:32.620 ","End":"04:37.130","Text":"Now it\u0027s minus 2 times the covariance of x and y."},{"Start":"04:37.130 ","End":"04:43.565","Text":"That\u0027s 240, so that equals to 145."},{"Start":"04:43.565 ","End":"04:47.045","Text":"This then is the variance of x minus y,"},{"Start":"04:47.045 ","End":"04:51.380","Text":"and this is the expectation of x minus y."},{"Start":"04:51.380 ","End":"04:54.820","Text":"In this section, we\u0027re given that the exam cost $2,000."},{"Start":"04:54.820 ","End":"04:56.960","Text":"Now, it\u0027s decided to reimburse the student"},{"Start":"04:56.960 ","End":"05:00.020","Text":"$1 for each point accumulated in the verbal section,"},{"Start":"05:00.020 ","End":"05:03.850","Text":"and $2 for each 1 accumulated on the quantitative section,"},{"Start":"05:03.850 ","End":"05:08.195","Text":"and we\u0027re asked what\u0027s the expectation in variance of the net cost of the exam?"},{"Start":"05:08.195 ","End":"05:11.830","Text":"That means that we\u0027re looking at the cost minus reimbursement."},{"Start":"05:11.830 ","End":"05:14.625","Text":"This looks like the linear combination."},{"Start":"05:14.625 ","End":"05:17.600","Text":"If that\u0027s the case, let\u0027s just define a new variable,"},{"Start":"05:17.600 ","End":"05:21.709","Text":"we\u0027ll call that w, and that\u0027ll be our net cost."},{"Start":"05:21.709 ","End":"05:24.080","Text":"Now, that\u0027s equal to the cost,"},{"Start":"05:24.080 ","End":"05:28.310","Text":"the cost is $2,000, minus the reimbursement."},{"Start":"05:28.310 ","End":"05:36.208","Text":"We reimbursed $2 for each point accumulated on the quantitative section,"},{"Start":"05:36.208 ","End":"05:42.035","Text":"and $1 for each point accumulated on the verbal section."},{"Start":"05:42.035 ","End":"05:46.235","Text":"This then is our expression for our net cost."},{"Start":"05:46.235 ","End":"05:48.860","Text":"Now, for linear combination,"},{"Start":"05:48.860 ","End":"05:51.308","Text":"the template for linear combination,"},{"Start":"05:51.308 ","End":"06:00.420","Text":"that equals to ax plus b plus cy plus d. In our case,"},{"Start":"06:00.420 ","End":"06:02.850","Text":"a x would be equal to minus 2,"},{"Start":"06:02.850 ","End":"06:05.990","Text":"that\u0027s our x multiplier,"},{"Start":"06:05.990 ","End":"06:08.990","Text":"C would be equal to minus 1,"},{"Start":"06:08.990 ","End":"06:10.945","Text":"that\u0027s our y multiplier."},{"Start":"06:10.945 ","End":"06:12.690","Text":"Now, what about b and d?"},{"Start":"06:12.690 ","End":"06:15.930","Text":"Let\u0027s take a look at the $2,000."},{"Start":"06:15.930 ","End":"06:20.750","Text":"We can allocate $2,000 to b and have d equals to 0,"},{"Start":"06:20.750 ","End":"06:26.600","Text":"or the other way round where b equals 0 and allocate all the $2,000 to d,"},{"Start":"06:26.600 ","End":"06:32.595","Text":"or we can split the $2,000 in any which way that we want to b and to d,"},{"Start":"06:32.595 ","End":"06:34.250","Text":"but let\u0027s keep things simple."},{"Start":"06:34.250 ","End":"06:38.128","Text":"Let\u0027s make b equal to $2,000,"},{"Start":"06:38.128 ","End":"06:41.665","Text":"and d will be equal to 0, really doesn\u0027t matter."},{"Start":"06:41.665 ","End":"06:44.049","Text":"Having said that,"},{"Start":"06:44.049 ","End":"06:46.485","Text":"what\u0027s the expectation of w?"},{"Start":"06:46.485 ","End":"06:48.720","Text":"Again, if we recall,"},{"Start":"06:48.720 ","End":"06:55.880","Text":"that equals to a times the expectation of x plus b plus c"},{"Start":"06:55.880 ","End":"07:02.820","Text":"times the expectation of y plus d. What\u0027s a?"},{"Start":"07:02.820 ","End":"07:04.883","Text":"A is minus 2,"},{"Start":"07:04.883 ","End":"07:07.650","Text":"and what\u0027s the expectation of x?"},{"Start":"07:07.650 ","End":"07:10.770","Text":"That\u0027s 100 plus b,"},{"Start":"07:10.770 ","End":"07:14.660","Text":"where b is 2,000 plus c,"},{"Start":"07:14.660 ","End":"07:18.665","Text":"where c is minus 1 times the expectation of y."},{"Start":"07:18.665 ","End":"07:22.760","Text":"That\u0027ll be 90 plus d that\u0027s 0."},{"Start":"07:22.760 ","End":"07:26.535","Text":"That equals to 1,710."},{"Start":"07:26.535 ","End":"07:35.120","Text":"The expectation of the net cost of the exam after reimbursement is $1,710."},{"Start":"07:35.120 ","End":"07:38.680","Text":"Now, what about the variance of w?"},{"Start":"07:38.680 ","End":"07:41.942","Text":"The variance of w,"},{"Start":"07:41.942 ","End":"07:43.214","Text":"if we recall,"},{"Start":"07:43.214 ","End":"07:49.610","Text":"that equals to a squared times the variance of x plus c squared times"},{"Start":"07:49.610 ","End":"07:53.495","Text":"the variance of y"},{"Start":"07:53.495 ","End":"08:01.510","Text":"plus 2 times a times c times the covariance of x and y."},{"Start":"08:01.510 ","End":"08:03.405","Text":"Let\u0027s just plug in the numbers."},{"Start":"08:03.405 ","End":"08:08.525","Text":"A is minus 2, so that\u0027s minus 2 squared times the variance of x,"},{"Start":"08:08.525 ","End":"08:12.885","Text":"that\u0027s 20 squared plus c squared,"},{"Start":"08:12.885 ","End":"08:16.340","Text":"that\u0027s minus 1 squared times the variance of y,"},{"Start":"08:16.340 ","End":"08:20.540","Text":"that\u0027s 15 squared plus 2 times,"},{"Start":"08:20.540 ","End":"08:22.835","Text":"a is minus 2,"},{"Start":"08:22.835 ","End":"08:30.510","Text":"times c is minus 1 times the covariance of x and y that\u0027s 240."},{"Start":"08:30.510 ","End":"08:35.070","Text":"That equals to 2,785,"},{"Start":"08:35.070 ","End":"08:39.960","Text":"and that will be in dollars squared units."},{"Start":"08:39.960 ","End":"08:43.530","Text":"This is the expectation of w,"},{"Start":"08:43.530 ","End":"08:48.130","Text":"and this would be the variance of w."}],"ID":13216},{"Watched":false,"Name":"Exercise 3","Duration":"5m 34s","ChapterTopicVideoID":12720,"CourseChapterTopicPlaylistID":245020,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12720.jpeg","UploadDate":"2018-07-30T15:08:20.9730000","DurationForVideoObject":"PT5M34S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.200","Text":"In this question we assume that the variance of x minus 2y equals 2,"},{"Start":"00:04.200 ","End":"00:07.080","Text":"and the variance of x plus 2y that equals to 3."},{"Start":"00:07.080 ","End":"00:10.800","Text":"We\u0027re asked to calculate the covariance of x and y."},{"Start":"00:10.800 ","End":"00:12.585","Text":"Well, in order to help us out,"},{"Start":"00:12.585 ","End":"00:17.130","Text":"let\u0027s use the expression for the variance of a linear combination."},{"Start":"00:17.130 ","End":"00:18.360","Text":"We\u0027ll call that w,"},{"Start":"00:18.360 ","End":"00:26.670","Text":"that equals to the variance of ax plus b plus cy plus d,"},{"Start":"00:26.670 ","End":"00:33.320","Text":"and that equals to a squared times the variance of x plus c squared times"},{"Start":"00:33.320 ","End":"00:41.120","Text":"the variance of y plus 2 times a times c times the covariance of x and y."},{"Start":"00:41.120 ","End":"00:43.490","Text":"Now, having said that,"},{"Start":"00:43.490 ","End":"00:46.475","Text":"let\u0027s take a look at this expression right here."},{"Start":"00:46.475 ","End":"00:50.240","Text":"What\u0027s the variance of x minus 2y?"},{"Start":"00:50.240 ","End":"00:52.820","Text":"Well, that equals to the following."},{"Start":"00:52.820 ","End":"00:57.230","Text":"Well, if we look at this linear combination, well,"},{"Start":"00:57.230 ","End":"00:59.450","Text":"the multiplier for x,"},{"Start":"00:59.450 ","End":"01:02.520","Text":"a that equals to 1 and the multiplier for y,"},{"Start":"01:02.520 ","End":"01:05.600","Text":"c that equals to minus 2."},{"Start":"01:05.600 ","End":"01:09.470","Text":"Let\u0027s just plug in these numbers here in this expression,"},{"Start":"01:09.470 ","End":"01:10.820","Text":"and we get this."},{"Start":"01:10.820 ","End":"01:14.510","Text":"That\u0027ll be 1 squared times the variance of x"},{"Start":"01:14.510 ","End":"01:20.420","Text":"plus minus 2 squared times the variance for y,"},{"Start":"01:20.420 ","End":"01:27.785","Text":"plus 2 times 1 times minus 2 times the covariance of x and y."},{"Start":"01:27.785 ","End":"01:30.515","Text":"Now, that has to be equal to 2."},{"Start":"01:30.515 ","End":"01:32.540","Text":"We\u0027re given that right here."},{"Start":"01:32.540 ","End":"01:34.850","Text":"Let\u0027s just clean that up a little bit."},{"Start":"01:34.850 ","End":"01:38.735","Text":"That\u0027ll be the variance of x plus"},{"Start":"01:38.735 ","End":"01:47.390","Text":"4 times the variance of y minus 4 times the covariance of x and y,"},{"Start":"01:47.390 ","End":"01:49.820","Text":"and that has to be equal to 2."},{"Start":"01:49.820 ","End":"01:53.180","Text":"Here we have 1 equation."},{"Start":"01:53.180 ","End":"01:55.895","Text":"Now let\u0027s take a look at the other equation."},{"Start":"01:55.895 ","End":"02:00.975","Text":"Variance of x plus 2y."},{"Start":"02:00.975 ","End":"02:03.090","Text":"Well, what is that equal to?"},{"Start":"02:03.090 ","End":"02:05.790","Text":"Again, a here equals to 1,"},{"Start":"02:05.790 ","End":"02:08.220","Text":"that\u0027s the x multiplier,"},{"Start":"02:08.220 ","End":"02:12.460","Text":"and c the y multiplier that equals to 2."},{"Start":"02:12.460 ","End":"02:14.070","Text":"As we can see here,"},{"Start":"02:14.070 ","End":"02:17.430","Text":"b and d do not play in this game at all."},{"Start":"02:17.430 ","End":"02:19.280","Text":"That equals to what?"},{"Start":"02:19.280 ","End":"02:21.410","Text":"Well, that\u0027s just plug in the numbers here again."},{"Start":"02:21.410 ","End":"02:29.435","Text":"That\u0027s 1 squared times the variance of x plus 2 squared times the variance for y,"},{"Start":"02:29.435 ","End":"02:36.220","Text":"plus 2 times 1 times 2 times the covariance of x and y."},{"Start":"02:36.220 ","End":"02:38.650","Text":"That has to be equal to 3,"},{"Start":"02:38.650 ","End":"02:41.065","Text":"and that\u0027s given to us right here."},{"Start":"02:41.065 ","End":"02:43.055","Text":"Again, let\u0027s clean that up."},{"Start":"02:43.055 ","End":"02:50.675","Text":"That\u0027ll be the variance of x plus 4 times the variance for y plus now,"},{"Start":"02:50.675 ","End":"02:54.725","Text":"4 times the covariance of x and y,"},{"Start":"02:54.725 ","End":"02:57.515","Text":"and that has to be equal to 3."},{"Start":"02:57.515 ","End":"03:00.565","Text":"Here we have 2 equations."},{"Start":"03:00.565 ","End":"03:05.569","Text":"We can see here that we have 2 equations with 3 unknowns."},{"Start":"03:05.569 ","End":"03:07.940","Text":"The first unknown is the variance of x,"},{"Start":"03:07.940 ","End":"03:09.650","Text":"the second is the variance of y,"},{"Start":"03:09.650 ","End":"03:12.605","Text":"and the third is the covariance of x and y."},{"Start":"03:12.605 ","End":"03:14.690","Text":"Now, how can we do that?"},{"Start":"03:14.690 ","End":"03:18.380","Text":"How can we solve and find out what"},{"Start":"03:18.380 ","End":"03:22.940","Text":"the covariance of x and y as if we have 3 unknowns and only 2 equations."},{"Start":"03:22.940 ","End":"03:25.250","Text":"Well, if we look closely,"},{"Start":"03:25.250 ","End":"03:29.090","Text":"we see that both equations have something in common."},{"Start":"03:29.090 ","End":"03:33.350","Text":"That\u0027s the variance of the x plus 4 times the variance of y."},{"Start":"03:33.350 ","End":"03:36.695","Text":"We see that in both equations right here,"},{"Start":"03:36.695 ","End":"03:38.480","Text":"it\u0027s right here and right here."},{"Start":"03:38.480 ","End":"03:41.255","Text":"Instead of this expression,"},{"Start":"03:41.255 ","End":"03:44.875","Text":"let\u0027s just use a letter to express this expression."},{"Start":"03:44.875 ","End":"03:49.080","Text":"We\u0027ll use the letter k. That\u0027ll be the following."},{"Start":"03:49.080 ","End":"03:58.235","Text":"That\u0027ll be k minus 4 times the covariance of x and y. That equals to 2."},{"Start":"03:58.235 ","End":"04:01.340","Text":"That\u0027s this equation right here."},{"Start":"04:01.340 ","End":"04:10.175","Text":"For this equation will write k plus 4 times the covariance of x and y,"},{"Start":"04:10.175 ","End":"04:12.454","Text":"and that equals to 3."},{"Start":"04:12.454 ","End":"04:15.200","Text":"Now, let\u0027s do the following."},{"Start":"04:15.200 ","End":"04:18.380","Text":"Let\u0027s just add these 2 equations up right here."},{"Start":"04:18.380 ","End":"04:24.025","Text":"There\u0027ll be 2k and the 4 covariance xy,"},{"Start":"04:24.025 ","End":"04:26.013","Text":"well, that cancels each other out,"},{"Start":"04:26.013 ","End":"04:27.860","Text":"and that equals to 5."},{"Start":"04:27.860 ","End":"04:32.220","Text":"That means that k then equals to 2.5."},{"Start":"04:32.230 ","End":"04:34.280","Text":"If that\u0027s the case,"},{"Start":"04:34.280 ","End":"04:39.170","Text":"let\u0027s just substitute 2.5 instead of k,"},{"Start":"04:39.170 ","End":"04:42.120","Text":"where k was this expression right here."},{"Start":"04:42.120 ","End":"04:44.840","Text":"The variance of x plus 4 times the variance of y."},{"Start":"04:44.840 ","End":"04:48.230","Text":"Now, it doesn\u0027t matter where we substitute this value."},{"Start":"04:48.230 ","End":"04:51.770","Text":"Let\u0027s just choose this equation."},{"Start":"04:51.770 ","End":"04:53.630","Text":"Now, if that\u0027s the case"},{"Start":"04:53.630 ","End":"04:58.115","Text":"then variance of x plus 4 times the variance of y. Well, that\u0027s 2.5."},{"Start":"04:58.115 ","End":"05:05.005","Text":"We\u0027ll write 2.5 minus 4 times the covariance of x and y,"},{"Start":"05:05.005 ","End":"05:07.370","Text":"but that has to be equal to 2."},{"Start":"05:07.370 ","End":"05:10.445","Text":"Now, let\u0027s do a little bit of manipulation here."},{"Start":"05:10.445 ","End":"05:14.525","Text":"Minus 4 times the covariance of x and y."},{"Start":"05:14.525 ","End":"05:17.600","Text":"That has to be equal to minus 0.5."},{"Start":"05:17.600 ","End":"05:21.320","Text":"That means that the covariance of x and y,"},{"Start":"05:21.320 ","End":"05:26.555","Text":"that has to be equal to 0.125."},{"Start":"05:26.555 ","End":"05:30.440","Text":"There we have it. That\u0027s the covariance that we were looking for,"},{"Start":"05:30.440 ","End":"05:34.020","Text":"and it equals to 0.125."}],"ID":13217},{"Watched":false,"Name":"Exercise 4","Duration":"5m 18s","ChapterTopicVideoID":12721,"CourseChapterTopicPlaylistID":245020,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12721.jpeg","UploadDate":"2018-07-30T15:08:44.3570000","DurationForVideoObject":"PT5M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.850","Text":"In this question, a dice thrown n times."},{"Start":"00:02.850 ","End":"00:05.160","Text":"Now we define the following variables."},{"Start":"00:05.160 ","End":"00:07.590","Text":"X is the number of times that 6 is thrown,"},{"Start":"00:07.590 ","End":"00:10.230","Text":"and y is the number of times that a 5 is thrown."},{"Start":"00:10.230 ","End":"00:13.440","Text":"We were asked to express the covariance of X and Y in"},{"Start":"00:13.440 ","End":"00:17.190","Text":"terms of n. Let\u0027s take a look at this."},{"Start":"00:17.190 ","End":"00:24.420","Text":"Now if we define success as getting a 6 and X is the number of times we get a 6."},{"Start":"00:24.420 ","End":"00:30.705","Text":"Then X has a binomial distribution with parameters n. We\u0027re throwing a die n times,"},{"Start":"00:30.705 ","End":"00:33.045","Text":"and the probability of success, well,"},{"Start":"00:33.045 ","End":"00:35.975","Text":"the probability of getting a 6 is 1/6."},{"Start":"00:35.975 ","End":"00:38.960","Text":"What about Y? Well, similarly,"},{"Start":"00:38.960 ","End":"00:41.960","Text":"Y is the number of times that a 5 is thrown out."},{"Start":"00:41.960 ","End":"00:44.525","Text":"If we define 5 is getting a success,"},{"Start":"00:44.525 ","End":"00:48.680","Text":"then Y has a binomial distribution with parameters n,"},{"Start":"00:48.680 ","End":"00:53.980","Text":"and the probability of getting a 5, that\u0027s also 1/6."},{"Start":"00:53.980 ","End":"00:57.590","Text":"Now we\u0027re asked to express the covariance of X and"},{"Start":"00:57.590 ","End":"01:01.340","Text":"Y in terms of n. We only have X and we only have Y,"},{"Start":"01:01.340 ","End":"01:03.230","Text":"and we have the distributions."},{"Start":"01:03.230 ","End":"01:10.580","Text":"But let\u0027s take a look at the event X plus Y. What are we doing here?"},{"Start":"01:10.580 ","End":"01:15.230","Text":"We\u0027re adding these 2 random variables together."},{"Start":"01:15.230 ","End":"01:18.095","Text":"That\u0027s the sum of 2 random variables."},{"Start":"01:18.095 ","End":"01:20.735","Text":"Now, what\u0027s the meaning of this?"},{"Start":"01:20.735 ","End":"01:25.715","Text":"This is the number of times of"},{"Start":"01:25.715 ","End":"01:31.590","Text":"getting at least 5."},{"Start":"01:31.590 ","End":"01:35.050","Text":"That\u0027s the interpretation of X plus Y."},{"Start":"01:35.050 ","End":"01:40.555","Text":"Now, that has a binomial distribution with parameters n,"},{"Start":"01:40.555 ","End":"01:44.615","Text":"and now what\u0027s the probability of getting at least a 5?"},{"Start":"01:44.615 ","End":"01:49.772","Text":"Well, that\u0027s 2 divided by 6 and that equals to 1/3."},{"Start":"01:49.772 ","End":"02:00.680","Text":"Having said that, let\u0027s take a look now at how we can express the covariance of X and Y."},{"Start":"02:00.680 ","End":"02:05.300","Text":"Well, let\u0027s take a look at this expression right here."},{"Start":"02:05.300 ","End":"02:08.570","Text":"The variance of X plus Y."},{"Start":"02:08.570 ","End":"02:09.710","Text":"What is that equal to?"},{"Start":"02:09.710 ","End":"02:13.940","Text":"That equals to the variance of X plus the variance of"},{"Start":"02:13.940 ","End":"02:19.715","Text":"Y plus 2 times the covariance of X and Y."},{"Start":"02:19.715 ","End":"02:22.475","Text":"Here we have our covariance that\u0027s right here."},{"Start":"02:22.475 ","End":"02:23.810","Text":"That\u0027s all we\u0027re looking for."},{"Start":"02:23.810 ","End":"02:25.685","Text":"Now, what don\u0027t we have now?"},{"Start":"02:25.685 ","End":"02:27.410","Text":"Well, we don\u0027t have variance of X,"},{"Start":"02:27.410 ","End":"02:28.880","Text":"we don\u0027t have the variance of Y,"},{"Start":"02:28.880 ","End":"02:31.610","Text":"and we don\u0027t have the variance of X plus Y,"},{"Start":"02:31.610 ","End":"02:33.745","Text":"but in actual fact we do,"},{"Start":"02:33.745 ","End":"02:36.630","Text":"because we know their distributions."},{"Start":"02:36.630 ","End":"02:39.125","Text":"The variance of X,"},{"Start":"02:39.125 ","End":"02:43.250","Text":"because X is distributed with the binomial distribution than the variance of X."},{"Start":"02:43.250 ","End":"02:51.885","Text":"Well, that\u0027s n times p times 1 minus p. In our case I\u0027ll be n times 1/6 times 5/6,"},{"Start":"02:51.885 ","End":"02:55.735","Text":"and that equals to n times 5/36."},{"Start":"02:55.735 ","End":"02:58.750","Text":"Now what about the variance of Y?"},{"Start":"02:58.750 ","End":"03:00.360","Text":"Well, that\u0027s the same."},{"Start":"03:00.360 ","End":"03:06.248","Text":"Because Y is also distributed with a binomial distribution with the same parameters."},{"Start":"03:06.248 ","End":"03:10.350","Text":"That\u0027ll be n times p times 1 minus p,"},{"Start":"03:10.350 ","End":"03:15.480","Text":"and that equals to n times 1/6 times 5/6,"},{"Start":"03:15.480 ","End":"03:19.150","Text":"and that equals to n times 5/36."},{"Start":"03:19.150 ","End":"03:23.435","Text":"Now, what about the variance of X plus Y?"},{"Start":"03:23.435 ","End":"03:28.235","Text":"Well, we know that that has a binomial distribution with parameters"},{"Start":"03:28.235 ","End":"03:33.515","Text":"n and the probability of being a 1/3 now,"},{"Start":"03:33.515 ","End":"03:43.215","Text":"so that equals to n times p times 1 minus p. That\u0027ll be n times 1/3 now times 2/3,"},{"Start":"03:43.215 ","End":"03:47.375","Text":"and that equals to n times 2 divided by 9."},{"Start":"03:47.375 ","End":"03:49.893","Text":"We have all the information that we need,"},{"Start":"03:49.893 ","End":"03:56.015","Text":"let\u0027s just plug in the numbers into this expression right here. Here we go."},{"Start":"03:56.015 ","End":"03:58.340","Text":"The variance of X plus Y,"},{"Start":"03:58.340 ","End":"04:03.355","Text":"that\u0027s this thing right here that\u0027s n times 2 divided by 9."},{"Start":"04:03.355 ","End":"04:05.480","Text":"That equals to the variance of X."},{"Start":"04:05.480 ","End":"04:09.050","Text":"Well, that\u0027s right here. N times 5 divided"},{"Start":"04:09.050 ","End":"04:13.505","Text":"by 36 plus the variance of Y. Well, that\u0027s here."},{"Start":"04:13.505 ","End":"04:22.340","Text":"N times 5 divided by 36 plus 2 times the covariance of X and Y."},{"Start":"04:22.340 ","End":"04:24.980","Text":"The unknown here."},{"Start":"04:24.980 ","End":"04:29.170","Text":"Now, let\u0027s just make a common denominator."},{"Start":"04:29.170 ","End":"04:31.635","Text":"There\u0027ll be n times,"},{"Start":"04:31.635 ","End":"04:36.410","Text":"that\u0027ll be 8 divided by 36 our common denominator is 36."},{"Start":"04:36.410 ","End":"04:38.780","Text":"That equals to the following."},{"Start":"04:38.780 ","End":"04:41.225","Text":"Let\u0027s just add these 2 things together."},{"Start":"04:41.225 ","End":"04:49.850","Text":"There\u0027ll be n times 10 divided by 36 plus 2 times the covariance of X and Y."},{"Start":"04:49.850 ","End":"04:53.260","Text":"Let\u0027s just move that over to this side."},{"Start":"04:53.260 ","End":"04:58.430","Text":"That means that we have here minus 2n divided by"},{"Start":"04:58.430 ","End":"05:04.430","Text":"36 and that equals to 2 times the covariance of X and Y."},{"Start":"05:04.430 ","End":"05:08.395","Text":"That means that the covariance of X and Y,"},{"Start":"05:08.395 ","End":"05:12.766","Text":"well that equals to minus n divided by 36."},{"Start":"05:12.766 ","End":"05:18.240","Text":"This then is the covariance of X and Y."}],"ID":13218}],"Thumbnail":null,"ID":245020},{"Name":"Summary Questions","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Introduction","Duration":"24s","ChapterTopicVideoID":12740,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12740.jpeg","UploadDate":"2019-02-11T14:37:06.5970000","DurationForVideoObject":"PT24S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.650","Text":"In this chapter, I present summary questions for the 2-dimensional discrete variable."},{"Start":"00:04.650 ","End":"00:09.537","Text":"I\u0027ve gathered all the theoretical materials that you\u0027ll need to solve these problems."},{"Start":"00:09.537 ","End":"00:12.870","Text":"I prefer that if you try to solve the problems by yourselves,"},{"Start":"00:12.870 ","End":"00:15.225","Text":"and only then, look at the recordings."},{"Start":"00:15.225 ","End":"00:19.140","Text":"The questions are ordered by their level of difficulty,"},{"Start":"00:19.140 ","End":"00:21.195","Text":"the easiest ones are first."},{"Start":"00:21.195 ","End":"00:24.520","Text":"Let\u0027s get to work. Good luck."}],"ID":13219},{"Watched":false,"Name":"Exercise 1 Part a","Duration":"2m 18s","ChapterTopicVideoID":12741,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12741.jpeg","UploadDate":"2019-02-01T01:25:40.3570000","DurationForVideoObject":"PT2M18S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.210","Text":"In this question, you are required to create a password using"},{"Start":"00:03.210 ","End":"00:06.045","Text":"3 characters from the following list of characters,"},{"Start":"00:06.045 ","End":"00:08.280","Text":"A, B, C, 1, and 2."},{"Start":"00:08.280 ","End":"00:10.515","Text":"The following variables are defined."},{"Start":"00:10.515 ","End":"00:14.790","Text":"X is the number of times that the character 1 appears in the password."},{"Start":"00:14.790 ","End":"00:19.680","Text":"Y is the number of times that the number 1 appears in the first and,"},{"Start":"00:19.680 ","End":"00:21.450","Text":"or last place in the password."},{"Start":"00:21.450 ","End":"00:25.710","Text":"We\u0027re asked to identify the marginal distributions of X and Y."},{"Start":"00:25.710 ","End":"00:31.305","Text":"Basically we\u0027re asked to identify the distributions of X and Y."},{"Start":"00:31.305 ","End":"00:34.000","Text":"Let\u0027s get to work."},{"Start":"00:34.700 ","End":"00:38.415","Text":"Let\u0027s take a look at the variable"},{"Start":"00:38.415 ","End":"00:44.715","Text":"X. X is the number of times that the character 1 appears in the password."},{"Start":"00:44.715 ","End":"00:48.500","Text":"Let\u0027s define a Bernoulli trial where"},{"Start":"00:48.500 ","End":"00:54.395","Text":"success in the Bernoulli trial is getting the number 1."},{"Start":"00:54.395 ","End":"00:57.320","Text":"What\u0027s the probability of getting the number 1?"},{"Start":"00:57.320 ","End":"00:59.120","Text":"We have 5 characters,"},{"Start":"00:59.120 ","End":"01:01.480","Text":"so the probability is a fifth."},{"Start":"01:01.480 ","End":"01:06.059","Text":"How many times do we have to choose in order to create the password?"},{"Start":"01:06.059 ","End":"01:10.600","Text":"We have to choose 3 times that\u0027s given to us right here."},{"Start":"01:10.600 ","End":"01:13.090","Text":"N here equals to 3."},{"Start":"01:13.090 ","End":"01:17.130","Text":"Each choice is independent of any other choice."},{"Start":"01:17.130 ","End":"01:18.995","Text":"If that\u0027s the case,"},{"Start":"01:18.995 ","End":"01:23.180","Text":"then X has a binomial distribution where n equals to 3"},{"Start":"01:23.180 ","End":"01:28.925","Text":"and the probability is equal to 1/5."},{"Start":"01:28.925 ","End":"01:33.680","Text":"What about Y? Y is"},{"Start":"01:33.680 ","End":"01:35.810","Text":"the number of times that the number 1 appears in"},{"Start":"01:35.810 ","End":"01:38.630","Text":"the first end or last place in the password."},{"Start":"01:38.630 ","End":"01:42.590","Text":"Here again, we have a Bernoulli trial where"},{"Start":"01:42.590 ","End":"01:47.495","Text":"the success is getting a 1, that\u0027s right here."},{"Start":"01:47.495 ","End":"01:51.800","Text":"The probability of getting 1 is the same, That\u0027s 1/5."},{"Start":"01:51.800 ","End":"01:57.890","Text":"But now we\u0027re interested only in the first and last place in the password."},{"Start":"01:57.890 ","End":"02:02.470","Text":"Here, for Y n equals to 2 now."},{"Start":"02:02.470 ","End":"02:05.630","Text":"Y similarly to X,"},{"Start":"02:05.630 ","End":"02:07.880","Text":"has a binomial distribution,"},{"Start":"02:07.880 ","End":"02:09.950","Text":"but now n here equals to 2,"},{"Start":"02:09.950 ","End":"02:14.360","Text":"not 3, but the probability is still 1/5."},{"Start":"02:14.360 ","End":"02:18.660","Text":"There you have it that\u0027s the marginal distribution of X and Y."}],"ID":13220},{"Watched":false,"Name":"Exercise 1 Part b","Duration":"8m 55s","ChapterTopicVideoID":12742,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12742.jpeg","UploadDate":"2019-02-01T01:26:06.3930000","DurationForVideoObject":"PT8M55S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.915","Text":"In this section we\u0027re asked to find the joint distribution of X and Y."},{"Start":"00:03.915 ","End":"00:05.700","Text":"Well, from the last section,"},{"Start":"00:05.700 ","End":"00:13.782","Text":"we know that X has a binomial distribution where n equals to 3 and p equals to 1/5,"},{"Start":"00:13.782 ","End":"00:17.550","Text":"and Y also has a binomial distribution but n here"},{"Start":"00:17.550 ","End":"00:21.840","Text":"equals to 2 and the probability equals to 1/5."},{"Start":"00:21.840 ","End":"00:24.480","Text":"Now, what are the values for X?"},{"Start":"00:24.480 ","End":"00:26.625","Text":"Well, because n equals 3,"},{"Start":"00:26.625 ","End":"00:29.460","Text":"X can have values of 0,"},{"Start":"00:29.460 ","End":"00:31.290","Text":"1, 2,"},{"Start":"00:31.290 ","End":"00:34.150","Text":"and 3. What about Y?"},{"Start":"00:34.150 ","End":"00:36.050","Text":"What are the valid values for Y?"},{"Start":"00:36.050 ","End":"00:37.985","Text":"Well, because n here is 2 now,"},{"Start":"00:37.985 ","End":"00:40.910","Text":"then Y can have values of 0,"},{"Start":"00:40.910 ","End":"00:43.200","Text":"1, and 2."},{"Start":"00:43.420 ","End":"00:50.750","Text":"Let\u0027s just recall the formula for the binomial distribution."},{"Start":"00:50.750 ","End":"00:53.000","Text":"The probability of X,"},{"Start":"00:53.000 ","End":"00:57.570","Text":"basically any random variable equaling some value."},{"Start":"00:57.570 ","End":"01:03.050","Text":"Well, that equals to n over k times p^k"},{"Start":"01:03.050 ","End":"01:09.970","Text":"times 1 minus p^n minus k. Now,"},{"Start":"01:09.970 ","End":"01:14.615","Text":"we have everything that we need to create the joint distribution of X and Y."},{"Start":"01:14.615 ","End":"01:16.855","Text":"Let\u0027s do that."