1 00:00:00,320 --> 00:00:04,019 In this last lecture of this unit, we continue with some of 2 00:00:04,019 --> 00:00:07,670 our earlier themes, and then introduce one new notion, the 3 00:00:07,670 --> 00:00:11,480 notion of independence of random variables. 4 00:00:11,480 --> 00:00:14,820 We will start by elaborating a bit more on the subject of 5 00:00:14,820 --> 00:00:17,890 conditional probability mass functions. 6 00:00:17,890 --> 00:00:21,070 We have already discussed the case where we condition a 7 00:00:21,070 --> 00:00:23,710 random variable on an event. 8 00:00:23,710 --> 00:00:27,690 Here we will talk about conditioning a random variable 9 00:00:27,690 --> 00:00:31,510 on another random variable, and we will develop yet 10 00:00:31,510 --> 00:00:34,420 another version of the total probability and total 11 00:00:34,420 --> 00:00:36,810 expectation theorems. 12 00:00:36,810 --> 00:00:40,920 There are no new concepts here, just new notation. 13 00:00:40,920 --> 00:00:45,070 I should say, however, that notation is important, because 14 00:00:45,070 --> 00:00:48,590 it guides you on how to think about problems in the most 15 00:00:48,590 --> 00:00:51,410 economical way. 16 00:00:51,410 --> 00:00:55,190 The one new concept that we will introduce is the notion 17 00:00:55,190 --> 00:00:57,960 of independence of random variables. 18 00:00:57,960 --> 00:01:01,200 It is actually not an entirely new concept. 19 00:01:01,200 --> 00:01:04,580 It is defined more or less the same way as independence of 20 00:01:04,580 --> 00:01:09,110 events, and has a similar intuitive interpretation. 21 00:01:09,110 --> 00:01:13,130 Two random variables are independent if information 22 00:01:13,130 --> 00:01:16,960 about the value of one of them does not change your model or 23 00:01:16,960 --> 00:01:20,090 beliefs about the other. 24 00:01:20,090 --> 00:01:23,130 On the mathematical side, we will see that independence 25 00:01:23,130 --> 00:01:25,740 leads to some additional nice properties 26 00:01:25,740 --> 00:01:27,010 of means and variances. 27 00:01:29,560 --> 00:01:32,640 We will conclude this lecture and this unit on discrete 28 00:01:32,640 --> 00:01:36,190 random variables by considering a rather difficult 29 00:01:36,190 --> 00:01:38,750 problem, the hat problem. 30 00:01:38,750 --> 00:01:41,910 We will see that by being systematic and using some of 31 00:01:41,910 --> 00:01:45,170 the tricks that we have learned, we can calculate the 32 00:01:45,170 --> 00:01:48,910 mean and variance of a rather complicated random variable.