1 00:00:00,550 --> 00:00:03,440 We will now go through an example, which is essentially 2 00:00:03,440 --> 00:00:06,290 a drill, to consolidate our understanding of the 3 00:00:06,290 --> 00:00:09,890 conditional expectation and the conditional variance. 4 00:00:09,890 --> 00:00:12,870 Consider a random variable X, which is continuous and is 5 00:00:12,870 --> 00:00:16,079 described by a PDF of this form. 6 00:00:16,079 --> 00:00:19,890 Whenever we have a PDF that seems to consist of different 7 00:00:19,890 --> 00:00:23,930 pieces, it's always useful to divide and conquer. 8 00:00:23,930 --> 00:00:26,490 And the way we will do that will be to consider two 9 00:00:26,490 --> 00:00:27,610 different scenarios. 10 00:00:27,610 --> 00:00:29,940 That X falls in this range. 11 00:00:29,940 --> 00:00:32,910 And in that scenario, we say that the certain random 12 00:00:32,910 --> 00:00:34,880 variable Y is equal to 1. 13 00:00:34,880 --> 00:00:38,450 And another scenario in which X falls in this range, and in 14 00:00:38,450 --> 00:00:42,850 that case, we say that Y is equal to 2. 15 00:00:42,850 --> 00:00:45,600 Let us now look at the conditional expectation of X 16 00:00:45,600 --> 00:00:47,760 given Y. What is it? 17 00:00:47,760 --> 00:00:51,210 Well, it is a random variable which can take a different 18 00:00:51,210 --> 00:00:53,960 values depending on what Y is. 19 00:00:53,960 --> 00:00:57,980 If Y happens to take a value of 1, then 20 00:00:57,980 --> 00:00:59,560 we are in this range. 21 00:00:59,560 --> 00:01:03,380 And the conditional PDF of X, given that Y falls in this 22 00:01:03,380 --> 00:01:06,270 range, keeps the same shape, it's uniform. 23 00:01:06,270 --> 00:01:10,390 And so it's mean is going to be equal to the midpoint of 24 00:01:10,390 --> 00:01:14,260 this interval, which is 1/2. 25 00:01:14,260 --> 00:01:18,320 And this is something that happens when Y is equal to 1. 26 00:01:18,320 --> 00:01:20,340 What is the probability of this happening? 27 00:01:20,340 --> 00:01:24,020 The probability that Y is equal to 1 is the area under 28 00:01:24,020 --> 00:01:26,610 the PDF in this range. 29 00:01:26,610 --> 00:01:29,730 And since the height of the PDF is 1/2, this 30 00:01:29,730 --> 00:01:31,260 probability is 1/2. 31 00:01:33,830 --> 00:01:37,180 The alternative scenario is that Y happens to take the 32 00:01:37,180 --> 00:01:38,020 value of 2. 33 00:01:38,020 --> 00:01:41,650 In which case, X lives in this interval. 34 00:01:41,650 --> 00:01:46,690 Given that X has fallen in this interval, the conditional 35 00:01:46,690 --> 00:01:50,509 expectation of X is the midpoint of this interval. 36 00:01:50,509 --> 00:01:56,229 And the midpoint of this interval is at 2. 37 00:01:56,229 --> 00:01:59,170 And this is an event that, again, happens with 38 00:01:59,170 --> 00:02:03,610 probability 1/2, because the area under the PDF in this 39 00:02:03,610 --> 00:02:06,520 region is equal to 1/2. 40 00:02:06,520 --> 00:02:09,720 So the conditional expectation is a random variable that 41 00:02:09,720 --> 00:02:12,730 takes these values with these probabilities. 42 00:02:12,730 --> 00:02:16,060 Since we now have a complete probabilistic description of 43 00:02:16,060 --> 00:02:19,250 this random variable, we're able to calculate the 44 00:02:19,250 --> 00:02:21,250 expectation of this random variable. 45 00:02:21,250 --> 00:02:22,230 What is it? 46 00:02:22,230 --> 00:02:25,840 With probability 1/2, the random variable takes the 47 00:02:25,840 --> 00:02:27,610 value of 1/2. 48 00:02:27,610 --> 00:02:31,970 And with probability 1/2, it takes a value of 2. 49 00:02:31,970 --> 00:02:34,500 And so the expected value of the conditional 50 00:02:34,500 --> 00:02:37,840 expectation is 5/4. 51 00:02:37,840 --> 00:02:40,720 But the law of iterated expectations tells us that 52 00:02:40,720 --> 00:02:45,620 this quantity is also the same as the expected value of X. So 53 00:02:45,620 --> 00:02:48,730 we have managed to find the expected value of X by the 54 00:02:48,730 --> 00:02:52,670 divide and conquer method, by considering different cases. 55 00:02:52,670 --> 00:02:55,820 Let us now turn to the conditional variance of X 56 00:02:55,820 --> 00:03:00,160 given Y. Once more, this quantity is a random variable. 