},{"Start":"01:16.855 ","End":"01:22.790","Text":"This then is the table in which we\u0027ll fill the probabilities of X and Y."},{"Start":"01:22.790 ","End":"01:27.200","Text":"Now, here we\u0027ll have the marginal probabilities of X and Y."},{"Start":"01:27.200 ","End":"01:30.970","Text":"The blue is the marginal probabilities for X and the orange for Y."},{"Start":"01:30.970 ","End":"01:34.008","Text":"The white cells, the cells within the table,"},{"Start":"01:34.008 ","End":"01:36.710","Text":"well, that\u0027ll be the joint probabilities of X and Y."},{"Start":"01:36.710 ","End":"01:38.765","Text":"Now, how do we start?"},{"Start":"01:38.765 ","End":"01:43.295","Text":"Well, since we already know what the distribution is of X and Y,"},{"Start":"01:43.295 ","End":"01:48.610","Text":"we\u0027ll start with the marginal probabilities. How do we do that?"},{"Start":"01:48.610 ","End":"01:55.235","Text":"Well, let\u0027s take a look at the probability of X equaling to 1."},{"Start":"01:55.235 ","End":"01:58.505","Text":"Well, that\u0027s this cell right here."},{"Start":"01:58.505 ","End":"02:01.130","Text":"Now, what\u0027s the probability of X equaling 1?"},{"Start":"02:01.130 ","End":"02:04.895","Text":"Well, X has a binomial distribution and k here equals to 1,"},{"Start":"02:04.895 ","End":"02:07.445","Text":"n equals to 3."},{"Start":"02:07.445 ","End":"02:16.970","Text":"That\u0027ll be 3 over 1 times 1/5^1 times 4/5^2."},{"Start":"02:16.970 ","End":"02:23.080","Text":"Now, that equals to 48 divided by 125."},{"Start":"02:23.080 ","End":"02:26.100","Text":"Let\u0027s write that probability."},{"Start":"02:26.100 ","End":"02:27.960","Text":"Let\u0027s write that here."},{"Start":"02:27.960 ","End":"02:32.265","Text":"That\u0027ll be 48 divided by 125."},{"Start":"02:32.265 ","End":"02:37.860","Text":"Now similarly, we\u0027ll do that for all other values of x."},{"Start":"02:38.210 ","End":"02:43.295","Text":"Here the other probabilities for the other values of X."},{"Start":"02:43.295 ","End":"02:46.460","Text":"Now, what about the marginal probabilities of Y?"},{"Start":"02:46.460 ","End":"02:51.235","Text":"Well, let\u0027s take a look at the probability of Y now equaling to 2."},{"Start":"02:51.235 ","End":"02:58.910","Text":"Well, n equals to 2 when we\u0027re talking about the random variable Y."},{"Start":"02:58.910 ","End":"03:08.990","Text":"That\u0027ll be 2 over 2 times 1/5^2 times 4/5^0."},{"Start":"03:08.990 ","End":"03:13.810","Text":"Now, that equals to 5 divided by 125."},{"Start":"03:13.810 ","End":"03:16.549","Text":"That\u0027s this cell right here."},{"Start":"03:16.549 ","End":"03:19.550","Text":"We\u0027ll input this probability right here."},{"Start":"03:19.550 ","End":"03:22.415","Text":"That\u0027ll be 5 divided by 125."},{"Start":"03:22.415 ","End":"03:28.080","Text":"Now similarly, we\u0027ll do that for all other values of Y as well."},{"Start":"03:28.460 ","End":"03:32.680","Text":"Now we have the marginal probabilities of X and Y."},{"Start":"03:32.680 ","End":"03:36.125","Text":"Now, what about the joint probabilities within the table?"},{"Start":"03:36.125 ","End":"03:38.760","Text":"Well, how do we do that?"},{"Start":"03:38.770 ","End":"03:42.950","Text":"In order to calculate the joint probabilities of X and Y,"},{"Start":"03:42.950 ","End":"03:49.160","Text":"let\u0027s first try to understand what\u0027s behind the definitions of X and Y."},{"Start":"03:49.160 ","End":"03:54.975","Text":"Now, we have a 3-character code that\u0027s given to us."},{"Start":"03:54.975 ","End":"03:58.845","Text":"This is the first character,"},{"Start":"03:58.845 ","End":"04:00.260","Text":"this is the second character,"},{"Start":"04:00.260 ","End":"04:02.660","Text":"and this is the third character."},{"Start":"04:02.660 ","End":"04:07.490","Text":"All these characters, they comprise the code."},{"Start":"04:07.490 ","End":"04:11.210","Text":"X counts the number of 1s in the code."},{"Start":"04:11.210 ","End":"04:15.710","Text":"So I can have 0, 1s in one of the 3 places,"},{"Start":"04:15.710 ","End":"04:17.315","Text":"either 1, 2,"},{"Start":"04:17.315 ","End":"04:20.285","Text":"or 3, 1s in the code."},{"Start":"04:20.285 ","End":"04:22.020","Text":"Now what about Y?"},{"Start":"04:22.020 ","End":"04:29.940","Text":"Well, Y counts the number of 1s in either the first or the third place."},{"Start":"04:29.940 ","End":"04:32.115","Text":"I can have 0, 1s,"},{"Start":"04:32.115 ","End":"04:35.480","Text":"I can have a 1 in the second place,"},{"Start":"04:35.480 ","End":"04:37.460","Text":"but not in the first and third place,"},{"Start":"04:37.460 ","End":"04:39.150","Text":"or I can have 1,"},{"Start":"04:39.150 ","End":"04:43.340","Text":"1 either in the first place or the third place, but not both."},{"Start":"04:43.340 ","End":"04:44.750","Text":"Or you can have 2,"},{"Start":"04:44.750 ","End":"04:47.435","Text":"1s in the first and in the third place."},{"Start":"04:47.435 ","End":"04:52.945","Text":"Let\u0027s now take a look at this cell right here where X equals 0 and Y equals 2."},{"Start":"04:52.945 ","End":"04:57.740","Text":"Y equals 2 means that I have 1s in the first and in the third place,"},{"Start":"04:57.740 ","End":"05:00.980","Text":"but X means that I have no 1s in the code whatsoever."},{"Start":"05:00.980 ","End":"05:03.035","Text":"So this is an impossibility."},{"Start":"05:03.035 ","End":"05:05.740","Text":"That means the probability here is 0."},{"Start":"05:05.740 ","End":"05:07.730","Text":"Now, what about here,"},{"Start":"05:07.730 ","End":"05:11.165","Text":"in this cell right here where X equals 0 and Y equals 1?"},{"Start":"05:11.165 ","End":"05:13.190","Text":"Well, when any X equals 0,"},{"Start":"05:13.190 ","End":"05:18.490","Text":"that means that I have no 1s in the code but Y says that I have at least 1,"},{"Start":"05:18.490 ","End":"05:21.860","Text":"1, either in the first or in the third place right here."},{"Start":"05:21.860 ","End":"05:23.900","Text":"Again, that\u0027s an impossibility,"},{"Start":"05:23.900 ","End":"05:25.775","Text":"so we\u0027ll put a 0 here."},{"Start":"05:25.775 ","End":"05:31.610","Text":"Now, what about this cell right here where X equals 1 but Y equals 2?"},{"Start":"05:31.610 ","End":"05:37.370","Text":"Well, Y equals 2 means that I have 1s in the first and in the third place,"},{"Start":"05:37.370 ","End":"05:39.935","Text":"but there\u0027s only 1, 1 in the code."},{"Start":"05:39.935 ","End":"05:42.455","Text":"Now again, that\u0027s an impossibility."},{"Start":"05:42.455 ","End":"05:46.340","Text":"Now, let\u0027s take a look at this cell right here,"},{"Start":"05:46.340 ","End":"05:48.710","Text":"where X equals 3 and Y equals 0."},{"Start":"05:48.710 ","End":"05:50.570","Text":"Well, when X equals 3,"},{"Start":"05:50.570 ","End":"05:52.700","Text":"I have 1 in the first place,"},{"Start":"05:52.700 ","End":"05:53.780","Text":"1 in the second place,"},{"Start":"05:53.780 ","End":"05:55.500","Text":"and 1 in the third place."},{"Start":"05:55.500 ","End":"05:59.915","Text":"But I have no 1s in the first and in the third place."},{"Start":"05:59.915 ","End":"06:01.340","Text":"Well, that\u0027s an impossibility,"},{"Start":"06:01.340 ","End":"06:03.260","Text":"so that also is a 0."},{"Start":"06:03.260 ","End":"06:05.660","Text":"What about this place right here,"},{"Start":"06:05.660 ","End":"06:07.205","Text":"this cell right here?"},{"Start":"06:07.205 ","End":"06:11.110","Text":"That\u0027s where X equals 3 and Y equals 1."},{"Start":"06:11.110 ","End":"06:13.830","Text":"Well, if X equals 3,"},{"Start":"06:13.830 ","End":"06:18.030","Text":"then I have 1 in the first place, 1 in the second place,"},{"Start":"06:18.030 ","End":"06:20.640","Text":"and 1 in the third, but I only have 1,"},{"Start":"06:20.640 ","End":"06:26.555","Text":"1 either in the first place or in the third place."},{"Start":"06:26.555 ","End":"06:29.210","Text":"That also is an impossibility."},{"Start":"06:29.210 ","End":"06:34.470","Text":"Now, what happens when X equals 2 and Y equals 0?"},{"Start":"06:34.470 ","End":"06:41.470","Text":"Well, if X equals 2 then I\u0027ll have 1 either here or here and not here."},{"Start":"06:41.470 ","End":"06:43.579","Text":"Here, not here,"},{"Start":"06:43.579 ","End":"06:46.385","Text":"and here, or not here,"},{"Start":"06:46.385 ","End":"06:47.935","Text":"here, and here."},{"Start":"06:47.935 ","End":"06:51.945","Text":"These are the 3 options where X equals 2."},{"Start":"06:51.945 ","End":"06:55.413","Text":"Now, what happens where Y equals 0?"},{"Start":"06:55.413 ","End":"07:02.825","Text":"Y equals 0 means that there are no 1s in the first and in the third position."},{"Start":"07:02.825 ","End":"07:07.400","Text":"Now, in no case here do I"},{"Start":"07:07.400 ","End":"07:14.040","Text":"have no 1s either in the first position or in the third position."},{"Start":"07:14.040 ","End":"07:17.850","Text":"That means that when X equals 2 and Y equals 0,"},{"Start":"07:17.850 ","End":"07:19.650","Text":"that\u0027s an impossibility,"},{"Start":"07:19.650 ","End":"07:22.830","Text":"so that also is a 0."},{"Start":"07:22.830 ","End":"07:28.100","Text":"Here we have it. We\u0027ve eliminated all the cells that are impossible."},{"Start":"07:28.100 ","End":"07:33.350","Text":"Let\u0027s try to calculate the joint probabilities of the remaining cells."},{"Start":"07:33.350 ","End":"07:41.960","Text":"Well, we know that if we add up all the joint probabilities in the row or in the column,"},{"Start":"07:41.960 ","End":"07:46.415","Text":"then the sum should equal the marginal probabilities."},{"Start":"07:46.415 ","End":"07:49.795","Text":"Here, when X equals 0,"},{"Start":"07:49.795 ","End":"07:53.380","Text":"the only remaining cell is this cell right here."},{"Start":"07:53.380 ","End":"07:55.330","Text":"All the other cells are 0."},{"Start":"07:55.330 ","End":"08:01.840","Text":"So that means that this all has to be equal to 64 divided by 125."},{"Start":"08:01.840 ","End":"08:04.315","Text":"Now, if that\u0027s the case,"},{"Start":"08:04.315 ","End":"08:11.575","Text":"then the sum of these rows have to add up to 80 divided by 125."},{"Start":"08:11.575 ","End":"08:17.710","Text":"That means that this has to be equal to 16 divided by 125."},{"Start":"08:17.710 ","End":"08:20.485","Text":"Now, what about this cell right here?"},{"Start":"08:20.485 ","End":"08:22.750","Text":"That has to add up to 48,"},{"Start":"08:22.750 ","End":"08:24.610","Text":"so divided by 125."},{"Start":"08:24.610 ","End":"08:28.680","Text":"This then is 32 divided by 125."},{"Start":"08:28.680 ","End":"08:33.105","Text":"This cell is 8 divided by 125,"},{"Start":"08:33.105 ","End":"08:36.620","Text":"this cell is 4 divided by 125,"},{"Start":"08:36.620 ","End":"08:39.170","Text":"and this is 1 divided by 125."},{"Start":"08:39.170 ","End":"08:44.205","Text":"So here we go. This is the joint distribution of X and Y,"},{"Start":"08:44.205 ","End":"08:49.545","Text":"and we saw that we didn\u0027t have to calculate anything,"},{"Start":"08:49.545 ","End":"08:55.950","Text":"all we needed to do was to understand the logic behind the definitions of X and Y."}],"ID":13221},{"Watched":false,"Name":"Exercise 1 Parts c-d","Duration":"7m 48s","ChapterTopicVideoID":12743,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12743.jpeg","UploadDate":"2019-02-11T14:25:35.9530000","DurationForVideoObject":"PT7M48S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.275","Text":"In this section, we\u0027re asked to find a correlation coefficient of X and Y."},{"Start":"00:04.275 ","End":"00:07.545","Text":"The correlation coefficient of X and Y,"},{"Start":"00:07.545 ","End":"00:11.400","Text":"well that equals to the covariance of X and Y"},{"Start":"00:11.400 ","End":"00:17.535","Text":"divided by the standard deviation of X times the standard deviation of Y."},{"Start":"00:17.535 ","End":"00:20.985","Text":"Now, what\u0027s the covariance of X and Y?"},{"Start":"00:20.985 ","End":"00:23.175","Text":"Covariance of X and Y."},{"Start":"00:23.175 ","End":"00:26.955","Text":"That equals to the expectation of X times Y"},{"Start":"00:26.955 ","End":"00:33.090","Text":"minus the expectation of X times the expectation of Y."},{"Start":"00:33.090 ","End":"00:40.470","Text":"Now, we know that X and Y have a binomial distribution."},{"Start":"00:40.470 ","End":"00:43.490","Text":"Let\u0027s just recall if we have a random variable,"},{"Start":"00:43.490 ","End":"00:48.470","Text":"let\u0027s say w and it has a binomial distribution with parameters and in p,"},{"Start":"00:48.470 ","End":"00:52.850","Text":"where the expectation of w that equals to n times p,"},{"Start":"00:52.850 ","End":"00:58.890","Text":"and the variance of w that equals to n times p times 1 minus"},{"Start":"00:58.890 ","End":"01:08.800","Text":"p. If X has a binomial distribution where n equals to 3 and p equals to 1/5."},{"Start":"01:08.800 ","End":"01:11.675","Text":"Well, the expectation of X,"},{"Start":"01:11.675 ","End":"01:13.910","Text":"that equals to n times p,"},{"Start":"01:13.910 ","End":"01:18.155","Text":"that\u0027s 3 times 1/5 and that equals to 0.6."},{"Start":"01:18.155 ","End":"01:20.300","Text":"What about the variance of X?"},{"Start":"01:20.300 ","End":"01:27.170","Text":"Well, that equals to n times p times 1 minus p. That\u0027s 3 times 1/5 times 4/5."},{"Start":"01:27.170 ","End":"01:32.735","Text":"Now, that equals to 0.48. Now, what about Y?"},{"Start":"01:32.735 ","End":"01:39.845","Text":"Y has a binomial distribution now where n equals to 2 and p equals to 1/5."},{"Start":"01:39.845 ","End":"01:41.930","Text":"The expectation of Y,"},{"Start":"01:41.930 ","End":"01:44.460","Text":"that equals to n times p,"},{"Start":"01:44.460 ","End":"01:48.810","Text":"that\u0027s 2 times 1/5. That equals to 0.4."},{"Start":"01:48.810 ","End":"01:50.880","Text":"Now, what about the variance of Y?"},{"Start":"01:50.880 ","End":"01:52.710","Text":"Well, that equals to 2,"},{"Start":"01:52.710 ","End":"01:59.010","Text":"n times p times 1 minus p that\u0027s 2 times 1/5 times 4/5,"},{"Start":"01:59.010 ","End":"02:03.340","Text":"and that equals to 0.32."},{"Start":"02:04.660 ","End":"02:09.410","Text":"Now, we have the expectation of Y,"},{"Start":"02:09.410 ","End":"02:10.895","Text":"the expectation of X,"},{"Start":"02:10.895 ","End":"02:12.500","Text":"we have the variance of X,"},{"Start":"02:12.500 ","End":"02:14.185","Text":"and the variance of Y."},{"Start":"02:14.185 ","End":"02:17.345","Text":"Expectation of X, expectation of Y,"},{"Start":"02:17.345 ","End":"02:19.190","Text":"the variance of X and Y."},{"Start":"02:19.190 ","End":"02:23.750","Text":"Well, the square root of the variance of the X and Y is the standard deviation."},{"Start":"02:23.750 ","End":"02:26.330","Text":"We have that. What are we missing?"},{"Start":"02:26.330 ","End":"02:28.910","Text":"Well, we\u0027re missing this element right here."},{"Start":"02:28.910 ","End":"02:31.235","Text":"The expectation of X times Y."},{"Start":"02:31.235 ","End":"02:34.400","Text":"Now, the expectation of X times Y, well,"},{"Start":"02:34.400 ","End":"02:42.545","Text":"that equals to the sum of X times Y times the probability of X and Y,"},{"Start":"02:42.545 ","End":"02:44.870","Text":"the joint probability of X and Y."},{"Start":"02:44.870 ","End":"02:47.255","Text":"Now, what is that equal to?"},{"Start":"02:47.255 ","End":"02:52.190","Text":"Well, for that we have to go as a joint probability table that\u0027s right here."},{"Start":"02:52.190 ","End":"02:54.080","Text":"Now, when we come here,"},{"Start":"02:54.080 ","End":"02:57.680","Text":"we can see that we can get rid of this column right here."},{"Start":"02:57.680 ","End":"03:02.645","Text":"Well, that\u0027s because X equals 0 and we have a 0 multiplier here."},{"Start":"03:02.645 ","End":"03:07.190","Text":"We can also get rid of this row right here again because Y"},{"Start":"03:07.190 ","End":"03:12.245","Text":"equals 0 and we have 0 multiply here for Y."},{"Start":"03:12.245 ","End":"03:18.075","Text":"We can get rid of these joint probabilities that equal 0,"},{"Start":"03:18.075 ","End":"03:20.730","Text":"that\u0027s a 0 multiplier here."},{"Start":"03:20.730 ","End":"03:24.770","Text":"All this thing on this expression was down to"},{"Start":"03:24.770 ","End":"03:31.605","Text":"4 basic elements where X equals 1 and Y equals 1"},{"Start":"03:31.605 ","End":"03:41.330","Text":"and the joint probability here is 32 divided by 125 plus X equals 2 times Y equals 1"},{"Start":"03:41.330 ","End":"03:51.770","Text":"times 8 divided by 125 plus X equals 2 times Y equals 2 times 4 divided by 125 plus X"},{"Start":"03:51.770 ","End":"03:56.090","Text":"equals 3 and Y equals 2 times 1 divided by"},{"Start":"03:56.090 ","End":"04:03.400","Text":"125 and that equals to 70 divided by 125."},{"Start":"04:03.740 ","End":"04:09.605","Text":"Now we can calculate the covariance of X and Y."},{"Start":"04:09.605 ","End":"04:13.490","Text":"Well, the covariance of X and Y equals the expectation of"},{"Start":"04:13.490 ","End":"04:17.840","Text":"X times Y minus the expectation of X times expectation of Y."},{"Start":"04:17.840 ","End":"04:20.840","Text":"Well, the expectation of X times Y, that\u0027s right here."},{"Start":"04:20.840 ","End":"04:26.635","Text":"That equals to 70 divided by 125 minus the expectation of"},{"Start":"04:26.635 ","End":"04:32.955","Text":"X that\u0027s 0.6 times the expectation of Y that\u0027s 0.4."},{"Start":"04:32.955 ","End":"04:37.845","Text":"Now, that comes out to 0.32."},{"Start":"04:37.845 ","End":"04:41.580","Text":"Now, if we have the covariance,"},{"Start":"04:41.580 ","End":"04:45.650","Text":"then we can calculate the correlation coefficient."},{"Start":"04:45.650 ","End":"04:48.560","Text":"The correlation coefficient between X and Y,"},{"Start":"04:48.560 ","End":"04:50.930","Text":"that equals to the covariance of X and Y."},{"Start":"04:50.930 ","End":"04:54.870","Text":"Well, that\u0027s 0.32, that\u0027s right here."},{"Start":"04:55.000 ","End":"04:58.175","Text":"Now, what\u0027s the standard deviation of X?"},{"Start":"04:58.175 ","End":"05:00.230","Text":"Well, that\u0027s the square root of the variance,"},{"Start":"05:00.230 ","End":"05:03.200","Text":"that\u0027s the square root of 0.48."},{"Start":"05:03.200 ","End":"05:09.580","Text":"That\u0027s right here, times the square root of 0.32."},{"Start":"05:09.580 ","End":"05:12.710","Text":"0.32 is the variance of Y."},{"Start":"05:12.710 ","End":"05:14.540","Text":"We want the standard deviation of Y,"},{"Start":"05:14.540 ","End":"05:17.885","Text":"so that\u0027s the square root of 0.32."},{"Start":"05:17.885 ","End":"05:23.015","Text":"Now, or this comes out to 0.816."},{"Start":"05:23.015 ","End":"05:27.215","Text":"There we have it. That\u0027s the correlation coefficient of X and Y."},{"Start":"05:27.215 ","End":"05:34.755","Text":"We see that there\u0027s a fairly strong connection between X and Y and it\u0027s positive."},{"Start":"05:34.755 ","End":"05:38.450","Text":"That makes sense because as Y increases,"},{"Start":"05:38.450 ","End":"05:42.634","Text":"the number of 1s in the first and 3rd positions increases."},{"Start":"05:42.634 ","End":"05:45.800","Text":"Well then so does X. X also increases."},{"Start":"05:45.800 ","End":"05:48.810","Text":"That\u0027s the total number of 1s in the code."},{"Start":"05:49.120 ","End":"05:56.135","Text":"In this section, we\u0027re asked what\u0027s the correlation coefficient between 2X and 3Y plus 5?"},{"Start":"05:56.135 ","End":"05:58.040","Text":"Well, if we recall,"},{"Start":"05:58.040 ","End":"06:02.120","Text":"the correlation coefficient of the linear transformation of a X plus"},{"Start":"06:02.120 ","End":"06:06.110","Text":"b and c Y plus d. Well,"},{"Start":"06:06.110 ","End":"06:12.980","Text":"that equals to the correlation coefficient between X and Y if a times c is greater than"},{"Start":"06:12.980 ","End":"06:21.210","Text":"0 and minus the correlation coefficient between X and Y if a times c is less than 0."},{"Start":"06:21.210 ","End":"06:22.790","Text":"What do we have here?"},{"Start":"06:22.790 ","End":"06:29.870","Text":"We have the correlation coefficient between 2X and 3Y plus 5."},{"Start":"06:29.870 ","End":"06:33.005","Text":"Now, if we use this template,"},{"Start":"06:33.005 ","End":"06:35.561","Text":"then a here equals to 2,"},{"Start":"06:35.561 ","End":"06:38.135","Text":"that\u0027s the X multiplier,"},{"Start":"06:38.135 ","End":"06:41.085","Text":"b that equals to 0,"},{"Start":"06:41.085 ","End":"06:43.110","Text":"c equals to 3,"},{"Start":"06:43.110 ","End":"06:44.310","Text":"the Y multiplier,"},{"Start":"06:44.310 ","End":"06:46.305","Text":"and d equals to 5."},{"Start":"06:46.305 ","End":"06:48.660","Text":"Now, a times c,"},{"Start":"06:48.660 ","End":"06:50.180","Text":"what is that equal to?"},{"Start":"06:50.180 ","End":"06:53.180","Text":"A times c, that equals to 2 times 3,"},{"Start":"06:53.180 ","End":"06:56.570","Text":"that equals to 6, and that\u0027s obviously greater than 0."},{"Start":"06:56.570 ","End":"07:01.610","Text":"That means that the correlation coefficients of this linear transformation of"},{"Start":"07:01.610 ","End":"07:07.225","Text":"the random variables that equals to the correlation coefficient of X and Y."},{"Start":"07:07.225 ","End":"07:12.725","Text":"We\u0027ve calculated that in the last section, that\u0027s 0.816."},{"Start":"07:12.725 ","End":"07:20.570","Text":"We can see that the correlation coefficient of a linear transformation remains the same."},{"Start":"07:20.570 ","End":"07:24.415","Text":"The strength of the connection remains the same between X and Y."},{"Start":"07:24.415 ","End":"07:29.975","Text":"The only thing that may change the direction of the dependency,"},{"Start":"07:29.975 ","End":"07:37.730","Text":"but that\u0027s dependent on the multipliers of the random variables a and c. But in our case,"},{"Start":"07:37.730 ","End":"07:44.340","Text":"a times c was greater than 0 so the correlation coefficient remains the same."}],"ID":13222},{"Watched":false,"Name":"Exercise 2 Part a","Duration":"10m 50s","ChapterTopicVideoID":12744,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12744.jpeg","UploadDate":"2019-02-11T14:26:46.2200000","DurationForVideoObject":"PT10M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.410","Text":"At a graduation party,"},{"Start":"00:01.410 ","End":"00:04.365","Text":"there\u0027s an ice-filled tube containing 7 bottles of beer,"},{"Start":"00:04.365 ","End":"00:07.620","Text":"4 Coors, 2 Budweiser, and 1 Miller."},{"Start":"00:07.620 ","End":"00:11.190","Text":"Now, Jane took 3 bottles of beer at random from the tub."},{"Start":"00:11.190 ","End":"00:13.845","Text":"We define the following variables."},{"Start":"00:13.845 ","End":"00:17.460","Text":"X is the number, of Coors bottles taken from the tub by Jane,"},{"Start":"00:17.460 ","End":"00:21.420","Text":"and Y is the number of Miller\u0027s bottles taken from the tub by Jane,"},{"Start":"00:21.420 ","End":"00:26.470","Text":"and we\u0027re asked to construct the joint distribution of X and Y."},{"Start":"00:26.540 ","End":"00:29.410","Text":"Now in this specific question,"},{"Start":"00:29.410 ","End":"00:34.040","Text":"we should start with the marginal distribution of X and Y. Why is that?"},{"Start":"00:34.040 ","End":"00:40.010","Text":"Well, it\u0027s easily seen that the marginal distribution of X and Y is a special one."},{"Start":"00:40.010 ","End":"00:42.665","Text":"It\u0027s a hypergeometric distribution."},{"Start":"00:42.665 ","End":"00:44.450","Text":"Now, why am I saying that?"},{"Start":"00:44.450 ","End":"00:46.835","Text":"Well, let\u0027s take a look at what\u0027s given."},{"Start":"00:46.835 ","End":"00:51.815","Text":"We have a population of size N,"},{"Start":"00:51.815 ","End":"00:53.950","Text":"and that equals to 7."},{"Start":"00:53.950 ","End":"00:57.335","Text":"We have a sample size,"},{"Start":"00:57.335 ","End":"01:01.190","Text":"that\u0027s n, that\u0027s 3."},{"Start":"01:01.190 ","End":"01:05.180","Text":"Also, we\u0027re interested in the number of"},{"Start":"01:05.180 ","End":"01:11.720","Text":"objects in the sample from a sub-population or subgroup."},{"Start":"01:11.720 ","End":"01:15.560","Text":"In our case, for X for example,"},{"Start":"01:15.560 ","End":"01:18.620","Text":"D here would be equal to 4."},{"Start":"01:18.620 ","End":"01:22.775","Text":"We\u0027re looking at the number of Coors bottles taken from the tub by Jane."},{"Start":"01:22.775 ","End":"01:27.365","Text":"That means that the size of the subpopulation here,"},{"Start":"01:27.365 ","End":"01:29.336","Text":"well that equals to 4,"},{"Start":"01:29.336 ","End":"01:35.420","Text":"and we\u0027re looking for the number of Coors bottles in our sample of 3."},{"Start":"01:35.420 ","End":"01:43.500","Text":"So X then has a hypergeometric distribution where N equals to 7,"},{"Start":"01:43.500 ","End":"01:47.795","Text":"D, the size of our subpopulation here,"},{"Start":"01:47.795 ","End":"01:53.285","Text":"that equals to 4, and our sample size n, that equals to 3."},{"Start":"01:53.285 ","End":"01:56.240","Text":"Now, what about Y? Well similarly,"},{"Start":"01:56.240 ","End":"02:03.350","Text":"Y has a hypergeometric distribution where the population size is the same."},{"Start":"02:03.350 ","End":"02:07.820","Text":"It\u0027s 7. Now, what about the sub-population size?"},{"Start":"02:07.820 ","End":"02:11.075","Text":"Well, we\u0027re looking now at Miller bottles."},{"Start":"02:11.075 ","End":"02:13.070","Text":"Now, how many Miller bottles do we have?"},{"Start":"02:13.070 ","End":"02:14.270","Text":"We have 1,"},{"Start":"02:14.270 ","End":"02:16.130","Text":"so D here equals to 1,"},{"Start":"02:16.130 ","End":"02:20.270","Text":"and we\u0027re still sampling 3 bottles,"},{"Start":"02:20.270 ","End":"02:23.670","Text":"so n here equals to 3."},{"Start":"02:23.920 ","End":"02:30.020","Text":"Having said that, what are the valid values for X and Y?"},{"Start":"02:30.020 ","End":"02:36.560","Text":"Well, X can have the values of 0 in our sample."},{"Start":"02:36.560 ","End":"02:40.205","Text":"We can sample any other bottles of beer,"},{"Start":"02:40.205 ","End":"02:42.100","Text":"except for Coors bottles."},{"Start":"02:42.100 ","End":"02:44.950","Text":"That means that we have 0 Coors bottles in our sample."},{"Start":"02:44.950 ","End":"02:46.640","Text":"We can have 1,"},{"Start":"02:46.640 ","End":"02:49.680","Text":"2, or 3 Coors bottles."},{"Start":"02:49.680 ","End":"02:53.025","Text":"Although there are 4 Coors bottles in the tub,"},{"Start":"02:53.025 ","End":"02:55.050","Text":"since we only sample 3,"},{"Start":"02:55.050 ","End":"02:59.220","Text":"then the maximum number of Coors bottles that we can have in our sample is 3."},{"Start":"02:59.220 ","End":"03:01.030","Text":"Now, what about Y?"},{"Start":"03:01.030 ","End":"03:03.890","Text":"Well, what\u0027s the valid values for Y?"},{"Start":"03:03.890 ","End":"03:07.325","Text":"Well, we can have 0 bottles in our sample,"},{"Start":"03:07.325 ","End":"03:10.790","Text":"and we can have 1 bottle of Miller\u0027s beer in our sample."},{"Start":"03:10.790 ","End":"03:15.185","Text":"We can\u0027t have anymore because there aren\u0027t anymore in the population."},{"Start":"03:15.185 ","End":"03:19.010","Text":"These are the valid values for X and Y."},{"Start":"03:19.010 ","End":"03:22.185","Text":"Now also, let\u0027s just remind ourselves,"},{"Start":"03:22.185 ","End":"03:27.650","Text":"what\u0027s the probability of a random variable, for example,"},{"Start":"03:27.650 ","End":"03:33.665","Text":"W equaling k, where W has a hypergeometric distribution."},{"Start":"03:33.665 ","End":"03:44.420","Text":"Well, that equals to D over k times N minus D over n"},{"Start":"03:44.420 ","End":"03:51.360","Text":"minus k divided by N over n. This"},{"Start":"03:51.360 ","End":"03:54.560","Text":"then is the way that we calculate"},{"Start":"03:54.560 ","End":"03:59.340","Text":"our probabilities when we have a hypergeometric distribution."},{"Start":"03:59.390 ","End":"04:03.065","Text":"We have, now, all the information that we need"},{"Start":"04:03.065 ","End":"04:07.230","Text":"to construct our joint probability distribution."},{"Start":"04:07.340 ","End":"04:12.530","Text":"Here\u0027s the table in which we\u0027ll put in all our probabilities."},{"Start":"04:12.530 ","End":"04:15.298","Text":"In the white cells, we\u0027ll have the joint probabilities,"},{"Start":"04:15.298 ","End":"04:18.620","Text":"and in the margins, we\u0027ll have the marginal probabilities."},{"Start":"04:18.620 ","End":"04:22.110","Text":"The blue cells are the marginal probabilities of"},{"Start":"04:22.110 ","End":"04:26.495","Text":"X and the orange are the marginal probabilities of Y."},{"Start":"04:26.495 ","End":"04:28.655","Text":"Let\u0027s get started."},{"Start":"04:28.655 ","End":"04:32.165","Text":"Let\u0027s first take a look at this right here,"},{"Start":"04:32.165 ","End":"04:33.500","Text":"where X equals 1."},{"Start":"04:33.500 ","End":"04:37.775","Text":"We\u0027re looking for the probability of X equaling 1."