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Mean (expected value) of a discrete random variable

Video transcript

- [Instructor] So, I'm defining the random variable x as the number of workouts that I will do in a given week. Now right over here, this table describes the probability distribution for x. And as you can see, x can take on only a finite number of values, zero, one, two, three, or four. And so, because there's a finite number of values here, we would call this a discrete random variable. And you can see that this is a valid probability distribution because the combined probability is one. .1 plus 0.15, plus 0.4, plus 0.25, plus 0.1 is one. And none of these are negative probabilities, which wouldn't have made sense. But what we care about in this video is the notion of an expected value of a discrete random variable, which we would just note this way. And one way to think about it is, once we calculate the expected value of this variable, of this random variable, that in a given week, that would give you a sense of the expected number of workouts. This is also sometimes referred to as the mean of a random variable. This, right over here, is the Greek letter mu, which is often used to denote the mean. So, this is the mean of the random variable x. But how do we actually compute it? To compute this, we essentially just take the weighted sum of the various outcomes, and we weight them by the probabilities. So, for example, this is going to be, the first outcome here is zero, and we'll weight it by its probability of 0.1. So, it's zero times 0.1. Plus, the next outcome is one, and it'd be weighted by its probability of 0.15. So, plus one times 0.15. Plus, the next outcome is two and has a probability of 0.4, plus two times 0.4. Plus, the outcome three has a probability of 0.25, plus three times 0.25. And then last but not least, we have the outcome four workouts in a week, that has a probability of 0.1, plus four times 0.1. Well, we can simplify this a little bit. Zero times anything is just zero. So, one times 0.15 is 0.15. Two times 0.4 is 0.8. Three times 0.25 is 0.75. And then four times .1 is 0.4. And so, we just have to add up these numbers. So, we get 0.15, plus .8, plus .75, plus .4, and let's say 0.4, 0.75, 0.8. Let's add 'em all together. And so, let's see, five plus five is 10. And then this is two plus eight is 10, plus seven is 17, plus four is 21. So, we get all of this is going to be equal to 2.1. So, one way to think about it is the expected value of x, the expected number of workouts for me in a week, given this probability distribution, is 2.1. Now you might be saying, wait, hold on a second. All of the outcomes here are whole numbers. How can you have 2.1 workouts in a week? What is .1 of a workout? Well, this isn't saying that in a given week, you would expect me to work out exactly 2.1 times. But this is valuable because you could say, well, in 10 weeks, you would expect me to do roughly 21 workouts. Sometimes I might do zero workouts, sometimes one, sometimes two, sometimes three, sometimes four. But in 100 weeks, you might expect me to do 210 workouts. So, even for a random variable that can only take on integer values, you can still have a non-integer expected value, and it is still useful.