57 00:03:00,160 --> 00:03:03,370 The value of that quantity depends on what Y 58 00:03:03,370 --> 00:03:04,780 turns out to be. 59 00:03:04,780 --> 00:03:06,930 And we have, again, the same two possibilities. 60 00:03:06,930 --> 00:03:10,510 Y could be equal to 1, or Y could be equal to 2. 61 00:03:10,510 --> 00:03:13,660 And these possibilities happen with equal probabilities. 62 00:03:16,820 --> 00:03:20,850 If Y is equal to 1, conditional on that event, X 63 00:03:20,850 --> 00:03:25,170 has a uniform PDF on this range, on an interval of 64 00:03:25,170 --> 00:03:25,970 length one. 65 00:03:25,970 --> 00:03:29,550 And we know that the variance of a uniform PDF on an 66 00:03:29,550 --> 00:03:32,440 interval of length one is 1/12. 67 00:03:32,440 --> 00:03:37,380 If on the other hand, Y takes a value of 2, then X is a 68 00:03:37,380 --> 00:03:42,030 uniform random variable on an interval of length 2. 69 00:03:42,030 --> 00:03:47,790 And the variance in this case is 2 squared, where this 2 70 00:03:47,790 --> 00:03:51,750 stands for the length of the interval, divided by 12, which 71 00:03:51,750 --> 00:03:53,755 is the same as 4/12. 72 00:03:57,880 --> 00:04:01,210 So we now have a complete probabilistic description of 73 00:04:01,210 --> 00:04:03,580 the conditional variance as a random variable. 74 00:04:03,580 --> 00:04:06,890 It's a random variable that with these probabilities, 75 00:04:06,890 --> 00:04:09,630 takes these two particular values. 76 00:04:09,630 --> 00:04:12,570 Since we know the distribution of this random variable, we 77 00:04:12,570 --> 00:04:15,640 can certainly calculate its expected value. 78 00:04:15,640 --> 00:04:17,790 And the expected value is found as follows. 79 00:04:17,790 --> 00:04:21,170 With probability 1/2, the random variable of interest 80 00:04:21,170 --> 00:04:24,000 takes a value of 1/12. 81 00:04:24,000 --> 00:04:28,000 And with probability 1/2, this random variable 82 00:04:28,000 --> 00:04:29,465 takes a value of 4/12. 83 00:04:32,690 --> 00:04:34,770 And this number happens to be 5/24. 84 00:04:38,590 --> 00:04:42,130 Finally, let us calculate the variance of the conditional 85 00:04:42,130 --> 00:04:43,480 expectation. 86 00:04:43,480 --> 00:04:45,790 Since we have complete information about the 87 00:04:45,790 --> 00:04:48,340 distribution of the conditional expectation, 88 00:04:48,340 --> 00:04:52,750 calculating its variance is not going to be difficult. 89 00:04:52,750 --> 00:04:54,380 So what is it? 90 00:04:54,380 --> 00:04:59,780 With probability 1/2, the conditional expectation takes 91 00:04:59,780 --> 00:05:02,380 a value of 1/2. 92 00:05:02,380 --> 00:05:05,990 We subtract from this is the mean of the conditional 93 00:05:05,990 --> 00:05:09,600 expectation, which is 5/4. 94 00:05:09,600 --> 00:05:12,120 And we take the square of that. 95 00:05:12,120 --> 00:05:18,360 So this term is the square or the deviation of the value of 96 00:05:18,360 --> 00:05:21,730 the random variable of 1/2 from the mean 97 00:05:21,730 --> 00:05:23,610 of that random variable. 98 00:05:23,610 --> 00:05:25,900 And we get a similar term. 99 00:05:25,900 --> 00:05:29,540 If Y happens to be equal to 2. 100 00:05:29,540 --> 00:05:34,470 With probability 1/2 half, our random variable takes a value 101 00:05:34,470 --> 00:05:38,500 of 2, which is so much away from the mean 102 00:05:38,500 --> 00:05:41,400 of the random variable. 103 00:05:41,400 --> 00:05:43,720 And then we square that quantity. 104 00:05:43,720 --> 00:05:46,770 If we carry out the algebra, the answer turns out 105 00:05:46,770 --> 00:05:50,240 to be 9 over 16. 106 00:05:50,240 --> 00:05:54,170 And now we can go back to the law of the total variance and 107 00:05:54,170 --> 00:05:57,560 calculate that the total variance is equal to the 108 00:05:57,560 --> 00:06:00,625 expected value of the variance, which is 5/24. 109 00:06:03,230 --> 00:06:05,960 And then we have the variance of the expected 110 00:06:05,960 --> 00:06:07,900 value, which is 9/16. 111 00:06:11,070 --> 00:06:14,653 And this number evaluates to 37/48. 112 00:06:19,860 --> 00:06:22,730 So we have managed to find the variance of this random 113 00:06:22,730 --> 00:06:26,930 variable using the divide and conquer methods and the law of 114 00:06:26,930 --> 00:06:28,180 the total variance.