},{"Start":"04:37.775 ","End":"04:41.840","Text":"Well, according to this expression,"},{"Start":"04:41.840 ","End":"04:43.580","Text":"when X equals 1, well,"},{"Start":"04:43.580 ","End":"04:45.890","Text":"d here, equals to 4."},{"Start":"04:45.890 ","End":"04:48.260","Text":"That\u0027ll be 4 over 1,"},{"Start":"04:48.260 ","End":"04:53.820","Text":"K equals 1 times n minus D. That\u0027ll be 7."},{"Start":"04:53.820 ","End":"04:56.280","Text":"N is 7 and D is 4,"},{"Start":"04:56.280 ","End":"05:01.260","Text":"so it\u0027ll be 7 minus 4 over n minus k. Well,"},{"Start":"05:01.260 ","End":"05:03.075","Text":"n is 3,"},{"Start":"05:03.075 ","End":"05:05.385","Text":"minus k is 1."},{"Start":"05:05.385 ","End":"05:09.860","Text":"That\u0027s the numerator divided by N,"},{"Start":"05:09.860 ","End":"05:13.475","Text":"which is 7 over n, which is 3."},{"Start":"05:13.475 ","End":"05:20.700","Text":"Now, that equals to 12 divided by 35."},{"Start":"05:20.810 ","End":"05:22.940","Text":"What we have here,"},{"Start":"05:22.940 ","End":"05:26.840","Text":"we have the probability of X equaling 1 that\u0027s 12 divided by 35."},{"Start":"05:26.840 ","End":"05:28.160","Text":"We put that here."},{"Start":"05:28.160 ","End":"05:29.669","Text":"That\u0027s right here,"},{"Start":"05:29.669 ","End":"05:32.145","Text":"12 divided by 35."},{"Start":"05:32.145 ","End":"05:39.505","Text":"Now, let\u0027s take a look at another example with the probability of X equaling 0."},{"Start":"05:39.505 ","End":"05:42.835","Text":"Well, again, k equals 0,"},{"Start":"05:42.835 ","End":"05:44.580","Text":"D equals 4,"},{"Start":"05:44.580 ","End":"05:46.110","Text":"n equals 3,"},{"Start":"05:46.110 ","End":"05:47.450","Text":"and N equals 7,"},{"Start":"05:47.450 ","End":"05:48.983","Text":"so that\u0027ll be D over k,"},{"Start":"05:48.983 ","End":"05:53.980","Text":"that\u0027ll be, 4 over 0 times N minus D. Well,"},{"Start":"05:53.980 ","End":"05:55.150","Text":"that\u0027s 7 minus 4,"},{"Start":"05:55.150 ","End":"05:59.440","Text":"that\u0027s 3 over n minus k. Well,"},{"Start":"05:59.440 ","End":"06:02.020","Text":"that\u0027s 3 minus 0, that\u0027s 3."},{"Start":"06:02.020 ","End":"06:06.070","Text":"Now, 7 over 3, well,"},{"Start":"06:06.070 ","End":"06:09.353","Text":"we know that\u0027s 35, so that\u0027s 35 right here,"},{"Start":"06:09.353 ","End":"06:12.785","Text":"and that equals to 1 divided by 35."},{"Start":"06:12.785 ","End":"06:14.850","Text":"When X equals 0,"},{"Start":"06:14.850 ","End":"06:19.355","Text":"the probability here in this cell will be 1 divided by 35."},{"Start":"06:19.355 ","End":"06:25.556","Text":"Now, similarly, we\u0027ll calculate the marginal probabilities for X equals 2 and X equals 3."},{"Start":"06:25.556 ","End":"06:33.515","Text":"That\u0027ll come out to 18 divided by 35 and 4 divided by 35."},{"Start":"06:33.515 ","End":"06:36.320","Text":"We have to make sure that"},{"Start":"06:36.320 ","End":"06:42.550","Text":"the sum of these probabilities have to add up to 1, and they do."},{"Start":"06:42.550 ","End":"06:46.550","Text":"What about the marginal probabilities of Y?"},{"Start":"06:46.550 ","End":"06:53.140","Text":"Well, let\u0027s take a look at the probability of Y equaling 0."},{"Start":"06:53.140 ","End":"06:55.665","Text":"That\u0027s this cell right here."},{"Start":"06:55.665 ","End":"06:58.305","Text":"Well, D, now,"},{"Start":"06:58.305 ","End":"07:02.235","Text":"is not 4, it\u0027s 1 because we\u0027re looking at Y."},{"Start":"07:02.235 ","End":"07:05.775","Text":"D here equals 1 over 0,"},{"Start":"07:05.775 ","End":"07:09.030","Text":"0 times n minus D. Well,"},{"Start":"07:09.030 ","End":"07:16.365","Text":"N is still 7 minus D. That\u0027s 7 minus 1, that\u0027s 6."},{"Start":"07:16.365 ","End":"07:21.855","Text":"Now, over n minus k. Well,"},{"Start":"07:21.855 ","End":"07:27.900","Text":"n minus k, n is 3 and k is 0."},{"Start":"07:27.900 ","End":"07:35.265","Text":"That\u0027ll be 3 divided by 35 again."},{"Start":"07:35.265 ","End":"07:42.490","Text":"N over n. Now that equals to 20 divided by 35."},{"Start":"07:42.490 ","End":"07:44.215","Text":"Let\u0027s just input that,"},{"Start":"07:44.215 ","End":"07:46.875","Text":"that\u0027ll be 20 divided by 35."},{"Start":"07:46.875 ","End":"07:50.730","Text":"Here, that\u0027ll be 15 divided by 35."},{"Start":"07:50.730 ","End":"07:56.875","Text":"Now we\u0027ve completed our marginal probabilities."},{"Start":"07:56.875 ","End":"08:00.530","Text":"Now let\u0027s take a look at the joint probabilities here."},{"Start":"08:00.530 ","End":"08:04.700","Text":"Well, let\u0027s look at the cells that are impossible,"},{"Start":"08:04.700 ","End":"08:07.400","Text":"that have a 0 value for the probability."},{"Start":"08:07.400 ","End":"08:13.905","Text":"Well, let\u0027s take a look at the cell where X equals 3,"},{"Start":"08:13.905 ","End":"08:15.535","Text":"and Y equals 1."},{"Start":"08:15.535 ","End":"08:17.915","Text":"Well, we have a sample size of 3."},{"Start":"08:17.915 ","End":"08:20.300","Text":"We\u0027ve chosen 3 bottles at random."},{"Start":"08:20.300 ","End":"08:23.485","Text":"Now, if all of the bottles,"},{"Start":"08:23.485 ","End":"08:27.295","Text":"our Coors bottles, that\u0027s what this 3 means."},{"Start":"08:27.295 ","End":"08:32.060","Text":"That means that there\u0027s no way that I can choose a Miller bottle as well."},{"Start":"08:32.060 ","End":"08:35.585","Text":"Because I can only choose 3 bottles."},{"Start":"08:35.585 ","End":"08:38.285","Text":"This here is 0."},{"Start":"08:38.285 ","End":"08:42.620","Text":"Now, what about when X equals 0 and Y equals 0?"},{"Start":"08:42.620 ","End":"08:46.835","Text":"Well, that means that I did not choose any Coors bottles,"},{"Start":"08:46.835 ","End":"08:50.375","Text":"and I didn\u0027t choose any Miller bottles."},{"Start":"08:50.375 ","End":"08:54.725","Text":"That means that I need to choose 3 bottles that are Budweiser,"},{"Start":"08:54.725 ","End":"08:58.340","Text":"but I only have 2 bottles in the population."},{"Start":"08:58.340 ","End":"09:00.950","Text":"So that\u0027s an impossibility as well,"},{"Start":"09:00.950 ","End":"09:03.985","Text":"and therefore that\u0027s 0 as well."},{"Start":"09:03.985 ","End":"09:07.610","Text":"Now, let\u0027s take a look at the remaining cells."},{"Start":"09:07.610 ","End":"09:13.520","Text":"Let\u0027s now take a look at the cell where X equals 2 and Y equals 1."},{"Start":"09:13.520 ","End":"09:15.080","Text":"That\u0027s this thing right here."},{"Start":"09:15.080 ","End":"09:21.080","Text":"The probability of X equals 2 and Y equals 1."},{"Start":"09:21.080 ","End":"09:28.435","Text":"Well, what I\u0027m doing is out of the 4 Coors bottle I\u0027m choosing 2,"},{"Start":"09:28.435 ","End":"09:31.725","Text":"and out of the Miller bottle,"},{"Start":"09:31.725 ","End":"09:35.045","Text":"I\u0027m choosing 1,"},{"Start":"09:35.045 ","End":"09:40.070","Text":"and I have a total number of combinations of 35."},{"Start":"09:40.070 ","End":"09:44.620","Text":"Now that equals to 6 divided by 35."},{"Start":"09:44.620 ","End":"09:48.590","Text":"Here, that\u0027ll be 6 divided by 35."},{"Start":"09:48.590 ","End":"09:52.070","Text":"Now, the minute I do that, well,"},{"Start":"09:52.070 ","End":"09:56.585","Text":"then I\u0027ve basically completed the table. Well, let\u0027s see why."},{"Start":"09:56.585 ","End":"10:02.435","Text":"Well, I know that the sum of these 2 cells have to add up to 4 divided by 35."},{"Start":"10:02.435 ","End":"10:05.890","Text":"So that\u0027s 4 right here divided by 35."},{"Start":"10:05.890 ","End":"10:11.510","Text":"Here, these 2 cells have to add up to 18 divided by 35."},{"Start":"10:11.510 ","End":"10:14.685","Text":"That means that that\u0027s 12 divided by 35."},{"Start":"10:14.685 ","End":"10:23.910","Text":"That means that here I have 4 divided by 35,"},{"Start":"10:23.910 ","End":"10:26.800","Text":"in order to make this equal to 20."},{"Start":"10:26.800 ","End":"10:31.520","Text":"The sum of these probabilities have to be equal to 20 divided by 35."},{"Start":"10:31.520 ","End":"10:37.862","Text":"Because this is 4 divided by 35 and this here must be 8 divided by 35,"},{"Start":"10:37.862 ","End":"10:41.210","Text":"and in order to make this 15 divided by 35,"},{"Start":"10:41.210 ","End":"10:45.080","Text":"well, here, that has to be 1 divided by 35."},{"Start":"10:45.080 ","End":"10:51.120","Text":"There we go. I\u0027ve completed the joint probability distribution of X and Y."}],"ID":13223},{"Watched":false,"Name":"Exercise 2 Parts b-c","Duration":"5m 57s","ChapterTopicVideoID":12745,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12745.jpeg","UploadDate":"2019-02-11T14:27:04.2430000","DurationForVideoObject":"PT5M57S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.500","Text":"In this section, we\u0027re asked to calculate the expectation and variance of X and Y."},{"Start":"00:04.500 ","End":"00:08.610","Text":"Now, I can do that using the joint probability table"},{"Start":"00:08.610 ","End":"00:12.600","Text":"and using the regular formulas for expectation,"},{"Start":"00:12.600 ","End":"00:16.545","Text":"that means I\u0027m summing up X times the probability of X,"},{"Start":"00:16.545 ","End":"00:19.560","Text":"or if I\u0027m doing the expectation of Y,"},{"Start":"00:19.560 ","End":"00:22.770","Text":"that\u0027s Y times expectation of Y,"},{"Start":"00:22.770 ","End":"00:27.060","Text":"and the same thing for the variance that the sum of X squared times"},{"Start":"00:27.060 ","End":"00:32.325","Text":"the probability minus the expectation squared of X and likewise for Y."},{"Start":"00:32.325 ","End":"00:38.370","Text":"But since I know that X and Y have a hypergeometric distribution,"},{"Start":"00:38.370 ","End":"00:39.765","Text":"well, I can use that."},{"Start":"00:39.765 ","End":"00:41.955","Text":"I know that if I have"},{"Start":"00:41.955 ","End":"00:51.260","Text":"a random variable W that has a hypergeometric distribution with parameters N,"},{"Start":"00:51.260 ","End":"00:55.650","Text":"D, and n then the expectation of W, well,"},{"Start":"00:55.650 ","End":"01:05.674","Text":"that equals to small n times D divided by N. The variance of W,"},{"Start":"01:05.674 ","End":"01:13.560","Text":"well, that equals to n times D divided by N times 1"},{"Start":"01:13.560 ","End":"01:22.980","Text":"minus D divided by N times N minus small n divided by N minus 1."},{"Start":"01:22.980 ","End":"01:27.950","Text":"These are the expressions for calculating the expectation"},{"Start":"01:27.950 ","End":"01:33.460","Text":"and variance of a random variable that has a hypergeometric distribution."},{"Start":"01:33.460 ","End":"01:39.380","Text":"If that\u0027s the case, let\u0027s just plug in the numbers for X and Y."},{"Start":"01:39.380 ","End":"01:41.660","Text":"Now, the expectation of X, well,"},{"Start":"01:41.660 ","End":"01:44.760","Text":"that equals to the following; well,"},{"Start":"01:44.760 ","End":"01:49.965","Text":"that\u0027s n times D divided by N. Well,"},{"Start":"01:49.965 ","End":"01:52.250","Text":"n equals 3,"},{"Start":"01:52.250 ","End":"01:56.610","Text":"D for the random variable X,"},{"Start":"01:56.610 ","End":"02:00.810","Text":"that\u0027s 4 divided by N, that\u0027s 7."},{"Start":"02:00.810 ","End":"02:08.670","Text":"Now, that equals to 12 divided by 7."},{"Start":"02:08.670 ","End":"02:11.935","Text":"Now, what about the expectation of Y?"},{"Start":"02:11.935 ","End":"02:16.480","Text":"Well, again, n equals 3,"},{"Start":"02:16.970 ","End":"02:21.030","Text":"D equals 1 now divided by 7,"},{"Start":"02:21.030 ","End":"02:25.935","Text":"that\u0027s N. That means that equals to 3 divided by 7."},{"Start":"02:25.935 ","End":"02:28.710","Text":"What about the variance of X?"},{"Start":"02:28.710 ","End":"02:33.675","Text":"Well, the variance of X is n times,"},{"Start":"02:33.675 ","End":"02:37.690","Text":"n is 3 times D divided by N. Well,"},{"Start":"02:37.690 ","End":"02:43.990","Text":"that\u0027s 4 divided by 7 times 1 minus 4 divided by 7,"},{"Start":"02:43.990 ","End":"02:49.500","Text":"times 7 minus 3 divided by 7 minus 1."},{"Start":"02:49.500 ","End":"02:54.195","Text":"Now, that comes out to 24 divided by 49,"},{"Start":"02:54.195 ","End":"02:56.520","Text":"and the variance of Y."},{"Start":"02:56.520 ","End":"03:04.130","Text":"Well, again, n equals 3 times D divided by N. Now D equals 1,"},{"Start":"03:04.130 ","End":"03:09.290","Text":"so that\u0027ll be 1 divided by 7 times 1 minus 1 divided by"},{"Start":"03:09.290 ","End":"03:16.160","Text":"7 times 7 minus 3 divided by 7 minus 1."},{"Start":"03:16.160 ","End":"03:21.600","Text":"That comes out to 12 divided by 49."},{"Start":"03:21.600 ","End":"03:25.610","Text":"Here we go, these are the expectation of X and Y,"},{"Start":"03:25.610 ","End":"03:29.585","Text":"and these are the variances of X and Y."},{"Start":"03:29.585 ","End":"03:33.965","Text":"In this section we\u0027re asked to find the covariance of X and Y."},{"Start":"03:33.965 ","End":"03:39.815","Text":"The covariance of X and Y that equals to the expectation of"},{"Start":"03:39.815 ","End":"03:46.580","Text":"X times Y minus the expectation of X times the expectation of Y."},{"Start":"03:46.580 ","End":"03:50.465","Text":"Now, we know what the expectation of X and Y are."},{"Start":"03:50.465 ","End":"03:52.955","Text":"We\u0027ve calculated them in the last section."},{"Start":"03:52.955 ","End":"03:56.675","Text":"The expectation of X that\u0027s 12 divided by 7,"},{"Start":"03:56.675 ","End":"03:59.195","Text":"and the expectation of Y,"},{"Start":"03:59.195 ","End":"04:02.825","Text":"that equals to 3 divided by 7."},{"Start":"04:02.825 ","End":"04:06.710","Text":"Now, what\u0027s missing? Well, that\u0027s the expectation of X times Y."},{"Start":"04:06.710 ","End":"04:08.525","Text":"Let\u0027s calculate that."},{"Start":"04:08.525 ","End":"04:11.995","Text":"The expectation of X times Y,"},{"Start":"04:11.995 ","End":"04:13.350","Text":"well that equals to what?"},{"Start":"04:13.350 ","End":"04:17.240","Text":"Every time that we have a 0,"},{"Start":"04:17.240 ","End":"04:23.360","Text":"either in the X or Y or in the joint probability where I can get rid of that."},{"Start":"04:23.360 ","End":"04:26.770","Text":"That means that I can get rid of this column right here,"},{"Start":"04:26.770 ","End":"04:29.915","Text":"and I can get rid of this row right here,"},{"Start":"04:29.915 ","End":"04:32.735","Text":"and I can get rid of this cell right here."},{"Start":"04:32.735 ","End":"04:37.190","Text":"That means all I\u0027m left with is the following."},{"Start":"04:37.190 ","End":"04:40.280","Text":"That\u0027s 1 times 1 times that probability,"},{"Start":"04:40.280 ","End":"04:47.030","Text":"so it\u0027s 1 times 1 times 8 divided by 35 plus 2 times 1"},{"Start":"04:47.030 ","End":"04:56.415","Text":"times 6 divided by 35 and that equals to 20 divided by 35."},{"Start":"04:56.415 ","End":"05:01.385","Text":"The covariance of X and Y, well,"},{"Start":"05:01.385 ","End":"05:05.315","Text":"that equals to the expectation of X times Y,"},{"Start":"05:05.315 ","End":"05:13.240","Text":"that equals to 20 divided by 35 minus the multiplication of the expectations of X and Y."},{"Start":"05:13.240 ","End":"05:19.910","Text":"That\u0027s 12 divided by 7 times 3 divided by 7."},{"Start":"05:19.910 ","End":"05:27.975","Text":"Now, after calculation, this comes out to minus 8 divided by 49."},{"Start":"05:27.975 ","End":"05:31.710","Text":"Now, this then is the covariance."},{"Start":"05:31.710 ","End":"05:32.900","Text":"Now, why is it negative?"},{"Start":"05:32.900 ","End":"05:39.995","Text":"Well, it\u0027s negative because there\u0027s an inverse relationship between X and Y."},{"Start":"05:39.995 ","End":"05:43.040","Text":"The more Coors bottle that I have,"},{"Start":"05:43.040 ","End":"05:45.915","Text":"the less Miller\u0027s bottle that I\u0027ll have,"},{"Start":"05:45.915 ","End":"05:48.770","Text":"or if I choose a Miller bottle,"},{"Start":"05:48.770 ","End":"05:53.105","Text":"they\u0027ll take away from the number of Coors bottles that are chosen."},{"Start":"05:53.105 ","End":"05:57.780","Text":"This then is the covariance of X and Y."}],"ID":13224},{"Watched":false,"Name":"Exercise 2 parts d-e","Duration":"7m 11s","ChapterTopicVideoID":12746,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12746.jpeg","UploadDate":"2019-02-11T14:27:27.2300000","DurationForVideoObject":"PT7M11S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.545","Text":"In this section, we define W as the number of Budweiser bottles taken by Jane."},{"Start":"00:04.545 ","End":"00:09.420","Text":"We\u0027re asked to express W in terms of X and Y and calculate the expectation and"},{"Start":"00:09.420 ","End":"00:14.520","Text":"variance of W based on the results from only Sections B and C above."},{"Start":"00:14.520 ","End":"00:17.070","Text":"Let\u0027s just review what we have."},{"Start":"00:17.070 ","End":"00:19.830","Text":"We have our sample size, that\u0027s 3."},{"Start":"00:19.830 ","End":"00:25.620","Text":"X is the number of Coors bottles that we have in the sample."},{"Start":"00:25.620 ","End":"00:30.780","Text":"Y is the number of Miller bottles that we have in the sample."},{"Start":"00:30.780 ","End":"00:36.500","Text":"W then would be the number of Budweiser bottles that we have in a sample."},{"Start":"00:36.500 ","End":"00:39.690","Text":"But the total number of bottles in the sample has to equal 3."},{"Start":"00:39.690 ","End":"00:42.710","Text":"The relationship between all these parameters is this,"},{"Start":"00:42.710 ","End":"00:52.295","Text":"that\u0027s 3 and that has to equal to X plus Y plus W. That means that W,"},{"Start":"00:52.295 ","End":"00:57.620","Text":"that equals to 3 minus X plus Y."},{"Start":"00:57.620 ","End":"01:01.249","Text":"Now, let\u0027s just write this out a little bit differently."},{"Start":"01:01.249 ","End":"01:09.080","Text":"That would equal to minus 1 times X plus Y plus 3."},{"Start":"01:09.080 ","End":"01:11.090","Text":"Now, why do I want to write it like this?"},{"Start":"01:11.090 ","End":"01:14.240","Text":"Because this has the form of a linear transformation."},{"Start":"01:14.240 ","End":"01:16.835","Text":"If we recall it,"},{"Start":"01:16.835 ","End":"01:24.285","Text":"let\u0027s say we have a random variable T. If we have a_T plus b,"},{"Start":"01:24.285 ","End":"01:26.775","Text":"that\u0027s a linear transformation."},{"Start":"01:26.775 ","End":"01:28.940","Text":"This is what we want to have."},{"Start":"01:28.940 ","End":"01:32.045","Text":"We have here a that would be equal to minus 1,"},{"Start":"01:32.045 ","End":"01:34.460","Text":"b that\u0027ll be equal to 3."},{"Start":"01:34.460 ","End":"01:36.110","Text":"Instead of X plus Y,"},{"Start":"01:36.110 ","End":"01:41.550","Text":"we wrote here T. This is just a quick reminder here."},{"Start":"01:41.550 ","End":"01:46.475","Text":"When we\u0027re looking at the expectation of W, then, well,"},{"Start":"01:46.475 ","End":"01:52.100","Text":"we\u0027re looking at the expectation of a linear transformation on X and Y."},{"Start":"01:52.100 ","End":"01:58.340","Text":"That\u0027ll be minus 1 times X plus Y plus 3."},{"Start":"01:58.340 ","End":"02:03.305","Text":"Now, what is the expectation of a linear transformation?"},{"Start":"02:03.305 ","End":"02:05.525","Text":"Well, if we recall again,"},{"Start":"02:05.525 ","End":"02:09.890","Text":"the expectation of a_T plus b,"},{"Start":"02:09.890 ","End":"02:15.480","Text":"well that equals to a times the expectation of T plus b."},{"Start":"02:15.480 ","End":"02:18.035","Text":"Let\u0027s just get back here."},{"Start":"02:18.035 ","End":"02:23.780","Text":"That equals to minus 1 times"},{"Start":"02:23.780 ","End":"02:29.930","Text":"the expectation of X plus Y plus 3."},{"Start":"02:29.930 ","End":"02:33.395","Text":"Now, what\u0027s the expectation of,"},{"Start":"02:33.395 ","End":"02:36.110","Text":"let\u0027s say S plus T, well,"},{"Start":"02:36.110 ","End":"02:41.465","Text":"that equals to the expectation of S plus the expectation of T,"},{"Start":"02:41.465 ","End":"02:43.565","Text":"where S and T are random variables."},{"Start":"02:43.565 ","End":"02:46.025","Text":"Again, let\u0027s just apply that here."},{"Start":"02:46.025 ","End":"02:48.290","Text":"That\u0027ll be equal to the following."},{"Start":"02:48.290 ","End":"02:52.040","Text":"Now let\u0027s just multiply minus 1 right in there."},{"Start":"02:52.040 ","End":"03:00.575","Text":"That\u0027ll be equal to minus the expectation of X minus the expectation of Y plus 3."},{"Start":"03:00.575 ","End":"03:04.580","Text":"Now, what are the expectation of X and expectation of Y?"},{"Start":"03:04.580 ","End":"03:07.835","Text":"We\u0027ve calculated that in the previous sections,"},{"Start":"03:07.835 ","End":"03:15.155","Text":"that\u0027ll be minus 12 divided by 7."},{"Start":"03:15.155 ","End":"03:20.580","Text":"That\u0027s expectation of X minus 3 divided by 7."},{"Start":"03:20.580 ","End":"03:22.950","Text":"That\u0027s the expectation of Y plus 3."},{"Start":"03:22.950 ","End":"03:27.465","Text":"Now, that equals to 6 divided by 7."},{"Start":"03:27.465 ","End":"03:37.010","Text":"That then is the expectation of W. Let\u0027s take a look now at the variance of W. Well,"},{"Start":"03:37.010 ","End":"03:46.010","Text":"that\u0027ll be equal to the variance of minus 1 times X plus Y plus 3."},{"Start":"03:46.010 ","End":"03:52.910","Text":"Now, let\u0027s recall the variance of a_T plus b."},{"Start":"03:52.910 ","End":"04:00.110","Text":"Well, that equals to a squared times the variance of T. This then,"},{"Start":"04:00.110 ","End":"04:08.335","Text":"would be equal to minus 1 squared times the variance of X plus Y."},{"Start":"04:08.335 ","End":"04:13.810","Text":"Now, what\u0027s the variance of a sum of random variables?"},{"Start":"04:13.810 ","End":"04:18.760","Text":"Well, that equals to the variance of the first variable plus"},{"Start":"04:18.760 ","End":"04:25.730","Text":"the variance of the second variable plus the covariance of the 2 variables."},{"Start":"04:25.730 ","End":"04:28.765","Text":"2 times the covariance of the 2 variables."},{"Start":"04:28.765 ","End":"04:35.031","Text":"Well, then that means that the variance of W will be equal to the following,"},{"Start":"04:35.031 ","End":"04:38.310","Text":"well, minus 1 squared is 1 so we\u0027ll get rid of that."},{"Start":"04:38.310 ","End":"04:42.250","Text":"That\u0027ll be the variance of X plus the variance of"},{"Start":"04:42.250 ","End":"04:49.020","Text":"Y plus 2 times the covariance of X and Y."},{"Start":"04:49.020 ","End":"04:51.755","Text":"Now, what does that equal to?"},{"Start":"04:51.755 ","End":"04:58.475","Text":"Well, the variance of X what was calculated at 24 divided by 49."},{"Start":"04:58.475 ","End":"05:00.320","Text":"The variance of Y, well,"},{"Start":"05:00.320 ","End":"05:07.190","Text":"that was 12 divided by 49 plus 2 times the covariance of X and Y."},{"Start":"05:07.190 ","End":"05:11.915","Text":"Well, that was calculated at minus 8 divided by 49."},{"Start":"05:11.915 ","End":"05:16.620","Text":"That equals to 20 divided by 49."},{"Start":"05:16.620 ","End":"05:21.350","Text":"This then is the variance of W. Here we\u0027ve"},{"Start":"05:21.350 ","End":"05:25.880","Text":"calculated the expectation of W. In this section,"},{"Start":"05:25.880 ","End":"05:29.510","Text":"we\u0027re asked to calculate the correlation coefficient between the number of"},{"Start":"05:29.510 ","End":"05:33.760","Text":"Coors bottles and the number of non-Coors bottles that were taken by Jane."},{"Start":"05:33.760 ","End":"05:39.500","Text":"Let\u0027s just review the sample size, n that equals to 3."},{"Start":"05:39.500 ","End":"05:44.285","Text":"X is the number of Coors bottles in the sample."},{"Start":"05:44.285 ","End":"05:51.775","Text":"Let\u0027s define R now as the number of non-Coors bottles in the sample."},{"Start":"05:51.775 ","End":"05:57.995","Text":"Now, what\u0027s the relationship between the sample size X and R?"},{"Start":"05:57.995 ","End":"06:01.024","Text":"Well, 3, the sample size,"},{"Start":"06:01.024 ","End":"06:06.305","Text":"the number of bonds in the sample that has to be equal to X plus R,"},{"Start":"06:06.305 ","End":"06:11.240","Text":"or R has to be equal to 3 minus X."},{"Start":"06:11.240 ","End":"06:18.530","Text":"Now here we have an equation that relates X to R. Now, what does that mean?"},{"Start":"06:18.530 ","End":"06:21.785","Text":"That means that if I know the value of X,"},{"Start":"06:21.785 ","End":"06:26.360","Text":"then I definitively know the value of R and vice versa."},{"Start":"06:26.360 ","End":"06:30.710","Text":"Now, what does that mean with respect to the correlation coefficient?"},{"Start":"06:30.710 ","End":"06:35.240","Text":"Well, the correlation coefficient between X and R or between"},{"Start":"06:35.240 ","End":"06:42.530","Text":"any 2 variables measures the linear dependency of these 2 variables."},{"Start":"06:42.530 ","End":"06:46.105","Text":"Well here, the linear dependency is absolute."},{"Start":"06:46.105 ","End":"06:47.655","Text":"That has to be 1."},{"Start":"06:47.655 ","End":"06:51.169","Text":"What about the direction of this linear dependency?"},{"Start":"06:51.169 ","End":"06:52.640","Text":"Well, that\u0027s minus 1."},{"Start":"06:52.640 ","End":"06:54.515","Text":"Now, why is that?"},{"Start":"06:54.515 ","End":"06:57.650","Text":"Well, the more Coors bottles that I have,"},{"Start":"06:57.650 ","End":"07:02.585","Text":"well then the less non-Coors bottles that I have in the sample and vice versa."},{"Start":"07:02.585 ","End":"07:08.176","Text":"That means that the correlation coefficient between X and R is minus 1,"},{"Start":"07:08.176 ","End":"07:11.520","Text":"we don\u0027t have to do any calculations."}],"ID":13225},{"Watched":false,"Name":"Exercise 3 Part a","Duration":"15m 49s","ChapterTopicVideoID":12747,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12747.jpeg","UploadDate":"2019-02-11T14:28:31.2670000","DurationForVideoObject":"PT15M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.045","Text":"In this question, there are 6 pairs of shoes in a drawer."},{"Start":"00:03.045 ","End":"00:06.630","Text":"Jack removed 4 individual shoes at random from the drawer."},{"Start":"00:06.630 ","End":"00:08.565","Text":"The following variables are defined."},{"Start":"00:08.565 ","End":"00:12.360","Text":"W is defined as the number of pairs of shoes that Jack"},{"Start":"00:12.360 ","End":"00:16.782","Text":"removed and R is defined as the number of left choose that Jack removed."},{"Start":"00:16.782 ","End":"00:22.860","Text":"We\u0027re asked to find the joint distribution of W and R. The first thing that"},{"Start":"00:22.860 ","End":"00:29.115","Text":"we want to do is to try to identify the relevant values for R and W. Well,"},{"Start":"00:29.115 ","End":"00:34.285","Text":"let\u0027s first start with R. R is the number of left shoes that Jack removed."},{"Start":"00:34.285 ","End":"00:39.185","Text":"Out of the 4 individual shoes that he removed,"},{"Start":"00:39.185 ","End":"00:42.960","Text":"he could have chosen 0 left shoes,"},{"Start":"00:42.960 ","End":"00:47.460","Text":"1, 2, 3, or 4 left shoes."},{"Start":"00:47.460 ","End":"00:49.890","Text":"Now, what about W?"},{"Start":"00:49.890 ","End":"00:53.869","Text":"W talks about the pairs of shoes that Jack removed."},{"Start":"00:53.869 ","End":"00:56.640","Text":"Out of the 4 individual shoes,"},{"Start":"00:56.640 ","End":"00:59.445","Text":"he could construct 0 pairs,"},{"Start":"00:59.445 ","End":"01:02.040","Text":"1 pair, or 2 pairs."},{"Start":"01:02.040 ","End":"01:03.495","Text":"Now, that\u0027s all."},{"Start":"01:03.495 ","End":"01:07.050","Text":"He can\u0027t have more than 2 pairs of shoes."},{"Start":"01:07.050 ","End":"01:10.440","Text":"2 pairs of shoes that 4 individuals shoes,"},{"Start":"01:10.440 ","End":"01:13.890","Text":"and he didn\u0027t remove more than 4 individuals shoes."},{"Start":"01:13.890 ","End":"01:18.485","Text":"Now, having identified the relevant values for R and W,"},{"Start":"01:18.485 ","End":"01:27.315","Text":"let\u0027s now construct our joint probability distribution of W and R. Here\u0027s our table,"},{"Start":"01:27.315 ","End":"01:32.280","Text":"and we have here W and the values of W in blue 0, 1, and 2."},{"Start":"01:32.280 ","End":"01:35.705","Text":"The marginal probabilities of W will be here."},{"Start":"01:35.705 ","End":"01:38.350","Text":"Now, here\u0027s R,"},{"Start":"01:38.350 ","End":"01:44.390","Text":"and these are the valid values of R and the marginal probabilities of R would be here."},{"Start":"01:44.390 ","End":"01:49.070","Text":"Now, here, the white cells would be"},{"Start":"01:49.070 ","End":"01:56.615","Text":"the joint probabilities of R and W. The first thing that we want to do,"},{"Start":"01:56.615 ","End":"02:00.145","Text":"is we want to eliminate all the impossible situations."},{"Start":"02:00.145 ","End":"02:02.480","Text":"Let\u0027s do that."},{"Start":"02:02.480 ","End":"02:06.715","Text":"Let\u0027s take a look at where W equals to 2."},{"Start":"02:06.715 ","End":"02:10.919","Text":"That means that in the sample we have 2 pairs."