WEBVTT 00:00:00.400 --> 00:00:03.200 Let us now consider a simple example. 00:00:03.200 --> 00:00:07.460 Let random variables X and Y be described by a joint PMF 00:00:07.460 --> 00:00:09.820 which is the one shown in this table. 00:00:09.820 --> 00:00:10.990 Question-- 00:00:10.990 --> 00:00:14.190 are X and Y independent? 00:00:14.190 --> 00:00:16.910 We can try to answer this question by using the 00:00:16.910 --> 00:00:18.670 definition of independence. 00:00:18.670 --> 00:00:22.320 But it is actually more instructive to proceed in a 00:00:22.320 --> 00:00:24.660 somewhat more intuitive way. 00:00:24.660 --> 00:00:29.320 We look at this table, and we observe that the value of one 00:00:29.320 --> 00:00:33.000 is possible for X. In particular, the probability 00:00:33.000 --> 00:00:36.950 that X takes the value of one, this is the marginal 00:00:36.950 --> 00:00:41.690 probability, this can be found by adding the probabilities of 00:00:41.690 --> 00:00:45.580 all of the outcomes in this column, which, in this case, 00:00:45.580 --> 00:00:48.710 is 3 over 20. 00:00:48.710 --> 00:00:52.560 Suppose now that somebody tells you the value of Y. For 00:00:52.560 --> 00:00:57.430 example, I tell you that Y happens to be equal to one, in 00:00:57.430 --> 00:01:02.100 which case you are transported into this universe. 00:01:02.100 --> 00:01:06.110 In this universe, the conditional probability that X 00:01:06.110 --> 00:01:10.510 takes a value of one, given that Y takes a value of one, 00:01:10.510 --> 00:01:12.100 what is it? 00:01:12.100 --> 00:01:14.800 In this universe, there's zero probability 00:01:14.800 --> 00:01:17.070 associated to this outcome. 00:01:17.070 --> 00:01:19.660 So this probability is zero, which is 00:01:19.660 --> 00:01:23.010 different than 3 over 20. 00:01:23.010 --> 00:01:26.789 And since these two numbers are different, this means that 00:01:26.789 --> 00:01:31.720 information from Y changes our beliefs about what's going to 00:01:31.720 --> 00:01:35.600 happen to X. And so, we do not have independence. 00:01:35.600 --> 00:01:40.009 So again, intuitively, in the beginning, we thought that X 00:01:40.009 --> 00:01:42.380 equal to one was possible. 00:01:42.380 --> 00:01:47.360 But information given by Y, namely that Y is equal to one, 00:01:47.360 --> 00:01:51.360 tells us that, actually, X equals to one is impossible. 00:01:51.360 --> 00:01:56.700 Information about Y changed our beliefs about X, so X and 00:01:56.700 --> 00:01:57.990 Y are dependent. 00:02:00.540 --> 00:02:03.510 Now, when we first introduced the notion of independence 00:02:03.510 --> 00:02:06.310 some time ago, we also introduced the notion of 00:02:06.310 --> 00:02:07.620 conditional independence. 00:02:07.620 --> 00:02:10.500 And we said that conditional independence is the same as 00:02:10.500 --> 00:02:15.940 ordinary independence, except that it would be applied to a 00:02:15.940 --> 00:02:18.880 conditional universe. 00:02:18.880 --> 00:02:21.480 Something similar can be done for the case of random 00:02:21.480 --> 00:02:22.960 variables as well. 00:02:22.960 --> 00:02:29.030 So suppose, for example, that someone tells us that the 00:02:29.030 --> 00:02:32.680 outcome of the experiment was such that it belongs 00:02:32.680 --> 00:02:35.250 to this blue set. 00:02:35.250 --> 00:02:38.960 This is the set where X is less than or equal to 2, and Y 00:02:38.960 --> 00:02:41.250 is larger than or equal to three. 00:02:41.250 --> 00:02:44.420 So we're given this information, and this is now 00:02:44.420 --> 00:02:47.050 our new conditional model. 00:02:47.050 --> 00:02:51.600 The question is, within this new conditional model are 00:02:51.600 --> 00:02:55.079 random variables X and Y independent? 00:02:55.079 --> 00:02:58.440 Let's just right down the conditional model, where I'm 00:02:58.440 --> 00:03:02.610 only showing the four possible outcomes that are allowed in 00:03:02.610 --> 00:03:03.850 the conditional model. 00:03:03.850 --> 00:03:06.810 All the others, of course, will have zero probability in 00:03:06.810 --> 00:03:08.540 the conditional model. 00:03:08.540 --> 00:03:12.110 So in the conditional model, probabilities will keep the 00:03:12.110 --> 00:03:16.260 same proportions as in the unconditional model-- 00:03:16.260 --> 00:03:19.260 and the proportions are 1, 2, 2, 4-- 00:03:19.260 --> 00:03:22.990 but then they need to be scaled, or normalized, so that 00:03:22.990 --> 00:03:24.810 they add to 1. 00:03:24.810 --> 00:03:30.030 And to make them add to 1, we need to divide them all by 9. 00:03:30.030 --> 00:03:34.520 In this conditional model, this is the joint PMF of the 00:03:34.520 --> 00:03:39.400 two random variables X and Y. Let us find the marginal PMFs. 00:03:39.400 --> 00:03:42.020 To find the marginal PMF of X, we add the 00:03:42.020 --> 00:03:43.300 entries in this column. 00:03:43.300 --> 00:03:47.829 And we get here 1/3, and here 2/3. 00:03:47.829 --> 00:03:50.829 And to find the marginal PMF of y, we add the 00:03:50.829 --> 00:03:52.110 entries in this [row] 00:03:52.110 --> 00:03:54.090 to find 2/3. 00:03:54.090 --> 00:03:55.940 And we adds the entries in that [row] 00:03:55.940 --> 00:03:58.020 to find 1/3. 00:03:58.020 --> 00:04:00.000 So this is the marginal PMF of x. 00:04:00.000 --> 00:04:04.900 That's the marginal PMF of Y. And now we notice that this 00:04:04.900 --> 00:04:08.670 entry of the joint PMF is 1/3 times 1/3, the 00:04:08.670 --> 00:04:10.300 product of the marginals. 00:04:10.300 --> 00:04:14.560 This entry is the product of 1/3 times 2/3, the product of 00:04:14.560 --> 00:04:18.070 the marginals, and so on for the remaining entries. 00:04:18.070 --> 00:04:22.670 So each entry of the joint PMF is equal to the product of the 00:04:22.670 --> 00:04:25.680 corresponding entries of the marginal PMFs. 00:04:25.680 --> 00:04:29.670 And this is the definition of independence. 00:04:29.670 --> 00:04:33.670 So in this conditional blue universe, we do have 00:04:33.670 --> 00:04:35.210 independence. 00:04:35.210 --> 00:04:38.909 And the way that this was established was to check that 00:04:38.909 --> 00:04:42.940 the joint PMF factors as a product of marginal PMFs.