1 00:00:00,200 --> 00:00:01,940 In this lecture, we introduce the 2 00:00:01,940 --> 00:00:04,170 notion of a random variable. 3 00:00:04,170 --> 00:00:08,890 A random variable is, loosely speaking, a numerical quantity 4 00:00:08,890 --> 00:00:12,390 whose value is determined by the outcome of a probabilistic 5 00:00:12,390 --> 00:00:13,760 experiment. 6 00:00:13,760 --> 00:00:15,640 The weight of a randomly selected 7 00:00:15,640 --> 00:00:18,520 student is one example. 8 00:00:18,520 --> 00:00:21,750 After giving a general definition, we will focus 9 00:00:21,750 --> 00:00:24,820 exclusively on discrete random variables. 10 00:00:24,820 --> 00:00:28,840 These are random variables that take values in finite or 11 00:00:28,840 --> 00:00:31,660 countably infinite sets. 12 00:00:31,660 --> 00:00:33,830 For example, random variables that take 13 00:00:33,830 --> 00:00:37,790 integer values are discrete. 14 00:00:37,790 --> 00:00:41,060 To any discrete random variable we will associate a 15 00:00:41,060 --> 00:00:45,210 probability mass function, which tells us the likelihood 16 00:00:45,210 --> 00:00:48,540 of each possible value of the random variable. 17 00:00:48,540 --> 00:00:52,360 Then we will go over a few examples and introduce some 18 00:00:52,360 --> 00:00:55,200 common types of random variables. 19 00:00:55,200 --> 00:00:59,680 And finally we will introduce a new concept- the expected 20 00:00:59,680 --> 00:01:02,490 value of the random variable, also called the 21 00:01:02,490 --> 00:01:05,519 expectation or mean. 22 00:01:05,519 --> 00:01:08,460 It is a weighted average of the values of the random 23 00:01:08,460 --> 00:01:11,990 variable, weighted according to their respective 24 00:01:11,990 --> 00:01:16,140 probabilities, and has an intuitive interpretation as 25 00:01:16,140 --> 00:01:20,310 the average value we expect to see if we repeat the same 26 00:01:20,310 --> 00:01:23,060 probabilistic experiment independently a 27 00:01:23,060 --> 00:01:25,870 large number of times. 28 00:01:25,870 --> 00:01:29,700 Expected values play a central role in probability theory. 29 00:01:29,700 --> 00:01:32,780 We will look into some of their properties. 30 00:01:32,780 --> 00:01:36,390 And we will also calculate the expected values of the example 31 00:01:36,390 --> 00:01:39,090 random variables that we will have introduced.