},{"Start":"02:10.919 ","End":"02:12.340","Text":"Now, if we have 2 pairs,"},{"Start":"02:12.340 ","End":"02:16.210","Text":"can we have 0 number of left shoes?"},{"Start":"02:16.210 ","End":"02:18.940","Text":"Well, of course not, that\u0027s an impossibility."},{"Start":"02:18.940 ","End":"02:21.135","Text":"Now, if we have 2 pairs,"},{"Start":"02:21.135 ","End":"02:23.605","Text":"can we have only 1 shoe?"},{"Start":"02:23.605 ","End":"02:24.875","Text":"That\u0027s a left shoe,"},{"Start":"02:24.875 ","End":"02:27.460","Text":"well again, that\u0027s an impossibility."},{"Start":"02:27.460 ","End":"02:30.820","Text":"What about if we have 2 pairs in the sample,"},{"Start":"02:30.820 ","End":"02:33.640","Text":"but we have 4 left shoes."},{"Start":"02:33.640 ","End":"02:37.520","Text":"Well, we\u0027ve sampled only 4 shoes."},{"Start":"02:38.030 ","End":"02:41.520","Text":"If all 4 of them are left shoes,"},{"Start":"02:41.520 ","End":"02:43.100","Text":"then we can\u0027t have 2 pairs,"},{"Start":"02:43.100 ","End":"02:45.050","Text":"so that\u0027s an impossibility."},{"Start":"02:45.050 ","End":"02:49.290","Text":"Now what about where we have 2 pairs of shoes,"},{"Start":"02:49.290 ","End":"02:53.700","Text":"but there are 3 left shoes in the sample,"},{"Start":"02:53.700 ","End":"02:56.085","Text":"and again that\u0027s an impossibility."},{"Start":"02:56.085 ","End":"03:01.490","Text":"Now, what about where we have 1 pair of shoes in the sample,"},{"Start":"03:01.490 ","End":"03:04.100","Text":"but we have no left shoes."},{"Start":"03:04.100 ","End":"03:06.370","Text":"Well again, that\u0027s an impossibility."},{"Start":"03:06.370 ","End":"03:09.320","Text":"Let\u0027s take a look at this cell right here,"},{"Start":"03:09.320 ","End":"03:13.145","Text":"where we have 1 pair of shoes in the sample,"},{"Start":"03:13.145 ","End":"03:16.380","Text":"but all 4 shoes are left shoes."},{"Start":"03:16.380 ","End":"03:20.320","Text":"Again, that\u0027s an impossibility."},{"Start":"03:20.420 ","End":"03:25.520","Text":"Now, what about the rest of the cells here?"},{"Start":"03:25.520 ","End":"03:29.615","Text":"Well, all the rest of the cells are valid cells."},{"Start":"03:29.615 ","End":"03:34.340","Text":"Now let\u0027s take a look at R and specifically"},{"Start":"03:34.340 ","End":"03:39.980","Text":"the marginal probabilities of R basically the distribution of R. Now,"},{"Start":"03:39.980 ","End":"03:46.970","Text":"we can see that R has a hypergeometric distribution."},{"Start":"03:46.970 ","End":"03:48.335","Text":"Now, why is that?"},{"Start":"03:48.335 ","End":"03:53.990","Text":"Let\u0027s take a look at the parameters of a hypergeometric distribution."},{"Start":"03:53.990 ","End":"03:59.265","Text":"Big N, that\u0027s the number of shoes in the drawer that equals to 12."},{"Start":"03:59.265 ","End":"04:02.895","Text":"Now, R talks about left shoes,"},{"Start":"04:02.895 ","End":"04:09.250","Text":"so this would be the subpopulation that is needed in the hypergeometric distribution."},{"Start":"04:09.250 ","End":"04:13.255","Text":"Subpopulation, that\u0027s the population of left shoes."},{"Start":"04:13.255 ","End":"04:16.731","Text":"That\u0027s D, and what\u0027s the size of our subpopulation,"},{"Start":"04:16.731 ","End":"04:18.570","Text":"that equals to 6."},{"Start":"04:18.570 ","End":"04:20.605","Text":"What\u0027s our sample size?"},{"Start":"04:20.605 ","End":"04:22.780","Text":"How many shoes did Jack sample?"},{"Start":"04:22.780 ","End":"04:24.910","Text":"Well, he sampled 4."},{"Start":"04:24.910 ","End":"04:34.630","Text":"Here we have a hypergeometric distribution for R. Now, if that\u0027s the case,"},{"Start":"04:34.630 ","End":"04:38.625","Text":"then the probability of R equaling some value k,"},{"Start":"04:38.625 ","End":"04:46.180","Text":"well that equals to D over k times big N minus D over small n"},{"Start":"04:46.180 ","End":"04:54.085","Text":"minus k/big N over small n. That\u0027s the probability here."},{"Start":"04:54.085 ","End":"04:56.005","Text":"Now, if that\u0027s the case,"},{"Start":"04:56.005 ","End":"05:03.355","Text":"let\u0027s take a look at the probability of R being equal to 1, for example."},{"Start":"05:03.355 ","End":"05:07.690","Text":"Let\u0027s plug in the numbers D is 6 and k is 1,"},{"Start":"05:07.690 ","End":"05:12.030","Text":"so that\u0027ll be 6 over 1 times N is 12,"},{"Start":"05:12.030 ","End":"05:15.585","Text":"D is 6, so that\u0027ll be 12 minus 6, that\u0027s 6."},{"Start":"05:15.585 ","End":"05:19.030","Text":"Small n, well that\u0027s 4 minus 1,"},{"Start":"05:19.030 ","End":"05:21.690","Text":"the k, so that\u0027ll be 4 minus 1,"},{"Start":"05:21.690 ","End":"05:26.580","Text":"that\u0027ll be 3./12 big N,"},{"Start":"05:26.580 ","End":"05:29.310","Text":"over small n, 12 over 4."},{"Start":"05:29.310 ","End":"05:38.945","Text":"Now that equals to 120/495."},{"Start":"05:38.945 ","End":"05:41.435","Text":"Now, let\u0027s just make a note of this,"},{"Start":"05:41.435 ","End":"05:46.335","Text":"that our sample size or the total amount of"},{"Start":"05:46.335 ","End":"05:55.140","Text":"permutations of taking 4 shoes out of a drawer of 12 shoes, well that\u0027s 495."},{"Start":"05:55.140 ","End":"05:59.235","Text":"That\u0027ll be our sample space."},{"Start":"05:59.235 ","End":"06:02.330","Text":"The probability of R equaling 1,"},{"Start":"06:02.330 ","End":"06:05.650","Text":"well that\u0027s 120/495 as we said,"},{"Start":"06:05.650 ","End":"06:13.915","Text":"so we\u0027ll put in this probability right here, that\u0027s 120/495."},{"Start":"06:13.915 ","End":"06:21.259","Text":"Let\u0027s take another example where we have the probability of R equaling 0."},{"Start":"06:21.259 ","End":"06:25.775","Text":"That\u0027ll be 6/0 times,"},{"Start":"06:25.775 ","End":"06:32.625","Text":"6/4 divided now by 495."},{"Start":"06:32.625 ","End":"06:34.740","Text":"We said that that was our sample space."},{"Start":"06:34.740 ","End":"06:42.510","Text":"Now that equals to 15/495."},{"Start":"06:42.510 ","End":"06:48.555","Text":"We\u0027ll put that in here, that\u0027s 15/495."},{"Start":"06:48.555 ","End":"06:54.905","Text":"We can continue calculating the marginal probabilities of R,"},{"Start":"06:54.905 ","End":"07:01.810","Text":"but let\u0027s take a closer look now at R. We see here that R is symmetrical."},{"Start":"07:01.810 ","End":"07:07.805","Text":"Why is that? The probability of choosing 0 left shoes,"},{"Start":"07:07.805 ","End":"07:14.560","Text":"well that means that that\u0027s the same probability of choosing 4 right shoes."},{"Start":"07:14.560 ","End":"07:23.870","Text":"The probability of choosing 4 left shoes is the probability of choosing 0 right shoes."},{"Start":"07:23.870 ","End":"07:32.945","Text":"We have here a symmetrical distribution that means that the probability of R equaling 0,"},{"Start":"07:32.945 ","End":"07:36.170","Text":"that equals to the probability of R equaling 4."},{"Start":"07:36.170 ","End":"07:38.020","Text":"Let\u0027s just write this down here."},{"Start":"07:38.020 ","End":"07:43.160","Text":"That\u0027ll be 15/495,"},{"Start":"07:43.160 ","End":"07:45.855","Text":"and here as well."},{"Start":"07:45.855 ","End":"07:48.770","Text":"When the probability of R equaling 1,"},{"Start":"07:48.770 ","End":"07:52.915","Text":"the probability of choosing only 1 left shoe."},{"Start":"07:52.915 ","End":"07:54.839","Text":"That means that you\u0027re choosing"},{"Start":"07:54.839 ","End":"07:59.570","Text":"3 right shoes and the probability of choosing 3 left shoes,"},{"Start":"07:59.570 ","End":"08:03.215","Text":"that\u0027s the same as the probability of choosing 1 right shoe."},{"Start":"08:03.215 ","End":"08:10.780","Text":"Again, here that\u0027ll be 120/495."},{"Start":"08:11.060 ","End":"08:15.190","Text":"We can calculate now this probability right here,"},{"Start":"08:15.190 ","End":"08:19.220","Text":"because the sum of all the probabilities have to add up to 1."},{"Start":"08:19.220 ","End":"08:25.145","Text":"This here is 225/495."},{"Start":"08:25.145 ","End":"08:30.775","Text":"Now, we know then that the sum of"},{"Start":"08:30.775 ","End":"08:38.305","Text":"the probabilities in the row have to add up to the marginal probability of that row."},{"Start":"08:38.305 ","End":"08:43.465","Text":"Here we have 15/495,"},{"Start":"08:43.465 ","End":"08:44.830","Text":"and the same thing here."},{"Start":"08:44.830 ","End":"08:52.630","Text":"We have 15/495."},{"Start":"08:52.630 ","End":"08:55.150","Text":"Let\u0027s now take a look at the joint probabilities here."},{"Start":"08:55.150 ","End":"09:00.010","Text":"Let\u0027s take a look at this cell where W equals 0 and R equals 1."},{"Start":"09:00.010 ","End":"09:04.720","Text":"We\u0027re looking at the probability of W equaling 0 and R equaling 1."},{"Start":"09:04.720 ","End":"09:09.295","Text":"Now, what does that mean? That means that we have no pairs of shoes in the sample,"},{"Start":"09:09.295 ","End":"09:12.700","Text":"and we have only 1 left on the sample."},{"Start":"09:12.700 ","End":"09:18.410","Text":"That means that we\u0027re dealing with 4 different pairs of shoes."},{"Start":"09:18.480 ","End":"09:21.085","Text":"What\u0027s this probability then?"},{"Start":"09:21.085 ","End":"09:24.970","Text":"Well, out of the 6 possible pairs of shoes in the drawer,"},{"Start":"09:24.970 ","End":"09:27.280","Text":"we have to choose 4 of them."},{"Start":"09:27.280 ","End":"09:29.320","Text":"Out of the 4 that were chosen,"},{"Start":"09:29.320 ","End":"09:32.920","Text":"we have to choose 1 pair that\u0027ll give me the left shoe."},{"Start":"09:32.920 ","End":"09:35.035","Text":"Out of the 3 remaining pairs,"},{"Start":"09:35.035 ","End":"09:39.430","Text":"I need to choose 3 of them to give me the 3 right shoes,"},{"Start":"09:39.430 ","End":"09:41.109","Text":"the 3 remaining right shoes."},{"Start":"09:41.109 ","End":"09:44.125","Text":"Now, we divide that by 495,"},{"Start":"09:44.125 ","End":"09:45.580","Text":"that\u0027s my sample size."},{"Start":"09:45.580 ","End":"09:49.150","Text":"That equals to 60/495."},{"Start":"09:49.150 ","End":"09:50.425","Text":"Let\u0027s put that here."},{"Start":"09:50.425 ","End":"09:53.530","Text":"That\u0027s 60/495."},{"Start":"09:53.530 ","End":"10:01.060","Text":"We can complete this probability right here, to be 60/495."},{"Start":"10:01.060 ","End":"10:06.040","Text":"Why is that? Well, 60 plus 60 gives me 120."},{"Start":"10:06.040 ","End":"10:09.070","Text":"We know that the joint probabilities in"},{"Start":"10:09.070 ","End":"10:14.000","Text":"a specific row have to add up to the marginal probability."},{"Start":"10:14.250 ","End":"10:19.555","Text":"Let\u0027s take a look at another probability"},{"Start":"10:19.555 ","End":"10:26.095","Text":"where W equals 0 and R equals to 2."},{"Start":"10:26.095 ","End":"10:32.155","Text":"Well, again, we have no pairs of shoes and we have 2 left shoes."},{"Start":"10:32.155 ","End":"10:35.290","Text":"Out of the 6 possible pairs of shoes,"},{"Start":"10:35.290 ","End":"10:37.735","Text":"we have to choose 4 pairs."},{"Start":"10:37.735 ","End":"10:40.450","Text":"Where out of the 4, we need 2 pair to give me"},{"Start":"10:40.450 ","End":"10:44.470","Text":"the 2 left shoes and out of the 2 remaining pairs of shoes,"},{"Start":"10:44.470 ","End":"10:50.850","Text":"I need to choose them to give me the 2 right shoes,"},{"Start":"10:50.850 ","End":"10:58.330","Text":"divided by 495 and that gives me 90/495."},{"Start":"10:58.330 ","End":"11:01.225","Text":"Where W equals 0 and R equals 2,"},{"Start":"11:01.225 ","End":"11:06.670","Text":"that\u0027s right here, that\u0027s 90/495."},{"Start":"11:06.670 ","End":"11:11.200","Text":"Let\u0027s now take a look at this cell right here."},{"Start":"11:11.200 ","End":"11:19.180","Text":"That\u0027s the probability of W equaling 1 and R equaling 2."},{"Start":"11:19.180 ","End":"11:24.445","Text":"Well, here we have 1 pair of shoes in our sample."},{"Start":"11:24.445 ","End":"11:27.220","Text":"That means that out of the 2 left shoes,"},{"Start":"11:27.220 ","End":"11:29.410","Text":"1 belongs to this pair."},{"Start":"11:29.410 ","End":"11:32.997","Text":"Now, we have here also 2 right shoes,"},{"Start":"11:32.997 ","End":"11:38.590","Text":"so 1 right shoe belongs to this pair and the other right shoe has to"},{"Start":"11:38.590 ","End":"11:45.320","Text":"come from a different pair than the paired of the remaining left shoe."},{"Start":"11:45.330 ","End":"11:47.530","Text":"How do we do this?"},{"Start":"11:47.530 ","End":"11:50.215","Text":"Well, that\u0027s 6 over 1."},{"Start":"11:50.215 ","End":"11:51.850","Text":"Out of the 6 possible pairs,"},{"Start":"11:51.850 ","End":"11:54.250","Text":"I\u0027ve chosen 1 pair."},{"Start":"11:54.250 ","End":"11:56.890","Text":"Now, out of the 5 remaining pairs,"},{"Start":"11:56.890 ","End":"11:59.704","Text":"I need 2 pairs of shoes,"},{"Start":"11:59.704 ","End":"12:08.380","Text":"where out of the 2 that were chosen I need 1 to be a left shoe."},{"Start":"12:08.380 ","End":"12:10.210","Text":"For the remaining pair,"},{"Start":"12:10.210 ","End":"12:13.744","Text":"I need that to be a right shoe,"},{"Start":"12:13.744 ","End":"12:15.385","Text":"divided by 495,"},{"Start":"12:15.385 ","End":"12:22.285","Text":"so that equals to 120/495."},{"Start":"12:22.285 ","End":"12:24.520","Text":"Here, W equals 1,"},{"Start":"12:24.520 ","End":"12:31.600","Text":"R equals 2, that\u0027s 120/495."},{"Start":"12:31.600 ","End":"12:35.545","Text":"Let\u0027s just complete this probability right here,"},{"Start":"12:35.545 ","End":"12:41.965","Text":"90 plus 120, we need here 15/495,"},{"Start":"12:41.965 ","End":"12:45.445","Text":"to sum up to 225."},{"Start":"12:45.445 ","End":"12:51.390","Text":"That means that here we have 15/495."},{"Start":"12:51.390 ","End":"12:58.330","Text":"Now, let\u0027s deal with these joint probabilities."},{"Start":"12:58.680 ","End":"13:05.800","Text":"Let\u0027s take a look at this joint probability right here with W equals 0 and R equals 3."},{"Start":"13:05.800 ","End":"13:11.275","Text":"That\u0027s the probability of W equaling 0 and R equaling 3."},{"Start":"13:11.275 ","End":"13:17.540","Text":"Now, let\u0027s just think about this probability right here."},{"Start":"13:19.170 ","End":"13:21.880","Text":"When R equals 3,"},{"Start":"13:21.880 ","End":"13:26.785","Text":"that means we have 3 left shoes and 1 right shoe."},{"Start":"13:26.785 ","End":"13:32.965","Text":"Now, that\u0027s symmetrical to the problem that we had here,"},{"Start":"13:32.965 ","End":"13:34.285","Text":"where R equals 1,"},{"Start":"13:34.285 ","End":"13:36.145","Text":"where we had 1 left shoe,"},{"Start":"13:36.145 ","End":"13:37.660","Text":"and 3 right shoes."},{"Start":"13:37.660 ","End":"13:42.985","Text":"That means that we have the same probability here where R equals 1,"},{"Start":"13:42.985 ","End":"13:47.280","Text":"as we have here where R equals 3. Why is that?"},{"Start":"13:47.280 ","End":"13:49.680","Text":"Well, as we said, that\u0027s because R,"},{"Start":"13:49.680 ","End":"13:52.800","Text":"the distribution of R is symmetrical."},{"Start":"13:52.800 ","End":"13:59.403","Text":"Here we should get 60/495,"},{"Start":"13:59.403 ","End":"14:01.660","Text":"but let\u0027s just calculate that right here."},{"Start":"14:01.660 ","End":"14:05.470","Text":"This should be the same thing as this thing right here."},{"Start":"14:05.470 ","End":"14:07.450","Text":"Now that equals again,"},{"Start":"14:07.450 ","End":"14:13.210","Text":"where we have 3 left shoes and we have no pairs of shoes."},{"Start":"14:13.210 ","End":"14:16.960","Text":"That means that we have 1 right shoe from 1 pair"},{"Start":"14:16.960 ","End":"14:21.400","Text":"and 3 left shoes from 3 other different pairs."},{"Start":"14:21.400 ","End":"14:25.150","Text":"Out of the 6 pairs of shoes that we have in the drawers,"},{"Start":"14:25.150 ","End":"14:27.220","Text":"we have to choose 4 of them."},{"Start":"14:27.220 ","End":"14:30.675","Text":"Out of the 4, we need to choose 1"},{"Start":"14:30.675 ","End":"14:40.405","Text":"that\u0027s a right shoe and out of the 3 remaining pairs of shoes,"},{"Start":"14:40.405 ","End":"14:45.085","Text":"we have 3 left shoes, divided by 495."},{"Start":"14:45.085 ","End":"14:46.825","Text":"Now, as we can see,"},{"Start":"14:46.825 ","End":"14:54.715","Text":"this is the same probability as we have here that equals to 60/495."},{"Start":"14:54.715 ","End":"14:59.140","Text":"We\u0027ll put that here, that\u0027s 60/495."},{"Start":"14:59.140 ","End":"15:03.580","Text":"We use logic to figure out what"},{"Start":"15:03.580 ","End":"15:08.800","Text":"this probability is on 1 hand and we also calculated the probability right here."},{"Start":"15:08.800 ","End":"15:15.430","Text":"But again, we can see that we have a symmetrical distribution for R. Now,"},{"Start":"15:15.430 ","End":"15:17.230","Text":"what about this guy right here?"},{"Start":"15:17.230 ","End":"15:26.440","Text":"Well, 60 plus 1 equals 120 and that equals to 60 as well, 60/495."},{"Start":"15:26.440 ","End":"15:32.959","Text":"Let\u0027s just sum up the probabilities here in order to get these values."},{"Start":"15:32.959 ","End":"15:37.660","Text":"Well that equals to 240/495."},{"Start":"15:37.660 ","End":"15:43.525","Text":"Here that also equals to 240/495."},{"Start":"15:43.525 ","End":"15:49.520","Text":"There we go. This is the joint distribution of W and R."}],"ID":13226},{"Watched":false,"Name":"Exercise 3 Parts b-d","Duration":"8m 29s","ChapterTopicVideoID":12748,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12748.jpeg","UploadDate":"2019-02-11T14:29:23.6370000","DurationForVideoObject":"PT8M29S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.290","Text":"In this section we\u0027re asked whether W and R are independent variables."},{"Start":"00:04.290 ","End":"00:08.895","Text":"Well, first let\u0027s recall what\u0027s the definition of independent variables."},{"Start":"00:08.895 ","End":"00:14.280","Text":"That\u0027s the probability of X equaling some value k and"},{"Start":"00:14.280 ","End":"00:20.190","Text":"Y equaling some value l. The joint probability of X equaling k and Y equaling l,"},{"Start":"00:20.190 ","End":"00:25.950","Text":"that equals to the multiplication of the marginal probabilities,"},{"Start":"00:25.950 ","End":"00:32.175","Text":"where the probability of X equaling k times the probability Y equaling some value l. Now,"},{"Start":"00:32.175 ","End":"00:35.910","Text":"that\u0027s for all values of X and Y,"},{"Start":"00:35.910 ","End":"00:39.610","Text":"for all k and l. Now,"},{"Start":"00:39.610 ","End":"00:41.045","Text":"again, what does that mean?"},{"Start":"00:41.045 ","End":"00:44.930","Text":"We\u0027re looking at the joint probability equaling to"},{"Start":"00:44.930 ","End":"00:50.345","Text":"the multiplication of the marginal probabilities of X and Y for all values of X and Y."},{"Start":"00:50.345 ","End":"00:52.205","Text":"Now, in our case,"},{"Start":"00:52.205 ","End":"00:58.095","Text":"it\u0027s easiest to look at joint probabilities equaling 0."},{"Start":"00:58.095 ","End":"00:59.465","Text":"As a rule of thumb,"},{"Start":"00:59.465 ","End":"01:02.525","Text":"if we have 0 in the joint probabilities,"},{"Start":"01:02.525 ","End":"01:06.575","Text":"then the variables are not independent. Now why is that?"},{"Start":"01:06.575 ","End":"01:09.185","Text":"Well, let\u0027s take a look at this,"},{"Start":"01:09.185 ","End":"01:16.710","Text":"probability of W equaling 2 and R equaling 0."},{"Start":"01:16.710 ","End":"01:22.655","Text":"Well, that should equal to the probability of W equaling"},{"Start":"01:22.655 ","End":"01:31.290","Text":"2 times the probability of R equaling 0 if W and R are independent."},{"Start":"01:31.290 ","End":"01:38.209","Text":"Let\u0027s take a look. The probability of W equaling 2 and R equaling 0, that equals 0."},{"Start":"01:38.209 ","End":"01:41.470","Text":"Now, the probability of W equaling 2,"},{"Start":"01:41.470 ","End":"01:48.335","Text":"that\u0027s 15/495, times the probability of R equaling 0,"},{"Start":"01:48.335 ","End":"01:51.785","Text":"well, that\u0027s also 15/495."},{"Start":"01:51.785 ","End":"01:58.470","Text":"Now, if we have a multiplication of 2 positive values,"},{"Start":"01:58.470 ","End":"02:03.095","Text":"then there\u0027s no way that the multiplication will be equal to 0."},{"Start":"02:03.095 ","End":"02:11.710","Text":"That means that W and R are dependent, they\u0027re not independent."},{"Start":"02:11.710 ","End":"02:14.720","Text":"In this section, we\u0027re asked to find the distribution of"},{"Start":"02:14.720 ","End":"02:17.330","Text":"R. That means the number of left shoes that were"},{"Start":"02:17.330 ","End":"02:19.880","Text":"removed from the drawer given that in"},{"Start":"02:19.880 ","End":"02:23.770","Text":"total only 1 pair of shoes were removed from the drawer."},{"Start":"02:23.770 ","End":"02:32.845","Text":"We\u0027re looking for the probability function of R given W equals to 1."},{"Start":"02:32.845 ","End":"02:37.010","Text":"Now, that means that we\u0027re looking at this column right here,"},{"Start":"02:37.010 ","End":"02:38.810","Text":"where W equals to 1."},{"Start":"02:38.810 ","End":"02:42.315","Text":"We\u0027re looking at the distribution of R. Well,"},{"Start":"02:42.315 ","End":"02:47.985","Text":"we can see that R can have values of 1,"},{"Start":"02:47.985 ","End":"02:51.160","Text":"2, and 3 only."},{"Start":"02:51.200 ","End":"02:59.315","Text":"Let\u0027s now calculate the probability of R equaling 1 given W equals 1."},{"Start":"02:59.315 ","End":"03:03.785","Text":"Well, that equals to the probability of R equaling 1,"},{"Start":"03:03.785 ","End":"03:12.400","Text":"intersect W equaling 1,/the probability of what\u0027s known, W equaling 1."},{"Start":"03:12.400 ","End":"03:14.420","Text":"Well, that equals to this."},{"Start":"03:14.420 ","End":"03:15.950","Text":"Well, in the denominator,"},{"Start":"03:15.950 ","End":"03:18.500","Text":"the probability of W equaling 1, well,"},{"Start":"03:18.500 ","End":"03:22.580","Text":"that\u0027s 240/495. Let\u0027s write that down."},{"Start":"03:22.580 ","End":"03:27.260","Text":"That\u0027s 240/495. Now,"},{"Start":"03:27.260 ","End":"03:31.840","Text":"what\u0027s the probability of R equals 1 and W equals 1?"},{"Start":"03:31.840 ","End":"03:33.785","Text":"Well, that\u0027s right here."},{"Start":"03:33.785 ","End":"03:42.605","Text":"That\u0027s 60/495, so that equals to 60/240,"},{"Start":"03:42.605 ","End":"03:45.680","Text":"and that equals to 1/4."},{"Start":"03:45.680 ","End":"03:53.925","Text":"Now, what about the probability of R equaling 2 given W equals 1?"},{"Start":"03:53.925 ","End":"03:58.490","Text":"Well, again, that\u0027s the probability of R equaling 2,"},{"Start":"03:58.490 ","End":"04:05.540","Text":"intersect W equaling 1 divided by the probability of W equaling 1."},{"Start":"04:05.540 ","End":"04:07.400","Text":"Well, in the denominator,"},{"Start":"04:07.400 ","End":"04:08.720","Text":"we know what that is."},{"Start":"04:08.720 ","End":"04:16.185","Text":"That\u0027s 240/495, and in the numerator,"},{"Start":"04:16.185 ","End":"04:19.040","Text":"well, R equals 2 and W equals 1."},{"Start":"04:19.040 ","End":"04:22.070","Text":"That\u0027s this joint probability right here."},{"Start":"04:22.070 ","End":"04:29.915","Text":"That\u0027s 120/495, that equals to 120/240,"},{"Start":"04:29.915 ","End":"04:33.685","Text":"and that equals to 2/4."},{"Start":"04:33.685 ","End":"04:40.175","Text":"I think we have enough to create the distribution function of R. Now, let\u0027s do that."},{"Start":"04:40.175 ","End":"04:44.820","Text":"We have R given W equals 1,"},{"Start":"04:44.820 ","End":"04:49.260","Text":"and we know that that can have the values of 1,"},{"Start":"04:49.260 ","End":"04:50.985","Text":"2, and 3."},{"Start":"04:50.985 ","End":"04:55.035","Text":"Now, what\u0027s the probability of R given W equals 1?"},{"Start":"04:55.035 ","End":"04:59.390","Text":"Well, when R equals 1 given W equals 1,"},{"Start":"04:59.390 ","End":"05:01.870","Text":"well, that\u0027s 1 quarter."},{"Start":"05:01.870 ","End":"05:05.040","Text":"When R equals 2 given W equals 1,"},{"Start":"05:05.040 ","End":"05:07.440","Text":"well, that\u0027s 2/4."},{"Start":"05:07.440 ","End":"05:12.655","Text":"Now, we can calculate the probability of R equals 3 given W equals 1,"},{"Start":"05:12.655 ","End":"05:17.410","Text":"but we can also use the fact that the sum of the probabilities have to add up to 1,"},{"Start":"05:17.410 ","End":"05:22.150","Text":"so therefore we can complete this probability that is also 1 quarter."},{"Start":"05:22.150 ","End":"05:25.610","Text":"This then is the answer."},{"Start":"05:25.610 ","End":"05:31.550","Text":"That\u0027s the distribution function of R given that W equals 1."},{"Start":"05:31.550 ","End":"05:35.990","Text":"In this section, we\u0027re given that at least 3 left shoes were removed,"},{"Start":"05:35.990 ","End":"05:41.390","Text":"and we\u0027re asked about the chances that at most 1 pair of shoes was removed."},{"Start":"05:41.390 ","End":"05:45.315","Text":"We have here a conditional probability."},{"Start":"05:45.315 ","End":"05:49.460","Text":"We\u0027re asked about the probability what\u0027s given that R,"},{"Start":"05:49.460 ","End":"05:51.500","Text":"the number of left shoes,"},{"Start":"05:51.500 ","End":"05:54.710","Text":"and we\u0027re given that at least 3 left shoes were removed,"},{"Start":"05:54.710 ","End":"05:57.005","Text":"that means that R is greater or equal to 3."},{"Start":"05:57.005 ","End":"06:01.370","Text":"We\u0027re asked about the chances that at most 1 pair of shoes was removed."},{"Start":"06:01.370 ","End":"06:03.690","Text":"Well, a pair of shoes that\u0027s W,"},{"Start":"06:03.690 ","End":"06:07.145","Text":"and at most that means that it\u0027s less than or equal to 1,"},{"Start":"06:07.145 ","End":"06:09.095","Text":"at most 1 pair."},{"Start":"06:09.095 ","End":"06:13.220","Text":"This then is the conditional probability that we have to calculate."},{"Start":"06:13.220 ","End":"06:16.095","Text":"Now, how do we do that?"},{"Start":"06:16.095 ","End":"06:17.209","Text":"Well, in the denominator,"},{"Start":"06:17.209 ","End":"06:19.625","Text":"we have the probability what\u0027s given."},{"Start":"06:19.625 ","End":"06:23.555","Text":"That\u0027s the probability of R being greater than or equal to 3."},{"Start":"06:23.555 ","End":"06:24.890","Text":"Then in the numerator,"},{"Start":"06:24.890 ","End":"06:26.960","Text":"we have the probability of the intersect."},{"Start":"06:26.960 ","End":"06:29.060","Text":"That\u0027s W being less than or equal to 1,"},{"Start":"06:29.060 ","End":"06:33.240","Text":"intersect R being greater or equal to 3."},{"Start":"06:33.490 ","End":"06:39.200","Text":"Now, let\u0027s take a look at the probability of the denominator,"},{"Start":"06:39.200 ","End":"06:44.150","Text":"where R is greater or equal to 3. Here it is."},{"Start":"06:44.150 ","End":"06:46.480","Text":"It\u0027s colored in yellow,"},{"Start":"06:46.480 ","End":"06:48.930","Text":"where R is greater or equal to 3,"},{"Start":"06:48.930 ","End":"06:52.350","Text":"that means that R equals 3 and R equals 4."},{"Start":"06:52.350 ","End":"06:55.500","Text":"Now, what about where W is less than or equal to 1?"},{"Start":"06:55.500 ","End":"06:57.360","Text":"Well, that\u0027s right here."},{"Start":"06:57.360 ","End":"07:01.890","Text":"Now, this is colored in blue,"},{"Start":"07:01.890 ","End":"07:04.635","Text":"where W is equal to 1,"},{"Start":"07:04.635 ","End":"07:06.920","Text":"or W is equal to 0."},{"Start":"07:06.920 ","End":"07:08.510","Text":"Now, what\u0027s the intersect?"},{"Start":"07:08.510 ","End":"07:11.930","Text":"Well, the intersect is in green right here."},{"Start":"07:11.930 ","End":"07:15.380","Text":"Let\u0027s now do the calculations."},{"Start":"07:15.380 ","End":"07:18.310","Text":"The intersect, the numerator,"},{"Start":"07:18.310 ","End":"07:23.840","Text":"that means that we have to add up all the probabilities in the green area right here."},{"Start":"07:23.840 ","End":"07:28.670","Text":"Well, that\u0027s 60 plus 60 plus 15 plus"},{"Start":"07:28.670 ","End":"07:35.680","Text":"0 divided by 495,"},{"Start":"07:35.680 ","End":"07:40.400","Text":"divided by the probability of R being greater or equal to 3."},{"Start":"07:40.400 ","End":"07:42.470","Text":"Well, that\u0027s this thing right here."},{"Start":"07:42.470 ","End":"07:49.070","Text":"That\u0027s 120 plus 15/495."},{"Start":"07:49.070 ","End":"07:58.040","Text":"Well, that equals to 135/495 divided by 135/495."},{"Start":"07:58.040 ","End":"07:59.300","Text":"Now, that equals to 1."},{"Start":"07:59.300 ","End":"08:00.955","Text":"We have a certainty."},{"Start":"08:00.955 ","End":"08:04.535","Text":"Now, let\u0027s see if that makes sense."},{"Start":"08:04.535 ","End":"08:09.735","Text":"Well, if I removed 3 left shoes,"},{"Start":"08:09.735 ","End":"08:12.990","Text":"then I obviously don\u0027t have 2 pairs."},{"Start":"08:12.990 ","End":"08:15.240","Text":"Now, in order to get 2 pairs in the sample,"},{"Start":"08:15.240 ","End":"08:19.625","Text":"I need 2 left shoes and 2 right shoes."},{"Start":"08:19.625 ","End":"08:24.770","Text":"That means that if I removed 3 left shoes,"},{"Start":"08:24.770 ","End":"08:29.400","Text":"then I have at most 1 pair of shoes."}],"ID":13227},{"Watched":false,"Name":"Exercise 4 Parts a-b","Duration":"9m 6s","ChapterTopicVideoID":12749,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12749.jpeg","UploadDate":"2019-02-11T14:30:34.5670000","DurationForVideoObject":"PT9M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.540","Text":"In this question, we\u0027re given a jar that contains 5 blue balls,"},{"Start":"00:03.540 ","End":"00:05.775","Text":"4 white balls, and 3 green balls."},{"Start":"00:05.775 ","End":"00:10.008","Text":"Now, 3 balls are randomly chosen 1 at a time without returns."},{"Start":"00:10.008 ","End":"00:12.060","Text":"We define the following variables."},{"Start":"00:12.060 ","End":"00:18.044","Text":"X has a value of 1 if at least 1 blue ball is chosen and 0 otherwise,"},{"Start":"00:18.044 ","End":"00:21.360","Text":"and Y is the number of white balls that were chosen."},{"Start":"00:21.360 ","End":"00:24.921","Text":"We\u0027re asked to calculate the probability of X equaling 1."},{"Start":"00:24.921 ","End":"00:29.250","Text":"Let\u0027s just review the information that we have."},{"Start":"00:29.250 ","End":"00:32.505","Text":"We have 5 blue balls,"},{"Start":"00:32.505 ","End":"00:35.790","Text":"we have 4 white balls,"},{"Start":"00:35.790 ","End":"00:39.255","Text":"and we have 3 green balls,"},{"Start":"00:39.255 ","End":"00:43.170","Text":"a total of 12 balls."},{"Start":"00:43.170 ","End":"00:47.720","Text":"Now, we sample 3 balls."},{"Start":"00:47.720 ","End":"00:50.015","Text":"That\u0027s our sample size."},{"Start":"00:50.015 ","End":"00:52.850","Text":"Now, what about the sample space?"},{"Start":"00:52.850 ","End":"01:01.160","Text":"How many permutations or how many combinations of 3 balls can we take out of 12 balls?"},{"Start":"01:01.160 ","End":"01:04.040","Text":"Well, that\u0027s 12 over 3."},{"Start":"01:04.040 ","End":"01:08.480","Text":"Now that equals to 220,"},{"Start":"01:08.480 ","End":"01:11.014","Text":"so that\u0027s our sample space."},{"Start":"01:11.014 ","End":"01:16.250","Text":"Now, we\u0027re asked about the probability of X equaling 1."},{"Start":"01:16.250 ","End":"01:22.204","Text":"That means that\u0027s the probability of getting at least 1 ball in our sample."},{"Start":"01:22.204 ","End":"01:27.740","Text":"Now, it\u0027s easier for us to calculate, the following probability,"},{"Start":"01:27.740 ","End":"01:32.185","Text":"not the probability of getting at least 1 ball,"},{"Start":"01:32.185 ","End":"01:34.125","Text":"but 1 minus that,"},{"Start":"01:34.125 ","End":"01:41.070","Text":"that means that\u0027s 1 minus the probability of getting no blue ball."},{"Start":"01:41.360 ","End":"01:45.780","Text":"Now, let\u0027s see what that equals to."},{"Start":"01:45.780 ","End":"01:49.565","Text":"Well, that\u0027s 1 minus now,"},{"Start":"01:49.565 ","End":"01:52.219","Text":"how many balls are there, that aren\u0027t blue?"},{"Start":"01:52.219 ","End":"01:53.855","Text":"That\u0027s 4 plus 7."},{"Start":"01:53.855 ","End":"01:57.460","Text":"Now, out of the 7 balls here,"},{"Start":"01:57.460 ","End":"01:59.565","Text":"we need to choose 3."},{"Start":"01:59.565 ","End":"02:01.515","Text":"That\u0027s 7 over 3,"},{"Start":"02:01.515 ","End":"02:05.820","Text":"divided by 220,"},{"Start":"02:05.820 ","End":"02:07.970","Text":"that\u0027s our sample space."},{"Start":"02:07.970 ","End":"02:14.900","Text":"Now, that equals to 185 divided by 220."},{"Start":"02:14.900 ","End":"02:19.670","Text":"This then is the probability of X equaling 1,"},{"Start":"02:19.670 ","End":"02:27.749","Text":"or the probability of getting at least 1 blue ball in our sample."},{"Start":"02:27.749 ","End":"02:32.870","Text":"In this section, we\u0027re asked to build a joint probability distribution of X and Y."},{"Start":"02:32.870 ","End":"02:36.710","Text":"Let\u0027s just recall the data that we were given."},{"Start":"02:36.710 ","End":"02:41.230","Text":"Well, we had 5 blue balls,"},{"Start":"02:41.230 ","End":"02:44.655","Text":"we had 4 white balls,"},{"Start":"02:44.655 ","End":"02:48.230","Text":"and we had 3 green balls."},{"Start":"02:48.230 ","End":"02:51.170","Text":"Now, we took a sample size of 3."},{"Start":"02:51.170 ","End":"02:57.844","Text":"That means that we chose 3 balls from a total of 12 balls, without returns."},{"Start":"02:57.844 ","End":"03:03.120","Text":"Now, we defined X receiving the value of 1 if there is"},{"Start":"03:03.120 ","End":"03:12.470","Text":"at least 1 blue ball and 0 if there wasn\u0027t any blue ball."},{"Start":"03:12.470 ","End":"03:18.760","Text":"Now Y was defined as the number of white balls."},{"Start":"03:18.760 ","End":"03:26.370","Text":"Now, what are the valid values of X and Y?"},{"Start":"03:26.370 ","End":"03:27.930","Text":"Well, for X it\u0027s simple."},{"Start":"03:27.930 ","End":"03:32.610","Text":"We\u0027re given the values X equals 0 or 1."},{"Start":"03:32.610 ","End":"03:38.040","Text":"1 whether we have at least 1 blue ball in the sample and 0 if we don\u0027t."},{"Start":"03:38.040 ","End":"03:41.770","Text":"Now Y is the number of white balls in the sample, so therefore,"},{"Start":"03:41.770 ","End":"03:44.570","Text":"Y can have values from 0,"},{"Start":"03:44.570 ","End":"03:47.715","Text":"1, 2 or 3."},{"Start":"03:47.715 ","End":"03:49.350","Text":"Great."},{"Start":"03:49.350 ","End":"03:51.235","Text":"Now that we have all this information,"},{"Start":"03:51.235 ","End":"03:56.360","Text":"we can start building the joint probability distribution."},{"Start":"03:56.450 ","End":"04:01.260","Text":"Here\u0027s the table and we have here for X, 0,"},{"Start":"04:01.260 ","End":"04:03.860","Text":"1 that\u0027s our valid values for X that\u0027s in"},{"Start":"04:03.860 ","End":"04:07.018","Text":"blue and the marginal probabilities of X that\u0027ll be right here."},{"Start":"04:07.018 ","End":"04:10.385","Text":"We have the valid values of Y, 0-3,"},{"Start":"04:10.385 ","End":"04:14.400","Text":"and the marginal probabilities of Y in orange right here,"},{"Start":"04:14.400 ","End":"04:17.150","Text":"and we have the joint probabilities right here."},{"Start":"04:17.150 ","End":"04:19.565","Text":"Now, all the probabilities have to add up to 1."},{"Start":"04:19.565 ","End":"04:21.935","Text":"Let\u0027s get started."},{"Start":"04:21.935 ","End":"04:26.600","Text":"We know from the last section that the probability of X equaling 1,"},{"Start":"04:26.600 ","End":"04:31.000","Text":"well that equal to 185 divided by 220,"},{"Start":"04:31.000 ","End":"04:34.925","Text":"and that means that the probability of X equaling 0,"},{"Start":"04:34.925 ","End":"04:39.520","Text":"that\u0027s 35 divided by 220."},{"Start":"04:39.520 ","End":"04:48.015","Text":"Now, let\u0027s take a look at the probability of Y equaling 0."},{"Start":"04:48.015 ","End":"04:50.660","Text":"This, we\u0027ll take a look at"},{"Start":"04:50.660 ","End":"04:54.305","Text":"the marginal probabilities and we\u0027ll look specifically at this cell."},{"Start":"04:54.305 ","End":"04:57.110","Text":"Well, what does that mean? Y equaling 0."},{"Start":"04:57.110 ","End":"04:59.930","Text":"That means that in our sample of 3,"},{"Start":"04:59.930 ","End":"05:02.540","Text":"we didn\u0027t choose any white balls."},{"Start":"05:02.540 ","End":"05:06.245","Text":"That means that out of the 8 remaining balls,"},{"Start":"05:06.245 ","End":"05:12.575","Text":"I had to choose 3 and that is divided by 220,"},{"Start":"05:12.575 ","End":"05:14.540","Text":"which is our sample space now,"},{"Start":"05:14.540 ","End":"05:19.375","Text":"that equals to 56 divided by 220."},{"Start":"05:19.375 ","End":"05:20.880","Text":"We\u0027ll write that down here,"},{"Start":"05:20.880 ","End":"05:23.700","Text":"56 divided by 220."},{"Start":"05:23.700 ","End":"05:27.605","Text":"Now, what about the probability of Y equaling 1?"},{"Start":"05:27.605 ","End":"05:35.265","Text":"Well, that means that I chose 1 white ball and out of the remaining 8 I chose 2."},{"Start":"05:35.265 ","End":"05:39.930","Text":"Out of the 4 white balls, I chose 1,"},{"Start":"05:39.930 ","End":"05:43.080","Text":"and out of the 8 balls that aren\u0027t white,"},{"Start":"05:43.080 ","End":"05:46.995","Text":"I chose 2, divided by 220."},{"Start":"05:46.995 ","End":"05:53.865","Text":"Now that equals to 112 divided by 220."},{"Start":"05:53.865 ","End":"05:55.905","Text":"We\u0027ll write that here."},{"Start":"05:55.905 ","End":"05:59.580","Text":"Now, for the other 2 cells right here,"},{"Start":"05:59.580 ","End":"06:01.860","Text":"we can do the calculations,"},{"Start":"06:01.860 ","End":"06:04.305","Text":"I\u0027ll let you do the calculations,"},{"Start":"06:04.305 ","End":"06:08.100","Text":"and I urge you to do that but I\u0027ll give you the answers right now,"},{"Start":"06:08.100 ","End":"06:11.730","Text":"that will be 48 divided by 220,"},{"Start":"06:11.730 ","End":"06:15.645","Text":"and that\u0027s 4 divided by 220."},{"Start":"06:15.645 ","End":"06:25.415","Text":"The next step is to see which of the joint probabilities form an impossibility."},{"Start":"06:25.415 ","End":"06:32.005","Text":"That means which 1 of the values the combinations of X and Y, form an impossibility?"},{"Start":"06:32.005 ","End":"06:34.200","Text":"Now, we have this cell right here,"},{"Start":"06:34.200 ","End":"06:37.070","Text":"where Y equals 3 and X equals 1."},{"Start":"06:37.070 ","End":"06:39.724","Text":"Now what does that mean? Well, remember,"},{"Start":"06:39.724 ","End":"06:41.225","Text":"our sample size is 3."},{"Start":"06:41.225 ","End":"06:42.410","Text":"If Y equals 3,"},{"Start":"06:42.410 ","End":"06:44.240","Text":"then we have 3 white balls,"},{"Start":"06:44.240 ","End":"06:48.717","Text":"but X means that I have at least 1 blue ball."},{"Start":"06:48.717 ","End":"06:49.920","Text":"If X equals 1,"},{"Start":"06:49.920 ","End":"06:51.380","Text":"now that\u0027s an impossibility."},{"Start":"06:51.380 ","End":"06:56.390","Text":"I can\u0027t have at least 1 blue ball if all the balls in my sample are white,"},{"Start":"06:56.390 ","End":"06:58.655","Text":"so that\u0027s 0 right here."},{"Start":"06:58.655 ","End":"07:04.110","Text":"Now, if that\u0027s 0, then that equals to 4 divided by 220."},{"Start":"07:04.110 ","End":"07:11.750","Text":"Now, let\u0027s take a look at the probability then of X equaling 0 and Y equaling 0."},{"Start":"07:11.750 ","End":"07:17.555","Text":"That means that I don\u0027t have any blue balls in my sample,"},{"Start":"07:17.555 ","End":"07:22.675","Text":"and Y equals 0 means I don\u0027t have any white balls as well."},{"Start":"07:22.675 ","End":"07:25.560","Text":"That means I\u0027m only left with green balls."},{"Start":"07:25.560 ","End":"07:28.080","Text":"Now, how many green balls in totals do I have?"},{"Start":"07:28.080 ","End":"07:29.535","Text":"I have 3. Now,"},{"Start":"07:29.535 ","End":"07:31.760","Text":"how many did I have to choose?"},{"Start":"07:31.760 ","End":"07:37.350","Text":"Well, 3 divided by 220."},{"Start":"07:37.350 ","End":"07:40.605","Text":"That means that that\u0027s 1 divided by 220."},{"Start":"07:40.605 ","End":"07:44.610","Text":"Right here, 1 divided by 220."},{"Start":"07:44.610 ","End":"07:50.575","Text":"That means that here I have 55 divided by 220."},{"Start":"07:50.575 ","End":"08:00.530","Text":"Now, what about the probability of X equaling 0 and Y equaling 1?"},{"Start":"08:00.530 ","End":"08:05.785","Text":"What does that mean? That means that I have no blue balls and I have 1 white ball."},{"Start":"08:05.785 ","End":"08:11.700","Text":"Now, that means that out of the 4 white balls, I chose 1,"},{"Start":"08:11.700 ","End":"08:14.775","Text":"and out of the 3 green balls,"},{"Start":"08:14.775 ","End":"08:18.900","Text":"I chose 2 divided by 220."},{"Start":"08:18.900 ","End":"08:24.120","Text":"Now that equals to 12 divided by 220."},{"Start":"08:24.120 ","End":"08:29.370","Text":"This then where P equals 0 and Y equals 1,"},{"Start":"08:29.370 ","End":"08:38.390","Text":"that\u0027s 12 divided by 220 and that means that this is 100 divided by 220."},{"Start":"08:38.440 ","End":"08:45.653","Text":"We can also fill in these 2 joint probabilities."},{"Start":"08:45.653 ","End":"08:48.993","Text":"Where Y equals 2 and x equals 1."},{"Start":"08:48.993 ","End":"08:55.470","Text":"That\u0027ll be 30 divided by 220,"},{"Start":"08:55.470 ","End":"09:00.630","Text":"and here that\u0027ll be 18 divided by 220,"},{"Start":"09:00.630 ","End":"09:01.770","Text":"so there we go."},{"Start":"09:01.770 ","End":"09:06.990","Text":"This is the joint probability distribution of X and Y."}],"ID":13228},{"Watched":false,"Name":"Exercise 4 Parts c-d","Duration":"8m 40s","ChapterTopicVideoID":12750,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12750.jpeg","UploadDate":"2019-02-11T14:31:45.8770000","DurationForVideoObject":"PT8M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.310","Text":"In this section we\u0027re asked what\u0027s the expectation of Y if no blue balls were removed?"},{"Start":"00:05.310 ","End":"00:07.890","Text":"That means that if X equal to 0."},{"Start":"00:07.890 ","End":"00:14.640","Text":"In essence we\u0027re looking for the expectation of Y given that X equals to 0."},{"Start":"00:14.640 ","End":"00:16.410","Text":"Now how do we solve this?"},{"Start":"00:16.410 ","End":"00:24.690","Text":"Well, we first need to build the probability distribution of Y given that X equals 0,"},{"Start":"00:24.690 ","End":"00:27.690","Text":"and then calculate the expectation."},{"Start":"00:27.690 ","End":"00:29.925","Text":"Let\u0027s go ahead and do that."},{"Start":"00:29.925 ","End":"00:38.200","Text":"What\u0027s the probability of y equaling 0 given that x equals 0?"},{"Start":"00:38.200 ","End":"00:40.460","Text":"Well, that equals 2."},{"Start":"00:40.460 ","End":"00:42.260","Text":"Now, what\u0027s in the denominator?"},{"Start":"00:42.260 ","End":"00:43.700","Text":"Well, that\u0027s the probability."},{"Start":"00:43.700 ","End":"00:47.390","Text":"What\u0027s given x equaling 0 and what\u0027s in the numerator?"},{"Start":"00:47.390 ","End":"00:48.890","Text":"Well, that\u0027s the intersect,"},{"Start":"00:48.890 ","End":"00:56.010","Text":"so that\u0027s the probability of x equaling 0 and y equaling 0."},{"Start":"00:56.010 ","End":"01:00.110","Text":"In the denominator, the probability of x equaling 0,"},{"Start":"01:00.110 ","End":"01:03.510","Text":"that\u0027s 35 divided by 220."},{"Start":"01:04.700 ","End":"01:09.680","Text":"What\u0027s the intersect of x equaling 0 and y equaling 0?"},{"Start":"01:09.680 ","End":"01:11.135","Text":"Well, that\u0027s right here."},{"Start":"01:11.135 ","End":"01:14.675","Text":"That\u0027s 1 divided by 220."},{"Start":"01:14.675 ","End":"01:18.215","Text":"Now, that equals to 1 divided by 35."},{"Start":"01:18.215 ","End":"01:24.935","Text":"What about the probability of y equaling 1,"},{"Start":"01:24.935 ","End":"01:27.020","Text":"given x equaling 0?"},{"Start":"01:27.020 ","End":"01:30.530","Text":"Well, we do the same procedure here."},{"Start":"01:30.530 ","End":"01:34.325","Text":"In the denominator we have the probability of"},{"Start":"01:34.325 ","End":"01:38.710","Text":"x equaling 0 and in the numerator we have the intersect,"},{"Start":"01:38.710 ","End":"01:42.930","Text":"where x equals 0 and y now equals 1."},{"Start":"01:42.930 ","End":"01:44.885","Text":"Well, in the denominator,"},{"Start":"01:44.885 ","End":"01:48.170","Text":"that\u0027ll be 35 divided by 220."},{"Start":"01:48.170 ","End":"01:54.275","Text":"Now what\u0027s the probability of the intersect when x equals 0 and y equals 1?"},{"Start":"01:54.275 ","End":"01:55.790","Text":"Well, that\u0027s right here."},{"Start":"01:55.790 ","End":"01:59.310","Text":"That\u0027s 12 divided by 220."},{"Start":"01:59.310 ","End":"02:04.720","Text":"This comes out to 12 divided by 35."},{"Start":"02:05.330 ","End":"02:09.380","Text":"What else? Well, we can go ahead and do this,"},{"Start":"02:09.380 ","End":"02:12.800","Text":"but I\u0027ll leave you to do the other calculations."},{"Start":"02:12.800 ","End":"02:14.795","Text":"Let\u0027s just now,"},{"Start":"02:14.795 ","End":"02:20.600","Text":"write our probability distribution function of y,"},{"Start":"02:20.600 ","End":"02:23.080","Text":"given x equals 0."},{"Start":"02:23.080 ","End":"02:27.765","Text":"The values of y are 0, 1,"},{"Start":"02:27.765 ","End":"02:31.260","Text":"2, and 3,"},{"Start":"02:31.260 ","End":"02:35.435","Text":"and what\u0027s the probability of y given x equals 0?"},{"Start":"02:35.435 ","End":"02:40.535","Text":"Well, the probability of y equaling 0 when x equals 0."},{"Start":"02:40.535 ","End":"02:43.955","Text":"That\u0027s right here, that\u0027s 1 divided by 35,"},{"Start":"02:43.955 ","End":"02:47.000","Text":"and here, when y equals 1,"},{"Start":"02:47.000 ","End":"02:50.495","Text":"well that\u0027s 12 divided by 35."},{"Start":"02:50.495 ","End":"02:57.155","Text":"As I said, I\u0027ll let you calculate these other probabilities right here,"},{"Start":"02:57.155 ","End":"02:58.880","Text":"but I\u0027ll give you the answer anyway,"},{"Start":"02:58.880 ","End":"03:01.640","Text":"that\u0027ll be 18 divided by 35,"},{"Start":"03:01.640 ","End":"03:05.845","Text":"and that\u0027s 4 divided by 35."},{"Start":"03:05.845 ","End":"03:12.710","Text":"What\u0027s the expectation then of y given x equaling 0?"},{"Start":"03:12.710 ","End":"03:14.060","Text":"Well, all that is,"},{"Start":"03:14.060 ","End":"03:19.130","Text":"is the value of y given equals 0 times the probability."},{"Start":"03:19.130 ","End":"03:22.265","Text":"That means it\u0027s 0 times 1 divided by 35."},{"Start":"03:22.265 ","End":"03:28.590","Text":"We won\u0027t write that down and that\u0027ll be 1 times 12 divided by 35,"},{"Start":"03:28.590 ","End":"03:33.580","Text":"plus 2 times 18 divided by 35,"},{"Start":"03:33.580 ","End":"03:39.275","Text":"plus 3 times 4 divided by 35."},{"Start":"03:39.275 ","End":"03:44.075","Text":"Now that equals to 1.714."},{"Start":"03:44.075 ","End":"03:46.595","Text":"This then is the expectation,"},{"Start":"03:46.595 ","End":"03:48.065","Text":"and what does that mean?"},{"Start":"03:48.065 ","End":"03:52.955","Text":"Well, that means that if we didn\u0027t choose any blue balls in our sample,"},{"Start":"03:52.955 ","End":"03:58.970","Text":"then the expected number of white balls would be 1.714."},{"Start":"03:58.970 ","End":"04:02.210","Text":"In this section, we\u0027re asked what\u0027s the variance of X if"},{"Start":"04:02.210 ","End":"04:05.885","Text":"it\u0027s known that at least 1 white ball was removed."},{"Start":"04:05.885 ","End":"04:10.040","Text":"We\u0027re looking for the variance of x,"},{"Start":"04:10.040 ","End":"04:13.905","Text":"given that y is greater or equal to 1."},{"Start":"04:13.905 ","End":"04:18.485","Text":"That expression means that at least 1 white ball was removed."},{"Start":"04:18.485 ","End":"04:21.950","Text":"Now, how do we calculate this variance?"},{"Start":"04:21.950 ","End":"04:24.365","Text":"Well, we first have to calculate"},{"Start":"04:24.365 ","End":"04:30.140","Text":"the probability distribution function of x given y being greater or equal to 1,"},{"Start":"04:30.140 ","End":"04:33.140","Text":"and from there we can calculate the variance."},{"Start":"04:33.140 ","End":"04:35.930","Text":"Now, what are we looking for?"},{"Start":"04:35.930 ","End":"04:39.425","Text":"What are the valid values then of x?"},{"Start":"04:39.425 ","End":"04:43.235","Text":"Well, let\u0027s take a look at this range for y first."},{"Start":"04:43.235 ","End":"04:45.665","Text":"That\u0027s this range right here,"},{"Start":"04:45.665 ","End":"04:48.485","Text":"where y equals 1, 2, and 3."},{"Start":"04:48.485 ","End":"04:52.670","Text":"Now, what are the valid values of x in this range?"},{"Start":"04:52.670 ","End":"04:57.035","Text":"Well, we can see that x has values for 0 and for 1."},{"Start":"04:57.035 ","End":"04:59.965","Text":"This would be the valid values."},{"Start":"04:59.965 ","End":"05:04.130","Text":"If that\u0027s the case, let\u0027s calculate the probability of x equaling"},{"Start":"05:04.130 ","End":"05:09.005","Text":"0 given that y is greater or equal to 1."},{"Start":"05:09.005 ","End":"05:13.600","Text":"That equals to the following where there\u0027s a conditional probability."},{"Start":"05:13.600 ","End":"05:18.025","Text":"In the denominator we have the probability of what\u0027s known and what\u0027s given,"},{"Start":"05:18.025 ","End":"05:20.810","Text":"and in the numerator we have the intersect."},{"Start":"05:20.810 ","End":"05:27.390","Text":"Well, that\u0027s the probability of x equaling 0 and y being greater or equal to 1."},{"Start":"05:27.590 ","End":"05:30.020","Text":"What does that equal to?"},{"Start":"05:30.020 ","End":"05:33.895","Text":"Well, let\u0027s take a look at this probability right here."},{"Start":"05:33.895 ","End":"05:38.585","Text":"The probability of the denominator means that I\u0027m taking them,"},{"Start":"05:38.585 ","End":"05:41.660","Text":"these marginal probabilities in the table for"},{"Start":"05:41.660 ","End":"05:47.010","Text":"the range of y being greater or equal to 1,"},{"Start":"05:47.010 ","End":"05:50.805","Text":"that means that we have 112 plus 48 plus 4."},{"Start":"05:50.805 ","End":"05:58.475","Text":"So 112 plus 48 plus 4 divided by 220."},{"Start":"05:58.475 ","End":"06:02.505","Text":"That\u0027s the probabilities for this range of y."},{"Start":"06:02.505 ","End":"06:05.115","Text":"What about the intersect?"},{"Start":"06:05.115 ","End":"06:09.485","Text":"When x equals 0, this is the column that we have to deal with."},{"Start":"06:09.485 ","End":"06:14.735","Text":"Again, we\u0027re dealing for this range of y,"},{"Start":"06:14.735 ","End":"06:18.530","Text":"that means that we\u0027re looking at these joint probabilities."},{"Start":"06:18.530 ","End":"06:25.355","Text":"That\u0027s 12 plus 18 plus 4 divided by 220."},{"Start":"06:25.355 ","End":"06:33.300","Text":"Now that equals to 34 divided by 164."},{"Start":"06:33.380 ","End":"06:35.705","Text":"Now that we have that,"},{"Start":"06:35.705 ","End":"06:40.280","Text":"we can build the probability distribution function"},{"Start":"06:40.280 ","End":"06:44.515","Text":"of x given that y is greater or equal to 1."},{"Start":"06:44.515 ","End":"06:46.770","Text":"Here we have the table."},{"Start":"06:46.770 ","End":"06:55.145","Text":"When x equals 0, then the probability here would be 34 divided by 164."},{"Start":"06:55.145 ","End":"06:57.170","Text":"We\u0027ve calculated that right here,"},{"Start":"06:57.170 ","End":"07:01.085","Text":"and that means that in order for the probabilities to add up to 1,"},{"Start":"07:01.085 ","End":"07:06.475","Text":"this has to be 130 divided by 164."},{"Start":"07:06.475 ","End":"07:08.575","Text":"Now that we have"},{"Start":"07:08.575 ","End":"07:15.050","Text":"the probability distribution function of x given y is greater or equal to 1,"},{"Start":"07:15.050 ","End":"07:16.730","Text":"then we can calculate the variance."},{"Start":"07:16.730 ","End":"07:20.065","Text":"Now, the variance of this."},{"Start":"07:20.065 ","End":"07:28.270","Text":"Now that equals to the expectation of x squared given y being greater or equal to 1,"},{"Start":"07:28.270 ","End":"07:35.720","Text":"minus the expectation squared of x given y being greater or equal to 1."},{"Start":"07:35.730 ","End":"07:40.890","Text":"What\u0027s the expectation of this expression right here?"},{"Start":"07:40.890 ","End":"07:46.050","Text":"Well the expectation of x given y being greater or equal to 1,"},{"Start":"07:46.050 ","End":"07:48.520","Text":"well, that\u0027s 0 times this probability,"},{"Start":"07:48.520 ","End":"07:51.140","Text":"plus 1 times this probability."},{"Start":"07:51.140 ","End":"07:56.225","Text":"Let\u0027s just write down the 1 times 130 divided by 164,"},{"Start":"07:56.225 ","End":"08:00.115","Text":"and that equals to 130 divided by 164."},{"Start":"08:00.115 ","End":"08:07.640","Text":"Now, what about the variance of x given y being greater or equal to 1?"},{"Start":"08:07.640 ","End":"08:10.025","Text":"Well that\u0027s the expectation squared,"},{"Start":"08:10.025 ","End":"08:13.080","Text":"so that\u0027s 0 squared times this,"},{"Start":"08:13.080 ","End":"08:14.925","Text":"plus 1 squared times this,"},{"Start":"08:14.925 ","End":"08:19.995","Text":"so the 1 squared times 130 divided by 164,"},{"Start":"08:19.995 ","End":"08:22.380","Text":"minus the expectation squared."},{"Start":"08:22.380 ","End":"08:27.170","Text":"That\u0027s 130 divided by 164 squared."},{"Start":"08:27.170 ","End":"08:32.220","Text":"Now that comes out to 0.164,"},{"Start":"08:32.220 ","End":"08:41.070","Text":"so this then is the variance of x given that at least 1 white ball was removed."}],"ID":13229},{"Watched":false,"Name":"Exercise 5 Part a","Duration":"11m 58s","ChapterTopicVideoID":12751,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12751.jpeg","UploadDate":"2019-02-11T14:33:31.1670000","DurationForVideoObject":"PT11M58S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.605","Text":"In this question, we\u0027ll be dealing with presents in birthday parties."},{"Start":"00:03.605 ","End":"00:05.000","Text":"On his 4th birthday,"},{"Start":"00:05.000 ","End":"00:07.770","Text":"Johnny gave out 3 different presents that were chosen at"},{"Start":"00:07.770 ","End":"00:11.055","Text":"random to 5 children who were also chosen at random."},{"Start":"00:11.055 ","End":"00:13.485","Text":"Each time that Johnny gave out a present,"},{"Start":"00:13.485 ","End":"00:18.149","Text":"he randomly chose a child independently with previous choices."},{"Start":"00:18.149 ","End":"00:22.350","Text":"The following variables are defined: X would be the number of presents given to Jane,"},{"Start":"00:22.350 ","End":"00:23.580","Text":"that\u0027s a girl at the party,"},{"Start":"00:23.580 ","End":"00:29.115","Text":"1 of 5 children and Y is the number of children who did not receive any presents at all."},{"Start":"00:29.115 ","End":"00:33.850","Text":"We\u0027re asked to build a joint probability distribution of X and Y."},{"Start":"00:33.950 ","End":"00:38.690","Text":"Let\u0027s now think of giving out a present as a trial."},{"Start":"00:38.690 ","End":"00:42.870","Text":"The trials are independent of each other and we"},{"Start":"00:42.870 ","End":"00:47.505","Text":"have 3 of them because we\u0027re handing out 3 presents."},{"Start":"00:47.505 ","End":"00:49.710","Text":"N here equals to 3,"},{"Start":"00:49.710 ","End":"00:51.645","Text":"the trials are independent."},{"Start":"00:51.645 ","End":"00:55.680","Text":"How do we define success?"},{"Start":"00:55.680 ","End":"01:03.180","Text":"A successful trial is that where Jane gets a present."},{"Start":"01:03.180 ","End":"01:07.170","Text":"What\u0027s the probability of Jane getting a present?"},{"Start":"01:07.170 ","End":"01:08.280","Text":"For each trial,"},{"Start":"01:08.280 ","End":"01:11.750","Text":"the probability is 1/5. Why is that?"},{"Start":"01:11.750 ","End":"01:17.835","Text":"Because we choose a child independently of the previous choices."},{"Start":"01:17.835 ","End":"01:19.785","Text":"If that\u0027s the case,"},{"Start":"01:19.785 ","End":"01:23.655","Text":"then X, the number of presents given to Jane,"},{"Start":"01:23.655 ","End":"01:27.710","Text":"that has a binomial distribution where n equals 3,"},{"Start":"01:27.710 ","End":"01:30.545","Text":"the number of trials or the number of presents given,"},{"Start":"01:30.545 ","End":"01:35.800","Text":"and the probability that Jane will be picked that\u0027s a 1/5."},{"Start":"01:35.800 ","End":"01:42.560","Text":"That means that we have here the probability of X equaling k,"},{"Start":"01:42.560 ","End":"01:45.615","Text":"because X has a binomial distribution."},{"Start":"01:45.615 ","End":"01:47.870","Text":"That equals to n over k,"},{"Start":"01:47.870 ","End":"01:57.550","Text":"p^k times 1 minus p^n minus k. What would be the valid values of X?"},{"Start":"01:57.550 ","End":"01:59.645","Text":"X can be equal to 0,"},{"Start":"01:59.645 ","End":"02:02.540","Text":"where Jane doesn\u0027t get any present whatsoever."},{"Start":"02:02.540 ","End":"02:06.020","Text":"She can get 1,2 or 3 presents,"},{"Start":"02:06.020 ","End":"02:08.620","Text":"or she gets all the presents."},{"Start":"02:08.620 ","End":"02:15.335","Text":"Let\u0027s now take a look at Y. Y doesn\u0027t have any special distribution,"},{"Start":"02:15.335 ","End":"02:19.345","Text":"but we can identify the valid values of Y."},{"Start":"02:19.345 ","End":"02:24.635","Text":"Let\u0027s do that. Let\u0027s take a look at the situation"},{"Start":"02:24.635 ","End":"02:30.695","Text":"where we have each present being given to 3 different children."},{"Start":"02:30.695 ","End":"02:34.550","Text":"That means that 3 children have presents."},{"Start":"02:34.550 ","End":"02:38.510","Text":"That means that Y equals to 2."},{"Start":"02:38.510 ","End":"02:42.760","Text":"That means that 2 children do not have any presents whatsoever."},{"Start":"02:42.760 ","End":"02:48.200","Text":"Let\u0027s take a look at the other end of the spectrum where 1 child gets"},{"Start":"02:48.200 ","End":"02:55.080","Text":"all 3 presents but that means that 4 children did not get any present whatsoever."},{"Start":"02:55.080 ","End":"02:58.215","Text":"The valid values for Y would be 2,"},{"Start":"02:58.215 ","End":"03:00.630","Text":"3, and 4."},{"Start":"03:00.630 ","End":"03:05.540","Text":"Having said that, let\u0027s just take a look at 1 more thing."},{"Start":"03:05.540 ","End":"03:08.225","Text":"What\u0027s our sample space?"},{"Start":"03:08.225 ","End":"03:16.165","Text":"How many combinations are there in giving out 3 different presents to 5 children?"},{"Start":"03:16.165 ","End":"03:19.050","Text":"That would be Omega,"},{"Start":"03:19.050 ","End":"03:25.740","Text":"our sample space that equals to 5 children and we have 3 different presents."},{"Start":"03:25.740 ","End":"03:31.235","Text":"That means that we have 100 and different combinations"},{"Start":"03:31.235 ","End":"03:38.535","Text":"of distributing 3 presents to 5 children done independently."},{"Start":"03:38.535 ","End":"03:41.015","Text":"This is the information that we have."},{"Start":"03:41.015 ","End":"03:44.760","Text":"Let\u0027s go and build a joint probability distribution."},{"Start":"03:44.900 ","End":"03:50.220","Text":"Here\u0027s a table and here we see X and the valid values of X from"},{"Start":"03:50.220 ","End":"03:54.560","Text":"0-3 in blue and the marginal probabilities of X,"},{"Start":"03:54.560 ","End":"03:55.865","Text":"that\u0027ll be right here."},{"Start":"03:55.865 ","End":"03:59.600","Text":"Here we have the valid values of Y in orange and"},{"Start":"03:59.600 ","End":"04:03.470","Text":"here we\u0027ll place the marginal probabilities of Y."},{"Start":"04:03.470 ","End":"04:05.450","Text":"In the middle and the white cells we\u0027ll have"},{"Start":"04:05.450 ","End":"04:09.495","Text":"the joint probabilities of X and Y. How do we start?"},{"Start":"04:09.495 ","End":"04:14.600","Text":"The best way to start or the easiest way to start is with the marginal probability of X."},{"Start":"04:14.600 ","End":"04:18.830","Text":"Why is that? Because we know what the probability of X is."},{"Start":"04:18.830 ","End":"04:26.050","Text":"X has a binomial distribution with n equals 3 and P the probability equals to 1/5."},{"Start":"04:26.050 ","End":"04:29.480","Text":"Let\u0027s get started with X equaling 0."},{"Start":"04:29.480 ","End":"04:32.820","Text":"The probability of X equaling 0,"},{"Start":"04:32.820 ","End":"04:34.950","Text":"well, that equals to n over k,"},{"Start":"04:34.950 ","End":"04:44.685","Text":"that\u0027s 3 over 0 times 1/5^0 times 4/5 to the power 3."},{"Start":"04:44.685 ","End":"04:49.575","Text":"That comes out to 64 divided by 125,"},{"Start":"04:49.575 ","End":"04:52.080","Text":"so let\u0027s just put that in here."},{"Start":"04:52.080 ","End":"04:55.940","Text":"What about the probability of X equaling 1?"},{"Start":"04:55.940 ","End":"05:03.455","Text":"That equals to 3 over 1, 1/5^1 times 4/5^2."},{"Start":"05:03.455 ","End":"05:09.940","Text":"That comes out to 48 divided by 125, put that here."},{"Start":"05:10.580 ","End":"05:16.820","Text":"I\u0027ll let you guys calculate the other 2 marginal probabilities."},{"Start":"05:16.820 ","End":"05:18.800","Text":"I urge you to do this by yourself,"},{"Start":"05:18.800 ","End":"05:21.365","Text":"but I\u0027ll just give you the answer right here."},{"Start":"05:21.365 ","End":"05:28.485","Text":"That\u0027s 12 divided by 125 and that\u0027s 1 divided by 125."},{"Start":"05:28.485 ","End":"05:32.135","Text":"Now that we have the marginal probabilities of X,"},{"Start":"05:32.135 ","End":"05:38.990","Text":"let\u0027s see if we have any joint probabilities that are 0,"},{"Start":"05:38.990 ","End":"05:45.445","Text":"that means that the combination of X and Y are an impossibility."},{"Start":"05:45.445 ","End":"05:48.080","Text":"Let\u0027s take a look at this cell right here,"},{"Start":"05:48.080 ","End":"05:51.110","Text":"where X equals 3 and Y equals 2."},{"Start":"05:51.110 ","End":"05:55.150","Text":"When X equals 3, Jane received all 3 presents."},{"Start":"05:55.150 ","End":"05:59.565","Text":"That means that 4 children didn\u0027t get any presents at all."},{"Start":"05:59.565 ","End":"06:03.694","Text":"That means that Y cannot be 2 and it cannot be 3"},{"Start":"06:03.694 ","End":"06:08.840","Text":"because Y is the number of children who did not receive any presents."},{"Start":"06:08.840 ","End":"06:11.960","Text":"Jane received all 3 presents and then, as I said,"},{"Start":"06:11.960 ","End":"06:14.765","Text":"4 children didn\u0027t receive any presence,"},{"Start":"06:14.765 ","End":"06:17.025","Text":"not 2 and not 3."},{"Start":"06:17.025 ","End":"06:21.450","Text":"What about when X equals 2 and Y equals 2,"},{"Start":"06:21.450 ","End":"06:24.060","Text":"Jane received 2 presents."},{"Start":"06:24.060 ","End":"06:28.145","Text":"That means that there\u0027s 1 present left to be given to another child."},{"Start":"06:28.145 ","End":"06:32.585","Text":"That means that 3 children didn\u0027t receive any presents."},{"Start":"06:32.585 ","End":"06:34.640","Text":"That means that here,"},{"Start":"06:34.640 ","End":"06:38.430","Text":"Y equals 2 is an impossibility."},{"Start":"06:38.430 ","End":"06:40.535","Text":"Same thing with here."},{"Start":"06:40.535 ","End":"06:46.725","Text":"If Jane received 2 presents and only 1 other child received a present,"},{"Start":"06:46.725 ","End":"06:51.650","Text":"so that means that 3 children did not receive any presence, not 4."},{"Start":"06:51.650 ","End":"06:56.820","Text":"What about here when X equals 1 and Y equals 4?"},{"Start":"06:56.820 ","End":"06:59.745","Text":"If Jane received 1 present,"},{"Start":"06:59.745 ","End":"07:07.830","Text":"then either 1 other child received 2 presents or 2 children received 1 present each."},{"Start":"07:07.830 ","End":"07:11.240","Text":"Again, there\u0027s no way that 4 children did not"},{"Start":"07:11.240 ","End":"07:16.275","Text":"receive any presents whatsoever so this also is 0."},{"Start":"07:16.275 ","End":"07:24.390","Text":"This basically completes the stage where we analyze impossibilities."},{"Start":"07:24.390 ","End":"07:27.770","Text":"Let\u0027s take a look at the joint probabilities here."},{"Start":"07:27.770 ","End":"07:29.860","Text":"What can we fill in?"},{"Start":"07:29.860 ","End":"07:35.420","Text":"Here the sum of the joint probabilities have to add up to 1 divided by 25,"},{"Start":"07:35.420 ","End":"07:38.825","Text":"so this here has to be 1 divided by 125."},{"Start":"07:38.825 ","End":"07:44.490","Text":"Likewise here, that should be 12 divided by 125."},{"Start":"07:44.860 ","End":"07:52.370","Text":"Let\u0027s see what we can do about the other joint probabilities."},{"Start":"07:52.370 ","End":"07:56.495","Text":"Let\u0027s take a look at this joint probability right here."},{"Start":"07:56.495 ","End":"08:03.470","Text":"The probability of X equaling 1 and Y equaling 3."},{"Start":"08:03.470 ","End":"08:07.710","Text":"What does that mean? That means that Jane got 1 present,"},{"Start":"08:07.710 ","End":"08:11.970","Text":"3 children didn\u0027t get any presents whatsoever and"},{"Start":"08:11.970 ","End":"08:16.395","Text":"that means that 1 child received 2 presents."},{"Start":"08:16.395 ","End":"08:20.715","Text":"How do we translate that into probabilities?"},{"Start":"08:20.715 ","End":"08:23.670","Text":"Out of the 3 presents that we have,"},{"Start":"08:23.670 ","End":"08:25.710","Text":"1 was given to Jane,"},{"Start":"08:25.710 ","End":"08:28.365","Text":"now for the other 2 presents,"},{"Start":"08:28.365 ","End":"08:36.855","Text":"only 1 child out of the remaining 4 received the 2 presents divided by 125,"},{"Start":"08:36.855 ","End":"08:38.760","Text":"that\u0027s our sample space,"},{"Start":"08:38.760 ","End":"08:42.600","Text":"and that equals to 12 divided by 125."},{"Start":"08:42.600 ","End":"08:44.505","Text":"Let\u0027s just write that down here,"},{"Start":"08:44.505 ","End":"08:47.910","Text":"that\u0027s 12 divided by 125."},{"Start":"08:47.910 ","End":"08:51.920","Text":"Let\u0027s take a look at this joint probability right here."},{"Start":"08:51.920 ","End":"08:57.995","Text":"That\u0027s the probability of X equaling 0 and Y equaling 4."},{"Start":"08:57.995 ","End":"09:02.220","Text":"This means that Jane didn\u0027t get any presents whatsoever."},{"Start":"09:02.220 ","End":"09:07.290","Text":"Y equals 4 means that 4 children didn\u0027t get any present,"},{"Start":"09:07.290 ","End":"09:12.075","Text":"that means that only 1 child received all 3 presents."},{"Start":"09:12.075 ","End":"09:16.245","Text":"All we have to do is out of the 4 remaining children, not Jane,"},{"Start":"09:16.245 ","End":"09:21.230","Text":"but out of the 4 remaining children we have to choose that 1 child that received"},{"Start":"09:21.230 ","End":"09:26.810","Text":"the 3 presents and that has to be divided by 125,"},{"Start":"09:26.810 ","End":"09:33.945","Text":"so that equals to 4 divided by 125."},{"Start":"09:33.945 ","End":"09:38.670","Text":"We\u0027ll put that here, that\u0027s 4 divided by 125."},{"Start":"09:38.670 ","End":"09:42.620","Text":"Let\u0027s take a look at another joint probability,"},{"Start":"09:42.620 ","End":"09:44.075","Text":"this 1 right here."},{"Start":"09:44.075 ","End":"09:51.000","Text":"That\u0027s the probability of X equaling 0 and Y equaling 3."},{"Start":"09:51.860 ","End":"09:56.730","Text":"X equals 0 means that Jane didn\u0027t get any presents whatsoever."},{"Start":"09:56.730 ","End":"10:03.905","Text":"Y equals 3 means that 3 children didn\u0027t get any presence whatsoever."},{"Start":"10:03.905 ","End":"10:09.550","Text":"That means that 2 children got prizes."},{"Start":"10:09.550 ","End":"10:14.295","Text":"2 children got the 3 prizes that means that 1 child got"},{"Start":"10:14.295 ","End":"10:20.325","Text":"2 prizes and the other child received only 1 prize."},{"Start":"10:20.325 ","End":"10:23.160","Text":"How do we translate that?"},{"Start":"10:23.160 ","End":"10:26.940","Text":"Out of the 4 children, not Jane,"},{"Start":"10:26.940 ","End":"10:34.055","Text":"we have to choose 1 child that received 2 out of the 3 presents,"},{"Start":"10:34.055 ","End":"10:40.515","Text":"and we have to choose another child that received only 1 present."},{"Start":"10:40.515 ","End":"10:42.880","Text":"We have to choose another present here."},{"Start":"10:42.880 ","End":"10:47.550","Text":"Of the 3 divided by 125,"},{"Start":"10:47.550 ","End":"10:52.435","Text":"and that equals to 36 divided by 125,"},{"Start":"10:52.435 ","End":"10:54.745","Text":"so we\u0027ll put that here."},{"Start":"10:54.745 ","End":"11:00.950","Text":"That\u0027ll be 36 divided by 125."},{"Start":"11:01.050 ","End":"11:06.765","Text":"Now we can just add up our probabilities here."},{"Start":"11:06.765 ","End":"11:11.905","Text":"If we add up the probabilities of the joint probabilities here,"},{"Start":"11:11.905 ","End":"11:15.370","Text":"it should add up to the marginal probability here."},{"Start":"11:15.370 ","End":"11:20.480","Text":"Here, we have 36 divided by 125,"},{"Start":"11:20.480 ","End":"11:26.360","Text":"and here we\u0027ll have 24 divided by 125."},{"Start":"11:26.360 ","End":"11:29.570","Text":"Excellent. Now that we have all the joint probabilities,"},{"Start":"11:29.570 ","End":"11:33.700","Text":"we can calculate the marginal probabilities of Y."},{"Start":"11:33.700 ","End":"11:41.885","Text":"The sum of these joint probabilities in this row add up to 60 divided by 125."},{"Start":"11:41.885 ","End":"11:48.930","Text":"Here, that also adds up to 60 divided by 125,"},{"Start":"11:48.930 ","End":"11:52.845","Text":"and here that\u0027ll be 5 divided by 125."},{"Start":"11:52.845 ","End":"11:59.070","Text":"There we go. This then is the joint probability distribution of X and Y."}],"ID":13230},{"Watched":false,"Name":"Exercise 5 Parts b-c","Duration":"5m 6s","ChapterTopicVideoID":12752,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12752.jpeg","UploadDate":"2019-02-11T14:34:16.4330000","DurationForVideoObject":"PT5M6S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.885","Text":"In this section, we\u0027re asked whether X and Y are uncorrelated."},{"Start":"00:03.885 ","End":"00:06.990","Text":"Well, in order for X and Y to be uncorrelated,"},{"Start":"00:06.990 ","End":"00:10.230","Text":"then the correlation coefficient has to be equal to 0."},{"Start":"00:10.230 ","End":"00:12.630","Text":"Now, in order for that to be equal to 0,"},{"Start":"00:12.630 ","End":"00:17.910","Text":"that means that the covariance of X and Y has to be equal to 0."},{"Start":"00:17.910 ","End":"00:19.665","Text":"Now, what does that mean?"},{"Start":"00:19.665 ","End":"00:22.455","Text":"What\u0027s the covariance of X and Y?"},{"Start":"00:22.455 ","End":"00:26.505","Text":"Well, that equals to the expectation of X times Y"},{"Start":"00:26.505 ","End":"00:32.310","Text":"minus the expectation of X times the expectation of Y."},{"Start":"00:32.310 ","End":"00:37.200","Text":"Okay, so that means I have to calculate the expectation of X,"},{"Start":"00:37.200 ","End":"00:40.555","Text":"the expectation of Y, and the expectation of XY."},{"Start":"00:40.555 ","End":"00:42.020","Text":"So let\u0027s get started."},{"Start":"00:42.020 ","End":"00:47.245","Text":"We know that X has a binomial distribution where n equals 3,"},{"Start":"00:47.245 ","End":"00:50.290","Text":"and p equals to 1/5."},{"Start":"00:50.290 ","End":"00:54.200","Text":"Now, because X has a binomial distribution,"},{"Start":"00:54.200 ","End":"00:56.195","Text":"then the expectation of X,"},{"Start":"00:56.195 ","End":"00:58.445","Text":"well that equals to n times p,"},{"Start":"00:58.445 ","End":"01:01.100","Text":"that equals to 3 times 1/5,"},{"Start":"01:01.100 ","End":"01:03.590","Text":"and that equals to 0.6."},{"Start":"01:03.590 ","End":"01:07.820","Text":"Now, what about the expectation of Y?"},{"Start":"01:07.820 ","End":"01:14.060","Text":"Well, here, that\u0027s the sum of Y times the probability of Y."},{"Start":"01:14.060 ","End":"01:16.235","Text":"Let\u0027s get to work here."},{"Start":"01:16.235 ","End":"01:22.405","Text":"That\u0027ll be 2 times 2 times 60/125,"},{"Start":"01:22.405 ","End":"01:25.080","Text":"2 times 60/125,"},{"Start":"01:25.080 ","End":"01:35.435","Text":"plus 3 times 60/125 plus 4 times 5/125."},{"Start":"01:35.435 ","End":"01:40.955","Text":"Now that equals to 2.56."},{"Start":"01:40.955 ","End":"01:44.750","Text":"So this then is the expectation of Y."},{"Start":"01:44.750 ","End":"01:48.785","Text":"Now, what about the expectation of XY?"},{"Start":"01:48.785 ","End":"01:57.035","Text":"Well, that\u0027s the sum of XY times the joint probability of X and Y."},{"Start":"01:57.035 ","End":"02:03.545","Text":"Now, we can get rid of this column right here because X equals 0."},{"Start":"02:03.545 ","End":"02:06.590","Text":"Let\u0027s just do the work right here."},{"Start":"02:06.590 ","End":"02:11.580","Text":"That means that it\u0027s 1 times 2 times the probability."},{"Start":"02:11.580 ","End":"02:17.540","Text":"So it\u0027ll be 1 times 2 times 36/125."},{"Start":"02:17.540 ","End":"02:19.910","Text":"Here, we have 0."},{"Start":"02:19.910 ","End":"02:22.575","Text":"So then we add up here."},{"Start":"02:22.575 ","End":"02:24.860","Text":"That\u0027ll be 1 times 3,"},{"Start":"02:24.860 ","End":"02:29.900","Text":"1 times 3 times 12/125"},{"Start":"02:29.900 ","End":"02:36.610","Text":"plus 2 times 3 times 12/125."},{"Start":"02:36.610 ","End":"02:38.500","Text":"Here we have zeros,"},{"Start":"02:38.500 ","End":"02:45.415","Text":"and here we have 3 times 4 times 1/125."},{"Start":"02:45.415 ","End":"02:51.630","Text":"Now that equals to 1.536."},{"Start":"02:51.630 ","End":"02:55.960","Text":"Okay, so we have all 3 expectations for the covariance."},{"Start":"02:55.960 ","End":"02:59.410","Text":"The covariance of X and Y,"},{"Start":"02:59.410 ","End":"03:01.990","Text":"what that equals to the expectation of X times Y,"},{"Start":"03:01.990 ","End":"03:07.780","Text":"that\u0027s 1.536 minus the expectation of X."},{"Start":"03:07.780 ","End":"03:13.690","Text":"So that\u0027s 0.6 times the expectation of Y. That\u0027s 2.56."},{"Start":"03:13.690 ","End":"03:17.165","Text":"Now, that equals 0,"},{"Start":"03:17.165 ","End":"03:23.460","Text":"that means that X and Y are uncorrelated."},{"Start":"03:24.550 ","End":"03:30.320","Text":"In this section, we\u0027re asked to calculate the expectation of X times Y squared."},{"Start":"03:30.320 ","End":"03:34.504","Text":"Well, the expectation of X times Y squared,"},{"Start":"03:34.504 ","End":"03:42.545","Text":"that equals to the sum of X times Y squared times the joint probability of X and Y."},{"Start":"03:42.545 ","End":"03:45.410","Text":"Now, if that\u0027s the case,"},{"Start":"03:45.410 ","End":"03:49.370","Text":"then we can disregard this column because here X equals"},{"Start":"03:49.370 ","End":"03:53.210","Text":"0 and that doesn\u0027t contribute anything to the expectation."},{"Start":"03:53.210 ","End":"03:58.100","Text":"Likewise, we have joint probabilities that are equal to 0."},{"Start":"03:58.100 ","End":"04:04.715","Text":"So we can disregard the combinations of X and Y where the joint probability is 0."},{"Start":"04:04.715 ","End":"04:06.950","Text":"So let\u0027s get started."},{"Start":"04:06.950 ","End":"04:14.940","Text":"They\u0027ll be 1, X equals 1 times 2 squared, Y equals 2."},{"Start":"04:14.940 ","End":"04:18.435","Text":"So we\u0027re looking at Y squared times the joint probability."},{"Start":"04:18.435 ","End":"04:24.720","Text":"That\u0027s 36/125 plus 1 times now 3"},{"Start":"04:24.720 ","End":"04:31.860","Text":"squared times 12/125 plus,"},{"Start":"04:31.860 ","End":"04:38.270","Text":"now we\u0027re here, 2 times 3 squared times"},{"Start":"04:38.270 ","End":"04:48.685","Text":"12/125 plus 3 times 4 squared times 1/125."},{"Start":"04:48.685 ","End":"04:51.600","Text":"All the rest of the combinations,"},{"Start":"04:51.600 ","End":"04:56.795","Text":"we have a 0 somewhere in the joint probability or where X equals 0."},{"Start":"04:56.795 ","End":"05:01.820","Text":"So that comes out to 4.128."},{"Start":"05:01.820 ","End":"05:06.540","Text":"So this then is the expectation of X times Y squared."}],"ID":13231},{"Watched":false,"Name":"Exercise 6","Duration":"2m 3s","ChapterTopicVideoID":12753,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12753.jpeg","UploadDate":"2019-02-11T14:34:33.2670000","DurationForVideoObject":"PT2M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.720","Text":"In this question, we\u0027re asked to determine which of the sentences"},{"Start":"00:02.720 ","End":"00:05.535","Text":"below are true and to explain our answer."},{"Start":"00:05.535 ","End":"00:08.040","Text":"Let\u0027s take a look at the first sentence."},{"Start":"00:08.040 ","End":"00:12.750","Text":"If 2 variables are correlated, then they\u0027re dependent."},{"Start":"00:12.750 ","End":"00:16.350","Text":"If 2 variables are correlated,"},{"Start":"00:16.350 ","End":"00:19.545","Text":"then they have some form of linear connection between them."},{"Start":"00:19.545 ","End":"00:22.185","Text":"That\u0027s the definition of correlated variables."},{"Start":"00:22.185 ","End":"00:25.920","Text":"If we can show some form of linear connection,"},{"Start":"00:25.920 ","End":"00:28.350","Text":"that means they\u0027re dependent,"},{"Start":"00:28.350 ","End":"00:30.525","Text":"we have a linear dependency."},{"Start":"00:30.525 ","End":"00:33.255","Text":"Therefore, this statement right here is true."},{"Start":"00:33.255 ","End":"00:35.320","Text":"What about the next statement?"},{"Start":"00:35.320 ","End":"00:40.300","Text":"If 2 variables are dependent, then they\u0027re correlated."},{"Start":"00:40.300 ","End":"00:43.745","Text":"If 2 variables are dependent,"},{"Start":"00:43.745 ","End":"00:47.090","Text":"then they have some form of connection between them."},{"Start":"00:47.090 ","End":"00:50.660","Text":"But it\u0027s not necessarily a linear 1."},{"Start":"00:50.660 ","End":"00:55.490","Text":"Therefore, this sentence isn\u0027t true because correlated"},{"Start":"00:55.490 ","End":"01:01.445","Text":"variable means that we have a linear connection between the 2 variables."},{"Start":"01:01.445 ","End":"01:04.625","Text":"What about the next statement?"},{"Start":"01:04.625 ","End":"01:09.764","Text":"If 2 variables are independent, then they\u0027re uncorrelated."},{"Start":"01:09.764 ","End":"01:13.325","Text":"If 2 variables are independent,"},{"Start":"01:13.325 ","End":"01:16.505","Text":"then they aren\u0027t connected in any way whatsoever."},{"Start":"01:16.505 ","End":"01:21.365","Text":"That means that there\u0027s also no linear connection whatsoever."},{"Start":"01:21.365 ","End":"01:25.235","Text":"Since there\u0027s no linear connection, they\u0027re uncorrelated."},{"Start":"01:25.235 ","End":"01:27.635","Text":"Therefore, this sentence is true."},{"Start":"01:27.635 ","End":"01:29.550","Text":"What about the last sentence?"},{"Start":"01:29.550 ","End":"01:32.345","Text":"If 2 variables are uncorrelated,"},{"Start":"01:32.345 ","End":"01:34.834","Text":"then they are independent."},{"Start":"01:34.834 ","End":"01:38.570","Text":"If we have uncorrelated variables,"},{"Start":"01:38.570 ","End":"01:41.570","Text":"that means that there\u0027s no linear connection whatsoever,"},{"Start":"01:41.570 ","End":"01:44.360","Text":"but that doesn\u0027t mean that the variables are independent."},{"Start":"01:44.360 ","End":"01:49.350","Text":"I\u0027ll give you an example of the function y equals x squared."},{"Start":"01:49.350 ","End":"01:56.284","Text":"We have here uncorrelated variables because this connection is not a linear connection,"},{"Start":"01:56.284 ","End":"01:59.695","Text":"but they\u0027re very much dependent on each other."},{"Start":"01:59.695 ","End":"02:03.820","Text":"Therefore, this statement is not true."}],"ID":13232},{"Watched":false,"Name":"Exercise 7","Duration":"5m 47s","ChapterTopicVideoID":12754,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12754.jpeg","UploadDate":"2019-02-11T14:34:52.7300000","DurationForVideoObject":"PT5M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.810","Text":"In this question, there are 50 employees working at the ACME factory,"},{"Start":"00:03.810 ","End":"00:06.045","Text":"25 men and 25 women."},{"Start":"00:06.045 ","End":"00:09.164","Text":"Each employee is asked to choose a present for Christmas."},{"Start":"00:09.164 ","End":"00:14.565","Text":"The employees have 5 different presents to choose from and must choose only 1 present."},{"Start":"00:14.565 ","End":"00:17.100","Text":"Each employee randomly chooses 1 present,"},{"Start":"00:17.100 ","End":"00:18.930","Text":"independent of other employees."},{"Start":"00:18.930 ","End":"00:21.675","Text":"The following variables are defined."},{"Start":"00:21.675 ","End":"00:25.395","Text":"Xi is the number of men who choose present i"},{"Start":"00:25.395 ","End":"00:29.565","Text":"and Yi is the number of women who choose present i."},{"Start":"00:29.565 ","End":"00:33.990","Text":"We\u0027re asked, are X1 and Y1 independent variables?"},{"Start":"00:33.990 ","End":"00:38.700","Text":"We\u0027re asked to explain our answer without doing any calculations."},{"Start":"00:38.700 ","End":"00:41.340","Text":"Let\u0027s assume, for example,"},{"Start":"00:41.340 ","End":"00:48.450","Text":"that present number 1 is a book."},{"Start":"00:48.450 ","End":"00:53.025","Text":"What\u0027s X1 and what\u0027s Y1?"},{"Start":"00:53.025 ","End":"00:56.300","Text":"X1 is the number of men who chose a book,"},{"Start":"00:56.300 ","End":"00:58.760","Text":"and Y1 is the number of women who chose a book."},{"Start":"00:58.760 ","End":"01:02.270","Text":"Now we\u0027re given that each employee randomly chooses"},{"Start":"01:02.270 ","End":"01:05.840","Text":"1 present independent of other employees."},{"Start":"01:05.840 ","End":"01:09.125","Text":"If we take a specific man,"},{"Start":"01:09.125 ","End":"01:16.655","Text":"he chooses a book independently of other men and also of other women, and vice versa."},{"Start":"01:16.655 ","End":"01:21.995","Text":"A woman chooses a book independent of other women and of other men."},{"Start":"01:21.995 ","End":"01:28.230","Text":"That means that there\u0027s no connection between X1 and Y1."},{"Start":"01:28.230 ","End":"01:34.350","Text":"That means that X1 and Y1 are independent."},{"Start":"01:36.610 ","End":"01:41.750","Text":"In this section, we\u0027re asked whether X1 and X2 are independent variables."},{"Start":"01:41.750 ","End":"01:44.825","Text":"We\u0027re asked to explain our answer without calculations."},{"Start":"01:44.825 ","End":"01:46.984","Text":"Well, If we recall,"},{"Start":"01:46.984 ","End":"01:52.290","Text":"the example for present number 1 was a book,"},{"Start":"01:52.290 ","End":"01:54.450","Text":"present number 1 was a book."},{"Start":"01:54.450 ","End":"01:58.950","Text":"What about present number 2?"},{"Start":"01:58.950 ","End":"02:04.505","Text":"Let\u0027s say the present number 2 is a bottle of wine."},{"Start":"02:04.505 ","End":"02:07.115","Text":"Okay, so what\u0027s X1?"},{"Start":"02:07.115 ","End":"02:13.655","Text":"X1 is the number of men who chose a book, and what\u0027s X2?"},{"Start":"02:13.655 ","End":"02:18.855","Text":"That\u0027s the number of men who chose the bottle of wine."},{"Start":"02:18.855 ","End":"02:23.300","Text":"Is there a connection between X1 and X2?"},{"Start":"02:23.300 ","End":"02:25.490","Text":"Do they influence 1 another?"},{"Start":"02:25.490 ","End":"02:30.150","Text":"We have a fixed amount of men in the ACME factory."},{"Start":"02:30.150 ","End":"02:31.965","Text":"We have 25 men."},{"Start":"02:31.965 ","End":"02:36.015","Text":"As more men choose present number 1,"},{"Start":"02:36.015 ","End":"02:40.020","Text":"the book then less men will choose wine and vice versa,"},{"Start":"02:40.020 ","End":"02:44.450","Text":"as more men choose the wine then less men will choose a book."},{"Start":"02:44.450 ","End":"02:48.820","Text":"That means that there\u0027s an influence of 1 variable on the other."},{"Start":"02:48.820 ","End":"02:50.405","Text":"If that\u0027s the case,"},{"Start":"02:50.405 ","End":"02:52.745","Text":"then X1 and X2,"},{"Start":"02:52.745 ","End":"02:56.370","Text":"they\u0027re dependent on each other."},{"Start":"02:57.940 ","End":"03:00.065","Text":"In this section, we\u0027re asked,"},{"Start":"03:00.065 ","End":"03:03.125","Text":"what\u0027s the distribution of X1 plus X2?"},{"Start":"03:03.125 ","End":"03:07.130","Text":"X1 plus X2. What does that mean?"},{"Start":"03:07.130 ","End":"03:12.360","Text":"That\u0027s the number of men who chose either present"},{"Start":"03:12.360 ","End":"03:19.150","Text":"1 or present number 2."},{"Start":"03:19.150 ","End":"03:23.540","Text":"Now, let\u0027s take a look at what they\u0027re doing,"},{"Start":"03:23.540 ","End":"03:27.920","Text":"each man chooses independently of every other man a present."},{"Start":"03:27.920 ","End":"03:30.380","Text":"Well, that\u0027s a Bernoulli trial."},{"Start":"03:30.380 ","End":"03:32.255","Text":"What\u0027s success?"},{"Start":"03:32.255 ","End":"03:37.385","Text":"Success is that we choose either present 1 or present 2."},{"Start":"03:37.385 ","End":"03:39.665","Text":"How many men do we have?"},{"Start":"03:39.665 ","End":"03:44.075","Text":"Well, we have 25, so we have n equals to 25."},{"Start":"03:44.075 ","End":"03:49.020","Text":"We have success defined as choosing present 1 or present"},{"Start":"03:49.020 ","End":"03:55.545","Text":"2 and the probability of success is 2/5."},{"Start":"03:55.545 ","End":"03:58.470","Text":"We have 5 presents and we choose either"},{"Start":"03:58.470 ","End":"04:02.370","Text":"present 1 or present 2 out of the 5 so that\u0027s 2/5."},{"Start":"04:02.370 ","End":"04:06.290","Text":"So X1 plus X2."},{"Start":"04:06.290 ","End":"04:12.140","Text":"Well, that has a binomial distribution where n equals to 25 and p,"},{"Start":"04:12.140 ","End":"04:16.080","Text":"probability equals to 2/5."},{"Start":"04:16.510 ","End":"04:19.310","Text":"In this section, we\u0027re asked whether there is"},{"Start":"04:19.310 ","End":"04:22.280","Text":"a full or partial correlation between X1 and X2,"},{"Start":"04:22.280 ","End":"04:24.970","Text":"whether it\u0027s positive or negative and we\u0027re asked to"},{"Start":"04:24.970 ","End":"04:28.055","Text":"explain our answer without the need for calculations."},{"Start":"04:28.055 ","End":"04:32.585","Text":"Okay, so let\u0027s just review our definition of X1 and X2."},{"Start":"04:32.585 ","End":"04:36.530","Text":"We see here that X1 is the number of men who"},{"Start":"04:36.530 ","End":"04:40.865","Text":"chose present number 1 and X2 is the number of men who chose present number 2."},{"Start":"04:40.865 ","End":"04:45.770","Text":"Now, there\u0027s no linear equation connecting X1 and X2."},{"Start":"04:45.770 ","End":"04:49.610","Text":"If we know the number of men who chose present number 1 we don\u0027t know for"},{"Start":"04:49.610 ","End":"04:53.855","Text":"sure that the number of men who chose present number 2."},{"Start":"04:53.855 ","End":"04:56.135","Text":"Why is that? Because there are other presents."},{"Start":"04:56.135 ","End":"04:59.815","Text":"Present number 2, present number 3, 4, and 5."},{"Start":"04:59.815 ","End":"05:05.960","Text":"So the correlation here is a partial correlation."},{"Start":"05:05.960 ","End":"05:07.760","Text":"What else?"},{"Start":"05:07.760 ","End":"05:16.815","Text":"Well, we know then that we have a fixed amount of men in the factory and"},{"Start":"05:16.815 ","End":"05:21.210","Text":"we expect that if more men choose present number 1"},{"Start":"05:21.210 ","End":"05:26.000","Text":"then less men will choose present number 2 and vice versa."},{"Start":"05:26.000 ","End":"05:30.755","Text":"The expectation is that if more men choose present number 2,"},{"Start":"05:30.755 ","End":"05:34.085","Text":"then less men will choose present number 1."},{"Start":"05:34.085 ","End":"05:38.380","Text":"The correlation is a negative 1."},{"Start":"05:38.380 ","End":"05:47.190","Text":"What we have here is a negative partial correlation between X1 and X2."}],"ID":13233},{"Watched":false,"Name":"Exercise 8","Duration":"4m 35s","ChapterTopicVideoID":12755,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12755.jpeg","UploadDate":"2019-02-11T14:35:08.9530000","DurationForVideoObject":"PT4M35S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:01.620","Text":"In this question, we\u0027re asked to prove"},{"Start":"00:01.620 ","End":"00:04.695","Text":"the following equations for the 3 variables X, Y, and Z."},{"Start":"00:04.695 ","End":"00:07.905","Text":"The covariance of X plus Y and Z,"},{"Start":"00:07.905 ","End":"00:13.545","Text":"that equals to the covariance of X and Z plus the covariance of Y and Z."},{"Start":"00:13.545 ","End":"00:17.400","Text":"If we remember the definition of covariance,"},{"Start":"00:17.400 ","End":"00:21.165","Text":"the covariance of let\u0027s say 2 variables W and T,"},{"Start":"00:21.165 ","End":"00:27.495","Text":"that equals to the expectation of W times T minus the expectation"},{"Start":"00:27.495 ","End":"00:34.560","Text":"of W times the expectation of T. Let\u0027s just use this definition here."},{"Start":"00:34.560 ","End":"00:38.303","Text":"Instead of W we\u0027ll plug in X plus Y,"},{"Start":"00:38.303 ","End":"00:40.170","Text":"and instead of T,"},{"Start":"00:40.170 ","End":"00:41.670","Text":"we\u0027ll plug in Z."},{"Start":"00:41.670 ","End":"00:43.740","Text":"That\u0027ll be equal to the following,"},{"Start":"00:43.740 ","End":"00:48.595","Text":"a covariance of x plus y and z."},{"Start":"00:48.595 ","End":"00:56.725","Text":"That equals to the expectation of x plus y times z"},{"Start":"00:56.725 ","End":"01:07.630","Text":"minus the expectation of x plus y times the expectation of z."},{"Start":"01:08.030 ","End":"01:13.070","Text":"Let\u0027s just open up these brackets right here and that\u0027ll be equal"},{"Start":"01:13.070 ","End":"01:17.360","Text":"to the expectation of xz"},{"Start":"01:17.360 ","End":"01:23.190","Text":"plus yz minus the expectation"},{"Start":"01:23.190 ","End":"01:28.150","Text":"of x plus y times the expectation of z."},{"Start":"01:28.150 ","End":"01:34.460","Text":"What\u0027s another characteristic of the expectation of the sum?"},{"Start":"01:34.460 ","End":"01:38.105","Text":"The expectation of W plus T,"},{"Start":"01:38.105 ","End":"01:42.530","Text":"that always equals to the expectation of W plus"},{"Start":"01:42.530 ","End":"01:47.510","Text":"the expectation of T. Let\u0027s use this characteristic here."},{"Start":"01:47.510 ","End":"01:56.940","Text":"We have this. We have the expectation of xz plus the expectation of yz"},{"Start":"01:56.940 ","End":"02:07.150","Text":"minus the expectation of x plus the expectation of y."},{"Start":"02:07.150 ","End":"02:13.050","Text":"This thing has to be multiplied by the expectation of z."},{"Start":"02:13.550 ","End":"02:22.260","Text":"That equals to the expectation of xz plus the expectation of yz."},{"Start":"02:22.260 ","End":"02:26.835","Text":"Let\u0027s multiply the expectation of z into these guys."},{"Start":"02:26.835 ","End":"02:28.760","Text":"Let\u0027s open up these brackets."},{"Start":"02:28.760 ","End":"02:35.690","Text":"Minus then the expectation of x times the expectation of"},{"Start":"02:35.690 ","End":"02:45.900","Text":"z plus the expectation of y times the expectation of z."},{"Start":"02:46.910 ","End":"02:52.710","Text":"The square brackets,"},{"Start":"02:52.710 ","End":"02:54.330","Text":"let\u0027s open these guys up."},{"Start":"02:54.330 ","End":"03:02.895","Text":"That\u0027ll be equal to the expectation of xz plus the expectation of yz minus"},{"Start":"03:02.895 ","End":"03:07.230","Text":"the expectation of x times the expectation of"},{"Start":"03:07.230 ","End":"03:14.175","Text":"z minus the expectation of y times the expectation of z."},{"Start":"03:14.175 ","End":"03:17.825","Text":"Let\u0027s just rearrange these guys here."},{"Start":"03:17.825 ","End":"03:21.880","Text":"That\u0027ll be equal to the expectation of x times z,"},{"Start":"03:21.880 ","End":"03:23.940","Text":"that\u0027s this guy right here,"},{"Start":"03:23.940 ","End":"03:29.895","Text":"minus the expectation of x times the expectation of z."},{"Start":"03:29.895 ","End":"03:32.105","Text":"That\u0027s these guys here."},{"Start":"03:32.105 ","End":"03:36.605","Text":"Plus the expectation of y times z,"},{"Start":"03:36.605 ","End":"03:40.370","Text":"that\u0027s this right here, minus this,"},{"Start":"03:40.370 ","End":"03:45.510","Text":"the expectation of y times the expectation of z."},{"Start":"03:46.280 ","End":"03:49.240","Text":"What do we have here?"},{"Start":"03:49.240 ","End":"03:52.610","Text":"This thing right here,"},{"Start":"03:52.610 ","End":"03:55.670","Text":"that\u0027s the definition of a covariance."},{"Start":"03:55.670 ","End":"04:01.455","Text":"That equals to the covariance of x and z."},{"Start":"04:01.455 ","End":"04:07.140","Text":"The expectation of x times z minus the expectation of x times the expectation of z,"},{"Start":"04:07.140 ","End":"04:10.945","Text":"that\u0027s the covariance of x and z plus,"},{"Start":"04:10.945 ","End":"04:13.700","Text":"likewise here, the expectation of y"},{"Start":"04:13.700 ","End":"04:17.315","Text":"times z minus the expectation of y times the expectation of z,"},{"Start":"04:17.315 ","End":"04:22.385","Text":"that equals to the covariance of y and z."},{"Start":"04:22.385 ","End":"04:24.650","Text":"This is what we proved."},{"Start":"04:24.650 ","End":"04:28.445","Text":"The covariance of X plus Y and Z,"},{"Start":"04:28.445 ","End":"04:35.550","Text":"that equals to the covariance of X and Z plus the covariance of Y and Z."}],"ID":13234},{"Watched":false,"Name":"Exercise 9 Part a","Duration":"10m 51s","ChapterTopicVideoID":12756,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12756.jpeg","UploadDate":"2019-02-11T14:35:42.9400000","DurationForVideoObject":"PT10M51S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.280","Text":"In this question we\u0027re given that in autumn,"},{"Start":"00:02.280 ","End":"00:04.260","Text":"the number of leaves falling at a garden has"},{"Start":"00:04.260 ","End":"00:08.295","Text":"a Poisson distribution with an expectation of 50 leaves per minute."},{"Start":"00:08.295 ","End":"00:10.785","Text":"Now, the following variables are defined."},{"Start":"00:10.785 ","End":"00:14.100","Text":"Y is the number of leaves falling between 12:00 and"},{"Start":"00:14.100 ","End":"00:19.800","Text":"12:10 and Q is the number of leaves falling between 12:05 and 12:30."},{"Start":"00:19.800 ","End":"00:25.080","Text":"We\u0027re asked to calculate the covariance of 4Y and Q plus 6."},{"Start":"00:25.080 ","End":"00:31.485","Text":"Let\u0027s take a look at Y. Y has a Poisson distribution."},{"Start":"00:31.485 ","End":"00:33.720","Text":"Now, what\u0027s the value of Lambda?"},{"Start":"00:33.720 ","End":"00:38.270","Text":"Well, we learned that the rate Lambda is proportional to time,"},{"Start":"00:38.270 ","End":"00:43.410","Text":"so that if leaves fall at a rate of 50 leaves per minute,"},{"Start":"00:43.490 ","End":"00:47.300","Text":"and we have a time frame of 10 minutes."},{"Start":"00:47.300 ","End":"00:51.440","Text":"Well, that\u0027ll be 50 leaves per minute times 10,"},{"Start":"00:51.440 ","End":"00:53.665","Text":"and that equals to 500."},{"Start":"00:53.665 ","End":"00:57.900","Text":"The value of Lambda here would be 500 for Y. Now, what about Q?"},{"Start":"00:57.900 ","End":"01:00.620","Text":"Well, Q also has a Poisson distribution."},{"Start":"01:00.620 ","End":"01:02.405","Text":"What\u0027s the value of its Lambda?"},{"Start":"01:02.405 ","End":"01:06.215","Text":"Well, again, we have 50 leaves per minute, that\u0027s right here."},{"Start":"01:06.215 ","End":"01:08.390","Text":"Now, what about the time frame?"},{"Start":"01:08.390 ","End":"01:10.590","Text":"Well, the time frame here\u0027s 25 minutes,"},{"Start":"01:10.590 ","End":"01:16.825","Text":"so that\u0027ll be times 25 and that would be equals to 1,250."},{"Start":"01:16.825 ","End":"01:19.400","Text":"Now, what are we looking for?"},{"Start":"01:19.400 ","End":"01:26.510","Text":"We\u0027re looking for the covariance of 4Y and Q plus 6."},{"Start":"01:26.510 ","End":"01:28.445","Text":"Now, these are linear transformations."},{"Start":"01:28.445 ","End":"01:33.170","Text":"Now, what have we learned about covariance of linear transformations?"},{"Start":"01:33.170 ","End":"01:41.745","Text":"Well, if we remember the covariance of aX plus b and cY plus d. Well,"},{"Start":"01:41.745 ","End":"01:48.315","Text":"that equal to a times c times the covariance of x, y."},{"Start":"01:48.315 ","End":"01:49.805","Text":"Now, in our case,"},{"Start":"01:49.805 ","End":"01:52.295","Text":"a here is equal to 4."},{"Start":"01:52.295 ","End":"01:54.720","Text":"That\u0027s the y multiplier here,"},{"Start":"01:54.720 ","End":"01:57.330","Text":"and c would be equal to 1."},{"Start":"01:57.330 ","End":"01:59.600","Text":"That\u0027s our Q multiplier here."},{"Start":"01:59.600 ","End":"02:07.620","Text":"Now, that\u0027ll be equal to 4 times 1 times the covariance between Y and Q."},{"Start":"02:07.620 ","End":"02:15.970","Text":"We\u0027re looking for 4 times the covariance of Y and Q."},{"Start":"02:16.160 ","End":"02:24.350","Text":"Here what I\u0027ve done is I\u0027ve mapped out the variables Y and Q on a timeline."},{"Start":"02:24.350 ","End":"02:26.510","Text":"I have here 12:00 o\u0027clock,"},{"Start":"02:26.510 ","End":"02:28.430","Text":"12:05, 12:10, and 12:30."},{"Start":"02:28.430 ","End":"02:32.910","Text":"So Y here we can see goes from 12:00 to 12:10"},{"Start":"02:32.910 ","End":"02:38.345","Text":"and Q goes from 12:05 to 12:30. That\u0027s given."},{"Start":"02:38.345 ","End":"02:41.555","Text":"Now, what do we know about variables with"},{"Start":"02:41.555 ","End":"02:45.140","Text":"overlapping timelines in a Poisson distribution?"},{"Start":"02:45.140 ","End":"02:51.485","Text":"Well, we know that if we have variables that do not have an overlapping timeline,"},{"Start":"02:51.485 ","End":"02:54.020","Text":"then these variables are independent."},{"Start":"02:54.020 ","End":"02:56.000","Text":"Now, these things right here,"},{"Start":"02:56.000 ","End":"02:59.040","Text":"Y and Q are not independent,"},{"Start":"02:59.040 ","End":"03:00.450","Text":"they are dependent and why is that?"},{"Start":"03:00.450 ","End":"03:05.030","Text":"Because they have an overlapping time frame right here from 12:05 to 12:10."},{"Start":"03:05.030 ","End":"03:08.660","Text":"In order to make things easier for us,"},{"Start":"03:08.660 ","End":"03:11.960","Text":"in order to calculate the covariance and so on and so forth,"},{"Start":"03:11.960 ","End":"03:15.860","Text":"we need to define Y and Q in terms"},{"Start":"03:15.860 ","End":"03:20.750","Text":"of new variables that do not have an overlapping timeline."},{"Start":"03:20.750 ","End":"03:24.665","Text":"So here we go. We\u0027ll define X_1,"},{"Start":"03:24.665 ","End":"03:26.810","Text":"X_2, and X_3."},{"Start":"03:26.810 ","End":"03:31.310","Text":"X_1 will be between 12:00 and 12:05,"},{"Start":"03:31.310 ","End":"03:35.713","Text":"X_2 will be between 12:05 and 12:10,"},{"Start":"03:35.713 ","End":"03:39.950","Text":"and X_3 will be between 12:10 and 12:30."},{"Start":"03:39.950 ","End":"03:42.510","Text":"Why did we do this?"},{"Start":"03:42.510 ","End":"03:47.870","Text":"Well now, we have 3 independent variables, X_1, X_2,"},{"Start":"03:47.870 ","End":"03:53.240","Text":"and X_3, and we can define Y in Q in terms of X_1,"},{"Start":"03:53.240 ","End":"03:54.343","Text":"X_2, and X_3,"},{"Start":"03:54.343 ","End":"03:55.445","Text":"and how do we do that?"},{"Start":"03:55.445 ","End":"04:02.205","Text":"Well, Y then is equal to X_1 plus X_2."},{"Start":"04:02.205 ","End":"04:04.695","Text":"X_1 plus X_2, well,"},{"Start":"04:04.695 ","End":"04:11.175","Text":"the time frame from these 2 guys is from 12:00 to 12:10."},{"Start":"04:11.175 ","End":"04:13.935","Text":"X_1 is from 12:00 to 12:05,"},{"Start":"04:13.935 ","End":"04:16.665","Text":"and X_2 is from 12:05 to 12:10."},{"Start":"04:16.665 ","End":"04:23.610","Text":"Q, well, that\u0027s X_2 plus X_3."},{"Start":"04:23.610 ","End":"04:27.135","Text":"Now, Q was from 12:05 to 12:30."},{"Start":"04:27.135 ","End":"04:30.650","Text":"Now, we define that as X_2 plus X_3."},{"Start":"04:30.650 ","End":"04:33.365","Text":"X_2 is from 12:05 to 12:10,"},{"Start":"04:33.365 ","End":"04:36.990","Text":"and X_3 is from 12:10 to 12:30."},{"Start":"04:36.990 ","End":"04:45.035","Text":"This then is the transformation of Y and Q into X_1,"},{"Start":"04:45.035 ","End":"04:47.305","Text":"X_2, and X_3."},{"Start":"04:47.305 ","End":"04:49.655","Text":"If that\u0027s the case,"},{"Start":"04:49.655 ","End":"04:52.160","Text":"what\u0027s the distribution of X_1?"},{"Start":"04:52.160 ","End":"04:54.290","Text":"Well, the distribution of X1,"},{"Start":"04:54.290 ","End":"04:56.930","Text":"well that has a Poisson distribution now,"},{"Start":"04:56.930 ","End":"04:59.855","Text":"where Lambda equals to what?"},{"Start":"04:59.855 ","End":"05:04.770","Text":"Well, we have a rate of 50 leaves per minute times,"},{"Start":"05:04.770 ","End":"05:06.060","Text":"what\u0027s the time frame rate here?"},{"Start":"05:06.060 ","End":"05:07.595","Text":"Well, that\u0027s 5 minutes."},{"Start":"05:07.595 ","End":"05:12.185","Text":"Lambda here for X_1 would be 250."},{"Start":"05:12.185 ","End":"05:14.165","Text":"What about X_2?"},{"Start":"05:14.165 ","End":"05:16.610","Text":"Well, that has a Poisson distribution."},{"Start":"05:16.610 ","End":"05:18.590","Text":"What\u0027s the Lambda for X_2?"},{"Start":"05:18.590 ","End":"05:20.780","Text":"Well, again, that\u0027s 50 leaves per minute."},{"Start":"05:20.780 ","End":"05:24.230","Text":"Now, X_2 also has a time frame of 5 minutes,"},{"Start":"05:24.230 ","End":"05:28.120","Text":"so that also has a value of 250."},{"Start":"05:28.120 ","End":"05:30.400","Text":"Lambda for X_2 is 250."},{"Start":"05:30.400 ","End":"05:32.655","Text":"What about X_3? Well,"},{"Start":"05:32.655 ","End":"05:38.075","Text":"that has a Poisson distribution where Lambda equals to 50 leaves per minute."},{"Start":"05:38.075 ","End":"05:43.390","Text":"Now, how many minutes do we have between 12:10 and 12:30? Well, that\u0027s 20."},{"Start":"05:43.390 ","End":"05:47.025","Text":"Now, that equals to 1,000,"},{"Start":"05:47.025 ","End":"05:48.960","Text":"so here we go."},{"Start":"05:48.960 ","End":"05:51.905","Text":"These are the distributions for X_1,"},{"Start":"05:51.905 ","End":"05:55.175","Text":"X_2, and X_3 and their appropriate Lambdas."},{"Start":"05:55.175 ","End":"05:57.380","Text":"Lambda for X_1 is 250,"},{"Start":"05:57.380 ","End":"06:03.680","Text":"Lambda for X_2 is 250 and Lambda for X_3 is 1,000."},{"Start":"06:03.680 ","End":"06:08.390","Text":"Now that we\u0027ve expressed Y and Q in terms of X_1,"},{"Start":"06:08.390 ","End":"06:09.755","Text":"X_2, and X_3,"},{"Start":"06:09.755 ","End":"06:11.735","Text":"well, let\u0027s continue from here."},{"Start":"06:11.735 ","End":"06:16.235","Text":"This then will continue from this right here,"},{"Start":"06:16.235 ","End":"06:19.886","Text":"that becomes 4 times the covariance."},{"Start":"06:19.886 ","End":"06:25.650","Text":"Now of Y and Y is X_1 plus X_2."},{"Start":"06:25.650 ","End":"06:29.560","Text":"Q that becomes X_2 plus X_3."},{"Start":"06:29.770 ","End":"06:37.130","Text":"Excellent. This now becomes much easier to deal with. Now, why is that?"},{"Start":"06:37.130 ","End":"06:44.660","Text":"Well, let\u0027s continue and see how we can calculate the covariance of this expression."},{"Start":"06:44.660 ","End":"06:50.435","Text":"Now, before we do that let\u0027s just recall what we did in the last section."},{"Start":"06:50.435 ","End":"06:55.760","Text":"We found out that the covariance of X plus Y was Z,"},{"Start":"06:55.760 ","End":"07:03.070","Text":"that equals to the covariance of X and Z plus the covariance of Y and Z."},{"Start":"07:03.070 ","End":"07:07.100","Text":"Now, let\u0027s use this characteristic here."},{"Start":"07:07.100 ","End":"07:10.580","Text":"Again, let\u0027s just take a look at the covariance right here."},{"Start":"07:10.580 ","End":"07:18.080","Text":"Now the covariance of X_1 plus X_2 with X_2 plus X_3."},{"Start":"07:18.080 ","End":"07:26.510","Text":"Well, that equals now to the covariance of X_1 with X_2 plus"},{"Start":"07:26.510 ","End":"07:36.090","Text":"X_3 plus the covariance of X_2 with X_2 plus X_3."},{"Start":"07:36.090 ","End":"07:41.410","Text":"Now, what else do we know about the characteristics of a covariance?"},{"Start":"07:41.410 ","End":"07:44.245","Text":"Well, the covariance of X and Y,"},{"Start":"07:44.245 ","End":"07:47.770","Text":"but that equals to the covariance of Y and X."},{"Start":"07:47.770 ","End":"07:49.240","Text":"We can switch things around."},{"Start":"07:49.240 ","End":"07:51.205","Text":"Let\u0027s do that here."},{"Start":"07:51.205 ","End":"07:57.985","Text":"Now, that equals to the covariance of X_2 plus X_3 with"},{"Start":"07:57.985 ","End":"08:06.130","Text":"X_1 plus the covariance of X_2 plus X_3 with X_2."},{"Start":"08:06.130 ","End":"08:10.035","Text":"Now again, let\u0027s use this characteristic right here."},{"Start":"08:10.035 ","End":"08:14.750","Text":"That equals to the covariance of X_2 and"},{"Start":"08:14.750 ","End":"08:21.880","Text":"X_1 plus the covariance of X_3 and X_1 plus,"},{"Start":"08:21.880 ","End":"08:23.360","Text":"now here we go here,"},{"Start":"08:23.360 ","End":"08:28.960","Text":"the covariance of X_2 with X_2,"},{"Start":"08:28.960 ","End":"08:36.480","Text":"plus the covariance of X_3 and X_2."},{"Start":"08:36.480 ","End":"08:40.940","Text":"Now, what do we know about X_1,"},{"Start":"08:40.940 ","End":"08:42.155","Text":"X_2, and X_3?"},{"Start":"08:42.155 ","End":"08:46.940","Text":"Well, again, they all have a Poisson distribution."},{"Start":"08:46.940 ","End":"08:49.010","Text":"Now, what else do we know?"},{"Start":"08:49.010 ","End":"08:52.805","Text":"We know that they do not have overlapping time frames."},{"Start":"08:52.805 ","End":"08:56.420","Text":"Now, one of the characteristics of a Poisson distribution is that we"},{"Start":"08:56.420 ","End":"08:59.930","Text":"have variables that do not have overlapping time frames,"},{"Start":"08:59.930 ","End":"09:01.760","Text":"then they are independent."},{"Start":"09:01.760 ","End":"09:03.590","Text":"Now, if they\u0027re independent,"},{"Start":"09:03.590 ","End":"09:05.690","Text":"then they\u0027re obviously uncorrelated."},{"Start":"09:05.690 ","End":"09:11.390","Text":"If they\u0027re uncorrelated, that means that their covariance is 0."},{"Start":"09:11.390 ","End":"09:15.110","Text":"What does this boil down to?"},{"Start":"09:15.110 ","End":"09:18.030","Text":"Well, the covariance of X_2 and X_1."},{"Start":"09:18.030 ","End":"09:22.340","Text":"Well that\u0027s 0, because X_1 and X_2 are independent,"},{"Start":"09:22.340 ","End":"09:25.415","Text":"therefore uncorrelated, therefore the covariance is 0."},{"Start":"09:25.415 ","End":"09:29.345","Text":"The covariance of X_3 and X_1 is also 0."},{"Start":"09:29.345 ","End":"09:32.240","Text":"The covariance of X_3 and X_2, well that\u0027s 0."},{"Start":"09:32.240 ","End":"09:37.940","Text":"This boils down to the covariance of X_2 with X_2."},{"Start":"09:37.940 ","End":"09:42.065","Text":"Now, the covariance of X with X,"},{"Start":"09:42.065 ","End":"09:45.635","Text":"that equals to the variance of X."},{"Start":"09:45.635 ","End":"09:51.965","Text":"That then equals to the variance of X_2."},{"Start":"09:51.965 ","End":"09:56.660","Text":"Now, let\u0027s take a look at the variable X_2."},{"Start":"09:56.660 ","End":"10:00.200","Text":"The variance of X_2 where X_2 has"},{"Start":"10:00.200 ","End":"10:05.000","Text":"a Poisson distribution with parameter Lambda equaling 250."},{"Start":"10:05.000 ","End":"10:11.285","Text":"Well, the variance of this thing equals to Lambda, 250."},{"Start":"10:11.285 ","End":"10:20.105","Text":"Now, we know that the covariance of X_1 plus X_2 with X_2 plus X_3,"},{"Start":"10:20.105 ","End":"10:24.350","Text":"that equals to the variance of X_2 and that equals to 250,"},{"Start":"10:24.350 ","End":"10:27.905","Text":"so 4 times this expression,"},{"Start":"10:27.905 ","End":"10:36.905","Text":"well, that equals to 4 times the variance of X_2."},{"Start":"10:36.905 ","End":"10:40.460","Text":"That means that this is 4 times 250."},{"Start":"10:40.460 ","End":"10:43.225","Text":"That means that this is 1,000."},{"Start":"10:43.225 ","End":"10:45.225","Text":"Here we go,"},{"Start":"10:45.225 ","End":"10:51.910","Text":"the covariance of Y and Q equals 1,000."}],"ID":13235},{"Watched":false,"Name":"Exercise 9 Part b","Duration":"2m 16s","ChapterTopicVideoID":12757,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12757.jpeg","UploadDate":"2019-02-11T14:35:49.6770000","DurationForVideoObject":"PT2M16S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.560","Text":"In this section, we\u0027re asked what\u0027s the correlation coefficient between Y and Q?"},{"Start":"00:04.560 ","End":"00:09.050","Text":"We\u0027re looking for the correlation coefficient between Y and Q."},{"Start":"00:09.050 ","End":"00:14.730","Text":"That\u0027s defined as the covariance between y and"},{"Start":"00:14.730 ","End":"00:22.845","Text":"q divided by the standard deviation of Y times the standard deviation of Q."},{"Start":"00:22.845 ","End":"00:31.010","Text":"Now, y has a plus N distribution where Lambda equals to 500."},{"Start":"00:31.010 ","End":"00:36.260","Text":"That means that the variance of Y,"},{"Start":"00:36.260 ","End":"00:39.440","Text":"well, that equals to 500."},{"Start":"00:39.440 ","End":"00:41.930","Text":"That means that the standard deviation of y,"},{"Start":"00:41.930 ","End":"00:44.855","Text":"that equals to the square root of 500."},{"Start":"00:44.855 ","End":"00:52.555","Text":"Now, likewise, Q has a plus N distribution where Lambda equals to 1,250."},{"Start":"00:52.555 ","End":"00:56.165","Text":"That means that the variance of q,"},{"Start":"00:56.165 ","End":"00:59.315","Text":"well, that equals to 1,250."},{"Start":"00:59.315 ","End":"01:02.150","Text":"That means that the standard deviation of q,"},{"Start":"01:02.150 ","End":"01:05.885","Text":"that equals to the square root of 1,250."},{"Start":"01:05.885 ","End":"01:08.765","Text":"Now, what about the covariance of Y and Q?"},{"Start":"01:08.765 ","End":"01:12.425","Text":"Again, let\u0027s recall what we did in the last section."},{"Start":"01:12.425 ","End":"01:20.205","Text":"We\u0027ve calculated the covariance of 4y and q plus 6."},{"Start":"01:20.205 ","End":"01:26.375","Text":"Now, that turned out to be 4 times the covariance of y and q."},{"Start":"01:26.375 ","End":"01:30.215","Text":"That equal to 4 times 250,"},{"Start":"01:30.215 ","End":"01:32.575","Text":"which equal to 1,000."},{"Start":"01:32.575 ","End":"01:34.550","Text":"Let\u0027s not get confused."},{"Start":"01:34.550 ","End":"01:37.445","Text":"1,000 is not the covariance of y and q,"},{"Start":"01:37.445 ","End":"01:42.155","Text":"it\u0027s the covariance of 4y and q plus 6."},{"Start":"01:42.155 ","End":"01:45.940","Text":"The covariance of y and q is 250."},{"Start":"01:45.940 ","End":"01:50.525","Text":"The correlation coefficient between y and q, well,"},{"Start":"01:50.525 ","End":"01:53.740","Text":"that equals to 250,"},{"Start":"01:53.740 ","End":"01:58.730","Text":"the covariance of y and q divided by the standard deviation of y."},{"Start":"01:58.730 ","End":"02:05.545","Text":"Well, that\u0027s the square root of 500 times the square root of 1,250."},{"Start":"02:05.545 ","End":"02:06.990","Text":"That\u0027s the standard deviation of q, now,"},{"Start":"02:06.990 ","End":"02:11.040","Text":"that equals to 0.316."},{"Start":"02:11.040 ","End":"02:16.440","Text":"This then is the correlation coefficient between Y and Q."}],"ID":13236},{"Watched":false,"Name":"Exercise 10","Duration":"8m 23s","ChapterTopicVideoID":12758,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12758.jpeg","UploadDate":"2019-02-11T14:36:13.4530000","DurationForVideoObject":"PT8M23S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.750","Text":"In this question, we\u0027re given a basket that contains 20 red balls,"},{"Start":"00:03.750 ","End":"00:06.015","Text":"20 green balls, and 20 blue balls."},{"Start":"00:06.015 ","End":"00:08.550","Text":"Now 20 balls are randomly chosen."},{"Start":"00:08.550 ","End":"00:11.670","Text":"We\u0027re asked to calculate the correlation coefficient between"},{"Start":"00:11.670 ","End":"00:16.120","Text":"the number of red balls and the number of green balls that were chosen."},{"Start":"00:16.160 ","End":"00:19.700","Text":"Let\u0027s, first of all, define our random variables."},{"Start":"00:19.700 ","End":"00:20.905","Text":"We have x,"},{"Start":"00:20.905 ","End":"00:24.665","Text":"that\u0027ll be the number of red balls,"},{"Start":"00:24.665 ","End":"00:30.520","Text":"and y would be the number of green balls."},{"Start":"00:30.520 ","End":"00:37.030","Text":"We\u0027re asked to find the correlation coefficient between x and y."},{"Start":"00:37.030 ","End":"00:41.390","Text":"Now that\u0027s defined as the covariance of x and y"},{"Start":"00:41.390 ","End":"00:47.045","Text":"divided by the standard deviation of x times the standard deviation of y."},{"Start":"00:47.045 ","End":"00:52.340","Text":"Now we can calculate the covariance of x and y,"},{"Start":"00:52.340 ","End":"00:53.870","Text":"the standard deviation of x,"},{"Start":"00:53.870 ","End":"00:59.120","Text":"and the standard deviation of y in the regular manner by building"},{"Start":"00:59.120 ","End":"01:03.320","Text":"the probability distribution function between x and"},{"Start":"01:03.320 ","End":"01:08.000","Text":"y and then calculating these parameters."},{"Start":"01:08.000 ","End":"01:14.720","Text":"But that\u0027s a very large table and it\u0027ll be very tedious to work things out."},{"Start":"01:14.720 ","End":"01:17.570","Text":"We need a different way of thinking."},{"Start":"01:17.570 ","End":"01:22.010","Text":"Now let\u0027s take a look at x and let\u0027s take a look at y."},{"Start":"01:22.010 ","End":"01:25.520","Text":"Now after studying these variables,"},{"Start":"01:25.520 ","End":"01:33.335","Text":"we see that x has a hyper-geometric distribution where n,"},{"Start":"01:33.335 ","End":"01:35.840","Text":"our population size, is 60."},{"Start":"01:35.840 ","End":"01:37.190","Text":"We have 20 red balls,"},{"Start":"01:37.190 ","End":"01:39.580","Text":"20 greenhouse, and 20 blue balls."},{"Start":"01:39.580 ","End":"01:43.475","Text":"The total number of balls in a basket is 60."},{"Start":"01:43.475 ","End":"01:48.080","Text":"Now here we\u0027re looking at a sub-population of red balls."},{"Start":"01:48.080 ","End":"01:50.015","Text":"Now, how many red balls do we have?"},{"Start":"01:50.015 ","End":"01:51.720","Text":"Well, we have 20,"},{"Start":"01:51.720 ","End":"01:53.895","Text":"so D here equals to 20."},{"Start":"01:53.895 ","End":"01:55.610","Text":"What about our sample size?"},{"Start":"01:55.610 ","End":"01:59.000","Text":"Well, 20 balls are randomly chosen so that means"},{"Start":"01:59.000 ","End":"02:05.050","Text":"its small n equals 20. Now, what about y?"},{"Start":"02:05.050 ","End":"02:10.910","Text":"Well, y also has a hyper-geometric distribution where n equals to 60."},{"Start":"02:10.910 ","End":"02:12.815","Text":"The population size is 60."},{"Start":"02:12.815 ","End":"02:15.765","Text":"Now the number of green balls is also 20."},{"Start":"02:15.765 ","End":"02:17.725","Text":"D here equals to 20."},{"Start":"02:17.725 ","End":"02:20.210","Text":"Now, what about the sample size?"},{"Start":"02:20.210 ","End":"02:23.585","Text":"Well, again that equals to 20 as well."},{"Start":"02:23.585 ","End":"02:28.580","Text":"Because we have 2 different random variables that"},{"Start":"02:28.580 ","End":"02:32.870","Text":"have the same distribution with the same parameters,"},{"Start":"02:32.870 ","End":"02:39.530","Text":"then we can say that the standard deviation of x equals to the standard deviation of y."},{"Start":"02:39.530 ","End":"02:46.380","Text":"Let\u0027s now take a look at our correlation coefficient."},{"Start":"02:46.430 ","End":"02:49.865","Text":"Our correlation coefficient now becomes this."},{"Start":"02:49.865 ","End":"02:54.170","Text":"That\u0027s the covariance of x and y divided by"},{"Start":"02:54.170 ","End":"02:56.630","Text":"the standard deviation of x times"},{"Start":"02:56.630 ","End":"03:00.840","Text":"the standard deviation of y equals the standard deviation of x."},{"Start":"03:00.840 ","End":"03:07.630","Text":"Now that becomes the covariance of x and y divided by the variance of x."},{"Start":"03:07.630 ","End":"03:11.305","Text":"Because the standard deviation of x times itself,"},{"Start":"03:11.305 ","End":"03:15.645","Text":"well that\u0027s the variance. Here we have it."},{"Start":"03:15.645 ","End":"03:19.900","Text":"Now what we need to do is we need to calculate the covariance of x and y."},{"Start":"03:19.900 ","End":"03:21.565","Text":"Now how do we do that?"},{"Start":"03:21.565 ","End":"03:25.690","Text":"Here we need a little bit of some special thinking."},{"Start":"03:25.690 ","End":"03:27.360","Text":"Now, how do we do that?"},{"Start":"03:27.360 ","End":"03:29.100","Text":"If we recall,"},{"Start":"03:29.100 ","End":"03:33.820","Text":"if we look at our arsenal of equations we have this,"},{"Start":"03:33.820 ","End":"03:37.705","Text":"we have the variance of x plus y."},{"Start":"03:37.705 ","End":"03:42.610","Text":"Now that equals to the variance of x plus the variance of"},{"Start":"03:42.610 ","End":"03:48.680","Text":"y plus 2 times the covariance of x and y."},{"Start":"03:48.680 ","End":"03:50.915","Text":"Now, this is what we\u0027re looking for."},{"Start":"03:50.915 ","End":"03:54.295","Text":"This is what we want to calculate right here."},{"Start":"03:54.295 ","End":"03:56.285","Text":"Let\u0027s do that."},{"Start":"03:56.285 ","End":"04:00.680","Text":"How do we calculate this covariance?"},{"Start":"04:00.680 ","End":"04:06.485","Text":"Well, let\u0027s take a look at this expression right here, x plus y."},{"Start":"04:06.485 ","End":"04:14.400","Text":"Well, that\u0027s the number of red plus number of green balls."},{"Start":"04:14.400 ","End":"04:17.930","Text":"Now we also have some blue balls here."},{"Start":"04:17.930 ","End":"04:24.425","Text":"We have w. Let\u0027s define w as the number of blue balls."},{"Start":"04:24.425 ","End":"04:26.060","Text":"Now if you take a look at w,"},{"Start":"04:26.060 ","End":"04:31.935","Text":"w also has a hyper-geometric distribution where big N equals 60."},{"Start":"04:31.935 ","End":"04:35.540","Text":"The sub-population number of blue balls here also equals"},{"Start":"04:35.540 ","End":"04:38.960","Text":"to 20 and the sample size is also 20."},{"Start":"04:38.960 ","End":"04:41.105","Text":"What we see here is that"},{"Start":"04:41.105 ","End":"04:48.920","Text":"the random variable w has the same distribution with the same parameters as x and y."},{"Start":"04:48.920 ","End":"04:51.215","Text":"The variance of w,"},{"Start":"04:51.215 ","End":"04:56.590","Text":"that equals to the variance of x and that equals to the variance of y."},{"Start":"04:56.590 ","End":"05:01.460","Text":"Now what\u0027s the relationship between w and x plus y?"},{"Start":"05:01.460 ","End":"05:07.485","Text":"Well, x plus y equals to the following;"},{"Start":"05:07.485 ","End":"05:10.550","Text":"we have a total of 20 balls in"},{"Start":"05:10.550 ","End":"05:15.570","Text":"our sample and w is the number of blue balls in her sample,"},{"Start":"05:15.570 ","End":"05:23.405","Text":"so 20 minus w will give us the number of red and green balls in our sample."},{"Start":"05:23.405 ","End":"05:25.670","Text":"Now the minute we have that,"},{"Start":"05:25.670 ","End":"05:29.540","Text":"then we can calculate this thing right here."},{"Start":"05:29.540 ","End":"05:33.625","Text":"The variance of x plus y,"},{"Start":"05:33.625 ","End":"05:41.820","Text":"what that equals to the variance of 20 minus w. Now that equals to"},{"Start":"05:41.820 ","End":"05:50.915","Text":"the variance of x plus the variance of y plus 2 times the covariance of x and y."},{"Start":"05:50.915 ","End":"05:55.080","Text":"Now let\u0027s take a look at this."},{"Start":"05:55.080 ","End":"05:56.760","Text":"That equals to the following."},{"Start":"05:56.760 ","End":"05:59.495","Text":"Well the variance of y equals to the variance of x."},{"Start":"05:59.495 ","End":"06:03.890","Text":"That means that this thing right here equals to 2 times the variance of"},{"Start":"06:03.890 ","End":"06:10.115","Text":"x plus 2 times the covariance of x and y."},{"Start":"06:10.115 ","End":"06:17.145","Text":"Now that equals to the variance of 20 minus w. Now,"},{"Start":"06:17.145 ","End":"06:21.645","Text":"the variance of ax plus b,"},{"Start":"06:21.645 ","End":"06:26.595","Text":"that\u0027s a linear equation right here and this is what we have here."},{"Start":"06:26.595 ","End":"06:31.580","Text":"That equals to a square times the variance of x."},{"Start":"06:31.580 ","End":"06:33.920","Text":"Now in our case,"},{"Start":"06:33.920 ","End":"06:37.770","Text":"a here, well that equals to minus 1."},{"Start":"06:38.720 ","End":"06:42.410","Text":"All we have to do is we have to substitute that in here."},{"Start":"06:42.410 ","End":"06:49.440","Text":"That\u0027ll be minus 1 squared times the variance of w. That"},{"Start":"06:49.440 ","End":"06:58.140","Text":"equals to 2 times the variance of x plus 2 times the covariance of x and y."},{"Start":"06:58.140 ","End":"07:05.330","Text":"Now that equals to the variance of w. Minus 1 squared,"},{"Start":"07:05.330 ","End":"07:07.190","Text":"that equals to 1,"},{"Start":"07:07.190 ","End":"07:09.785","Text":"so 1 times the variance of w. Now,"},{"Start":"07:09.785 ","End":"07:11.210","Text":"what\u0027s the variance of w?"},{"Start":"07:11.210 ","End":"07:13.640","Text":"Would that equals to the variance of x?"},{"Start":"07:13.640 ","End":"07:16.645","Text":"That equals to the variance of x."},{"Start":"07:16.645 ","End":"07:19.550","Text":"That equals to this expression right here."},{"Start":"07:19.550 ","End":"07:27.155","Text":"That equals to 2 times the variance of x plus 2 times the covariance of x and y."},{"Start":"07:27.155 ","End":"07:29.660","Text":"Let\u0027s just do a little bit of manipulation here."},{"Start":"07:29.660 ","End":"07:34.050","Text":"That becomes minus the variance of x divided by"},{"Start":"07:34.050 ","End":"07:39.635","Text":"2 and that becomes the covariance of x and y. Here we are."},{"Start":"07:39.635 ","End":"07:43.130","Text":"We\u0027ve calculated the covariance of x and y."},{"Start":"07:43.130 ","End":"07:49.865","Text":"Great. Now we can come back and calculate the correlation coefficient."},{"Start":"07:49.865 ","End":"07:55.550","Text":"The correlation coefficient of x and y as we said,"},{"Start":"07:55.550 ","End":"07:58.910","Text":"that is the covariance of x and y,"},{"Start":"07:58.910 ","End":"08:01.880","Text":"divided by the variance of x."},{"Start":"08:01.880 ","End":"08:05.390","Text":"Now the covariance of x and y well that\u0027s right here."},{"Start":"08:05.390 ","End":"08:12.140","Text":"That\u0027s minus the variance of x divided by 2 divided by the variance of x."},{"Start":"08:12.140 ","End":"08:16.640","Text":"Now, that equals to minus 1/2."},{"Start":"08:16.640 ","End":"08:23.130","Text":"There we go. This is the correlation coefficient between x and y."}],"ID":13237},{"Watched":false,"Name":"Exercise 11","Duration":"9m 3s","ChapterTopicVideoID":12759,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12759.jpeg","UploadDate":"2019-02-11T14:36:41.3600000","DurationForVideoObject":"PT9M3S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.510","Text":"In this question, we\u0027re given that Y has a binomial distribution where"},{"Start":"00:03.510 ","End":"00:07.260","Text":"n equals 1 and a probability between 0 and 1."},{"Start":"00:07.260 ","End":"00:11.010","Text":"We\u0027re asked to prove that if this equality holds true"},{"Start":"00:11.010 ","End":"00:14.940","Text":"for the conditional probabilities for all values of X,"},{"Start":"00:14.940 ","End":"00:18.345","Text":"then X and Y are independent variables."},{"Start":"00:18.345 ","End":"00:21.810","Text":"First of all, let\u0027s take a look at this random variable."},{"Start":"00:21.810 ","End":"00:24.120","Text":"Now, these, as we said,"},{"Start":"00:24.120 ","End":"00:26.940","Text":"are the binomial distribution where n equals 1 and p. Well,"},{"Start":"00:26.940 ","End":"00:31.020","Text":"this is sometimes called a Bernoulli distribution and this"},{"Start":"00:31.020 ","End":"00:35.400","Text":"is an indicator variable where Y has 2 values,"},{"Start":"00:35.400 ","End":"00:37.275","Text":"either 0 or 1."},{"Start":"00:37.275 ","End":"00:40.430","Text":"Y equals 1 with a probability of p,"},{"Start":"00:40.430 ","End":"00:42.845","Text":"and Y equals 0 with a probability of"},{"Start":"00:42.845 ","End":"00:50.930","Text":"1 minus p. Let\u0027s take a look now at the conditional probabilities here,"},{"Start":"00:50.930 ","End":"00:54.320","Text":"and we want to prove that if this equality holds true,"},{"Start":"00:54.320 ","End":"00:56.480","Text":"then X and Y are independent."},{"Start":"00:56.480 ","End":"01:06.335","Text":"Let\u0027s start. We have here the probability of X equaling some value given Y equals 0."},{"Start":"01:06.335 ","End":"01:10.100","Text":"We said that that equals to the probability of X"},{"Start":"01:10.100 ","End":"01:14.270","Text":"equaling to some value given that Y equals to 1."},{"Start":"01:14.270 ","End":"01:15.980","Text":"Now, first of all,"},{"Start":"01:15.980 ","End":"01:18.515","Text":"what are the valid values of X and Y?"},{"Start":"01:18.515 ","End":"01:19.940","Text":"Well, for Y it\u0027s simple,"},{"Start":"01:19.940 ","End":"01:21.515","Text":"that\u0027s either a 0, or 1."},{"Start":"01:21.515 ","End":"01:22.820","Text":"Now what about for X?"},{"Start":"01:22.820 ","End":"01:25.145","Text":"Well, it could be any value X_1,"},{"Start":"01:25.145 ","End":"01:26.330","Text":"X_2, and X_3."},{"Start":"01:26.330 ","End":"01:35.255","Text":"Let\u0027s take a look at the probability distribution function of X and Y. Here we go."},{"Start":"01:35.255 ","End":"01:38.640","Text":"We see here Y and the values for Y with"},{"Start":"01:38.640 ","End":"01:43.590","Text":"a probability of p and 1 minus p for Y equaling 1 and 0."},{"Start":"01:43.590 ","End":"01:46.290","Text":"The values of X, X_1, X_2,"},{"Start":"01:46.290 ","End":"01:49.735","Text":"and X_3 and so on and so forth with the marginal probabilities,"},{"Start":"01:49.735 ","End":"01:52.030","Text":"for the values of X."},{"Start":"01:52.030 ","End":"01:56.090","Text":"Now, in the white cells right here within the table,"},{"Start":"01:56.090 ","End":"02:01.210","Text":"that\u0027ll be the joint probabilities of X and Y."},{"Start":"02:01.210 ","End":"02:06.110","Text":"Now, we\u0027ll take a look at this table in a minute,"},{"Start":"02:06.110 ","End":"02:11.330","Text":"but let\u0027s just extend these conditional probabilities."},{"Start":"02:11.330 ","End":"02:14.060","Text":"Well, here on this side,"},{"Start":"02:14.060 ","End":"02:19.240","Text":"that would be the probability of X equaling some value and Y"},{"Start":"02:19.240 ","End":"02:24.635","Text":"equaling 0 divided by the probability of Y equaling 0."},{"Start":"02:24.635 ","End":"02:27.530","Text":"That\u0027s the definition of a conditional probability,"},{"Start":"02:27.530 ","End":"02:33.155","Text":"and that equals to the probability of X equaling some value and Y"},{"Start":"02:33.155 ","End":"02:39.780","Text":"equaling 1 divided by the probability of Y equaling 1."},{"Start":"02:39.830 ","End":"02:44.870","Text":"Now, what\u0027s the probability of Y equals 0,"},{"Start":"02:44.870 ","End":"02:46.340","Text":"that\u0027s 1 minus p,"},{"Start":"02:46.340 ","End":"02:48.445","Text":"and the probability Y equals 1,"},{"Start":"02:48.445 ","End":"02:52.415","Text":"that\u0027s p. Let\u0027s write these guys down."},{"Start":"02:52.415 ","End":"03:03.665","Text":"That\u0027ll be equal to the probability of X equaling x and Y equaling 0"},{"Start":"03:03.665 ","End":"03:10.250","Text":"divided by 1 minus p and"},{"Start":"03:10.250 ","End":"03:16.655","Text":"that has to equal to the probability of X equaling x and Y equaling 1 now,"},{"Start":"03:16.655 ","End":"03:19.760","Text":"divided by the probability of y equaling 1, well,"},{"Start":"03:19.760 ","End":"03:23.915","Text":"that\u0027s p. If that\u0027s the case,"},{"Start":"03:23.915 ","End":"03:26.375","Text":"let\u0027s just cross multiply,"},{"Start":"03:26.375 ","End":"03:29.430","Text":"and then we\u0027ll simplify."},{"Start":"03:29.810 ","End":"03:33.815","Text":"That\u0027ll be p times this probability,"},{"Start":"03:33.815 ","End":"03:38.630","Text":"and that should be equal to 1 minus p times this probability. Let\u0027s write that down."},{"Start":"03:38.630 ","End":"03:47.105","Text":"That\u0027ll be p times the probability of X equaling some value and Y equaling 0,"},{"Start":"03:47.105 ","End":"03:52.550","Text":"and that has to be equal to 1 minus p times this probability,"},{"Start":"03:52.550 ","End":"03:58.440","Text":"times the probability of X equaling some value and Y equals 1."},{"Start":"03:58.490 ","End":"04:01.255","Text":"Now, another thing."},{"Start":"04:01.255 ","End":"04:04.810","Text":"Now let\u0027s take a look at this probability right here."},{"Start":"04:04.810 ","End":"04:09.595","Text":"The probability of X equaling some value and Y equaling 0."},{"Start":"04:09.595 ","End":"04:11.185","Text":"Now, let\u0027s, for example,"},{"Start":"04:11.185 ","End":"04:14.160","Text":"take this cell right here,"},{"Start":"04:14.160 ","End":"04:18.005","Text":"where x equals X_3."},{"Start":"04:18.005 ","End":"04:21.130","Text":"Well, this cell right here is"},{"Start":"04:21.130 ","End":"04:26.305","Text":"the joint probability where x equals this value X_3 and Y equals 0."},{"Start":"04:26.305 ","End":"04:30.130","Text":"But we know that the sum of"},{"Start":"04:30.130 ","End":"04:34.810","Text":"these joint probabilities have to add up to the marginal probability."},{"Start":"04:34.810 ","End":"04:39.820","Text":"We can express this probability as the difference between"},{"Start":"04:39.820 ","End":"04:47.580","Text":"the marginal probability and the joint probability where Y equals 1."},{"Start":"04:47.580 ","End":"04:52.664","Text":"That\u0027ll be equal to right here to p times."},{"Start":"04:52.664 ","End":"04:59.425","Text":"Now, that will be the probability of X equaling some value."},{"Start":"04:59.425 ","End":"05:03.205","Text":"We just given example where x equals X_3."},{"Start":"05:03.205 ","End":"05:08.125","Text":"But it could be for any value of x minus"},{"Start":"05:08.125 ","End":"05:14.355","Text":"the probability of X equaling x and Y equaling 1, now."},{"Start":"05:14.355 ","End":"05:16.690","Text":"As we said, let\u0027s do that again."},{"Start":"05:16.690 ","End":"05:23.065","Text":"That this probability right here equals to this marginal probability minus"},{"Start":"05:23.065 ","End":"05:30.765","Text":"this guy right here and that\u0027s for all values of the joint probabilities."},{"Start":"05:30.765 ","End":"05:33.080","Text":"Now, that equals,"},{"Start":"05:33.080 ","End":"05:34.820","Text":"and again to this side now that\u0027s"},{"Start":"05:34.820 ","End":"05:43.235","Text":"1 minus p times the probability of X equaling some value and Y equaling 1."},{"Start":"05:43.235 ","End":"05:49.070","Text":"Let\u0027s just expand this expression right here."},{"Start":"05:49.070 ","End":"05:56.735","Text":"That\u0027ll be equal to p times the probability X equaling x"},{"Start":"05:56.735 ","End":"06:05.305","Text":"minus p times the probability of X equaling x and Y equaling 1,"},{"Start":"06:05.305 ","End":"06:08.100","Text":"and that should equal to, well,"},{"Start":"06:08.100 ","End":"06:10.160","Text":"that\u0027s 1 minus p times this guy."},{"Start":"06:10.160 ","End":"06:16.999","Text":"Well, that\u0027s the probability of X equaling x and Y equaling 1 minus"},{"Start":"06:16.999 ","End":"06:26.285","Text":"the probability of minus p times the probability of X equaling x and Y equaling 1."},{"Start":"06:26.285 ","End":"06:28.490","Text":"Now, if we take a look,"},{"Start":"06:28.490 ","End":"06:31.640","Text":"this expression, and this expression is the same,"},{"Start":"06:31.640 ","End":"06:33.290","Text":"so they cancel each other out."},{"Start":"06:33.290 ","End":"06:35.705","Text":"What do we have here? Well, what\u0027s left?"},{"Start":"06:35.705 ","End":"06:41.570","Text":"That\u0027s p times the probability of X equaling some value,"},{"Start":"06:41.570 ","End":"06:48.245","Text":"and that equals to the probability of X equaling some value and Y equaling 1."},{"Start":"06:48.245 ","End":"06:50.740","Text":"Well, what\u0027s p?"},{"Start":"06:50.740 ","End":"06:58.920","Text":"P is the probability of Y equaling 1 times the probability of X equaling some value,"},{"Start":"06:58.920 ","End":"07:05.505","Text":"and that equals to the joint probability of X equaling some value and Y equaling 1."},{"Start":"07:05.505 ","End":"07:08.375","Text":"Now, what\u0027s this?"},{"Start":"07:08.375 ","End":"07:11.900","Text":"This is the joint probability and"},{"Start":"07:11.900 ","End":"07:15.830","Text":"this is the multiplication of the marginal probabilities."},{"Start":"07:15.830 ","End":"07:19.340","Text":"Now let\u0027s recall the definition of independence."},{"Start":"07:19.340 ","End":"07:24.695","Text":"We say that the probability of X equaling some value,"},{"Start":"07:24.695 ","End":"07:28.940","Text":"and Y equaling another value with a joint probability, well,"},{"Start":"07:28.940 ","End":"07:36.060","Text":"that equals to the multiplication of the marginal probabilities."},{"Start":"07:36.060 ","End":"07:39.360","Text":"This is the definition."},{"Start":"07:39.360 ","End":"07:42.515","Text":"This is exactly what we have here."},{"Start":"07:42.515 ","End":"07:46.850","Text":"We have here a joint probability, that\u0027s right here,"},{"Start":"07:46.850 ","End":"07:52.895","Text":"and that equals to the multiplication of the marginal probabilities right here."},{"Start":"07:52.895 ","End":"07:58.150","Text":"Now, we\u0027ve done this for Y equaling 1."},{"Start":"07:58.150 ","End":"08:02.295","Text":"Now, what about Y equaling 0?"},{"Start":"08:02.295 ","End":"08:05.655","Text":"Well, here again is our table,"},{"Start":"08:05.655 ","End":"08:09.740","Text":"and what we\u0027ve done is we\u0027ve true that"},{"Start":"08:09.740 ","End":"08:13.760","Text":"the multiplication of the marginal probability where Y equals"},{"Start":"08:13.760 ","End":"08:16.760","Text":"1 and for all values"},{"Start":"08:16.760 ","End":"08:22.120","Text":"of the marginal probabilities of X with the multiplication of those things,"},{"Start":"08:22.120 ","End":"08:30.770","Text":"they equal to the joint probabilities of Y equals 1 and all values for X."},{"Start":"08:30.770 ","End":"08:35.240","Text":"Now, since Y is an indicator variable,"},{"Start":"08:35.240 ","End":"08:41.600","Text":"it\u0027s very easy to show that the multiplication of 1 minus p,"},{"Start":"08:41.600 ","End":"08:46.310","Text":"the marginal probability of Y equaling 0 and all values of X, well,"},{"Start":"08:46.310 ","End":"08:51.350","Text":"that equals to the joint probability for all values of X and Y equals 0."},{"Start":"08:51.350 ","End":"08:52.655","Text":"Now this is so simple."},{"Start":"08:52.655 ","End":"08:56.250","Text":"I\u0027ll let you do this by yourself."},{"Start":"08:56.250 ","End":"09:03.870","Text":"There we have it. We\u0027ve proved that X and Y are independent variables."}],"ID":13238},{"Watched":false,"Name":"Exercise 12","Duration":"7m 54s","ChapterTopicVideoID":12760,"CourseChapterTopicPlaylistID":245021,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/12760.jpeg","UploadDate":"2019-02-11T14:37:05.3130000","DurationForVideoObject":"PT7M54S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.255","Text":"In this question, we\u0027re given 2 independent random variables,"},{"Start":"00:03.255 ","End":"00:07.185","Text":"X and Y, where X has a binomial distribution with parameters,"},{"Start":"00:07.185 ","End":"00:12.090","Text":"n and p. Y has a binomial distribution with parameters m and p."},{"Start":"00:12.090 ","End":"00:18.120","Text":"Given X plus Y equals k,"},{"Start":"00:18.120 ","End":"00:19.950","Text":"we\u0027re asked to prove that that has"},{"Start":"00:19.950 ","End":"00:24.270","Text":"a hypergeometric distribution with parameters n plus m, n,"},{"Start":"00:24.270 ","End":"00:30.810","Text":"and k. So the probability of X equaling to some value,"},{"Start":"00:30.810 ","End":"00:34.500","Text":"given that X plus Y equals to k,"},{"Start":"00:34.500 ","End":"00:37.725","Text":"that\u0027s the probability of this thing,"},{"Start":"00:37.725 ","End":"00:41.369","Text":"we\u0027re asked to prove that this has a hypergeometric distribution."},{"Start":"00:41.369 ","End":"00:45.225","Text":"That\u0027s fine. This is a conditional probability."},{"Start":"00:45.225 ","End":"00:48.485","Text":"That means that in the denominator,"},{"Start":"00:48.485 ","End":"00:51.170","Text":"we have the probability of what\u0027s given,"},{"Start":"00:51.170 ","End":"00:54.650","Text":"that\u0027s X plus Y equaling k,"},{"Start":"00:54.650 ","End":"00:55.790","Text":"and in the numerator,"},{"Start":"00:55.790 ","End":"00:58.283","Text":"we have the probability of the intersect,"},{"Start":"00:58.283 ","End":"01:04.055","Text":"so that\u0027s the probability of X equaling to some value intersect"},{"Start":"01:04.055 ","End":"01:11.315","Text":"X plus Y equaling to k. Let\u0027s just write this out a little bit better."},{"Start":"01:11.315 ","End":"01:19.414","Text":"In the numerator, we\u0027ll have the probability of X equaling to some value x intersect,"},{"Start":"01:19.414 ","End":"01:21.655","Text":"here I\u0027ll write Y,"},{"Start":"01:21.655 ","End":"01:30.800","Text":"and that equals to k minus some value of x divided by this probability right here,"},{"Start":"01:30.800 ","End":"01:35.060","Text":"the probability of X plus Y equaling"},{"Start":"01:35.060 ","End":"01:41.590","Text":"k. Let\u0027s take first a look at this probability right here."},{"Start":"01:41.590 ","End":"01:45.010","Text":"That\u0027s the probability of a sum of random variables,"},{"Start":"01:45.010 ","End":"01:48.070","Text":"and we know what the distribution is of each random variable."},{"Start":"01:48.070 ","End":"01:51.475","Text":"It has a binomial distribution with different parameters."},{"Start":"01:51.475 ","End":"01:59.170","Text":"If we have X and Y having a binomial distribution with different parameters,"},{"Start":"01:59.170 ","End":"02:01.810","Text":"then the sum X plus Y,"},{"Start":"02:01.810 ","End":"02:05.020","Text":"that also has a binomial distribution,"},{"Start":"02:05.020 ","End":"02:12.610","Text":"but now the parameter would be n plus m with the same probability."},{"Start":"02:13.070 ","End":"02:18.280","Text":"What\u0027s the probability of, let\u0027s say,"},{"Start":"02:18.280 ","End":"02:23.040","Text":"w equaling some value of k,"},{"Start":"02:23.040 ","End":"02:25.400","Text":"where w has a binomial distribution."},{"Start":"02:25.400 ","End":"02:28.265","Text":"That will be n/k,"},{"Start":"02:28.265 ","End":"02:34.010","Text":"p^k times 1 minus p^n minus"},{"Start":"02:34.010 ","End":"02:41.085","Text":"k. Let\u0027s use this formula right here for this probability."},{"Start":"02:41.085 ","End":"02:43.235","Text":"That will be equal to what?"},{"Start":"02:43.235 ","End":"02:54.130","Text":"Instead of n, we have here n plus m. That\u0027ll be n plus m/k,"},{"Start":"02:54.200 ","End":"02:56.715","Text":"that\u0027s this thing right here,"},{"Start":"02:56.715 ","End":"03:01.640","Text":"times p^k times 1 minus"},{"Start":"03:01.640 ","End":"03:08.540","Text":"p^n plus m minus k. Here in the numerator,"},{"Start":"03:08.540 ","End":"03:12.770","Text":"that will be the probability of X equaling to some value x"},{"Start":"03:12.770 ","End":"03:17.970","Text":"intersect Y equaling to k minus x."},{"Start":"03:17.970 ","End":"03:22.170","Text":"What do we know about X and Y?"},{"Start":"03:22.170 ","End":"03:24.765","Text":"We know that they\u0027re independent."},{"Start":"03:24.765 ","End":"03:30.620","Text":"So the probability of the intersect is basically the multiplication of the probabilities."},{"Start":"03:30.620 ","End":"03:31.940","Text":"Let\u0027s write that down."},{"Start":"03:31.940 ","End":"03:38.330","Text":"That\u0027ll be the probability of x equaling to some value times the probability of"},{"Start":"03:38.330 ","End":"03:45.140","Text":"Y equaling to k minus x divided by this expression right here."},{"Start":"03:45.140 ","End":"03:49.070","Text":"That\u0027ll be n plus m/k,"},{"Start":"03:49.070 ","End":"03:56.780","Text":"p^k times 1 minus p^n plus m minus"},{"Start":"03:56.780 ","End":"04:03.570","Text":"k. We\u0027ll use this equation"},{"Start":"04:03.570 ","End":"04:07.050","Text":"for these probabilities where X equals small x,"},{"Start":"04:07.050 ","End":"04:10.485","Text":"and Y equals to k minus small x."},{"Start":"04:10.485 ","End":"04:16.790","Text":"What\u0027s the probability of X equaling to small x?"},{"Start":"04:16.790 ","End":"04:19.730","Text":"That will be n/x,"},{"Start":"04:19.730 ","End":"04:25.610","Text":"p^x times 1 minus p^n minus x."},{"Start":"04:25.610 ","End":"04:30.260","Text":"What\u0027s the probability Y equaling k minus x?"},{"Start":"04:30.260 ","End":"04:36.285","Text":"That will be m/k minus x,"},{"Start":"04:36.285 ","End":"04:47.765","Text":"p^k minus x times 1 minus p^m minus k plus x"},{"Start":"04:47.765 ","End":"04:54.005","Text":"divided by n plus m/k,"},{"Start":"04:54.005 ","End":"04:59.510","Text":"p^k times 1 minus p^n"},{"Start":"04:59.510 ","End":"05:05.720","Text":"plus m minus k. This is a little bit monstrous,"},{"Start":"05:05.720 ","End":"05:08.345","Text":"so let\u0027s try to simplify this."},{"Start":"05:08.345 ","End":"05:12.140","Text":"We\u0027ll take out these coefficients to the side."},{"Start":"05:12.140 ","End":"05:17.775","Text":"We\u0027ll have here n/x times"},{"Start":"05:17.775 ","End":"05:23.420","Text":"m/k minus x divided"},{"Start":"05:23.420 ","End":"05:29.165","Text":"by n plus m/k times,"},{"Start":"05:29.165 ","End":"05:34.373","Text":"now we have p^x,"},{"Start":"05:34.373 ","End":"05:36.835","Text":"and we have here p^k minus x."},{"Start":"05:36.835 ","End":"05:44.410","Text":"So that\u0027ll be x plus k minus x times 1 minus"},{"Start":"05:44.410 ","End":"05:53.790","Text":"p^n minus x plus m minus k plus x,"},{"Start":"05:53.790 ","End":"05:57.995","Text":"this is this guy, divided by"},{"Start":"05:57.995 ","End":"06:07.195","Text":"p^k times 1 minus p^n plus m minus k. Let\u0027s simplify that."},{"Start":"06:07.195 ","End":"06:09.935","Text":"That will be n/x"},{"Start":"06:09.935 ","End":"06:19.775","Text":"m/k minus x divided by m plus n/k."},{"Start":"06:19.775 ","End":"06:24.335","Text":"Now x plus k minus x,"},{"Start":"06:24.335 ","End":"06:29.075","Text":"that\u0027s p^k times 1 minus"},{"Start":"06:29.075 ","End":"06:35.360","Text":"p^n minus x plus x,"},{"Start":"06:35.360 ","End":"06:36.980","Text":"they cancel each other out,"},{"Start":"06:36.980 ","End":"06:41.365","Text":"so we have n plus m minus k,"},{"Start":"06:41.365 ","End":"06:47.780","Text":"and all that\u0027s divided by p^k times 1 minus p^n"},{"Start":"06:47.780 ","End":"06:54.020","Text":"plus m minus k. We see that this equals to 1,"},{"Start":"06:54.020 ","End":"06:57.860","Text":"so that equals to n/x,"},{"Start":"06:57.860 ","End":"07:02.640","Text":"m/k minus x,"},{"Start":"07:02.640 ","End":"07:11.730","Text":"and that has to be divided by n plus m/k."},{"Start":"07:11.730 ","End":"07:17.870","Text":"This is exactly the probability in a hypergeometric distribution."},{"Start":"07:17.870 ","End":"07:28.650","Text":"This means that X given X plus Y equals to k,"},{"Start":"07:28.650 ","End":"07:32.645","Text":"now that has a hypergeometric distribution,"},{"Start":"07:32.645 ","End":"07:36.290","Text":"where the population size N,"},{"Start":"07:36.290 ","End":"07:44.495","Text":"that equals to m plus n. The sub-population that equals to n,"},{"Start":"07:44.495 ","End":"07:51.110","Text":"and the sample size that equals to k. So there we go."},{"Start":"07:51.110 ","End":"07:54.240","Text":"This is the proof that we were looking for."}],"ID":13239}],"Thumbnail":null,"ID":245021}]

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