Mean (expected value) of a discrete random variable (video) | Khan Academy

Video transcript

– [ Instructor ] So, I’m defining the random variable ten as the issue of workouts that I will do in a given workweek. nowadays good over here, this board describes the probability distribution for adam. And as you can see, x can take on entirely a finite issue of values, zero, one, two, three, or four. And indeed, because there ‘s a finite number of values here, we would call this a discrete random varying. And you can see that this is a valid probability distribution because the compound 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 would n’t have made sense. But what we care about in this video recording is the notion of an expect prize of a discrete random variable, which we would merely note this way. And one way to think about it is, once we calculate the expect respect of this varying, of this random variable star, that in a given week, that would give you a sense of the expected number of workouts. This is besides sometimes referred to as the bastardly of a random variable star. This, right over here, is the greek letter mu, which is much used to denote the mean. thus, this is the intend of the random variable x. But how do we actually compute it ? To compute this, we basically just take the leaden sum of the diverse outcomes, and we weight them by the probabilities. thus, for example, this is going to be, the first consequence here is zero, and we ‘ll weight it by its probability of 0.1. so, it ‘s zero times 0.1. Plus, the adjacent result is one, and it ‘d be weighted by its probability of 0.15. thus, plus one times 0.15. Plus, the following consequence is two and has a probability of 0.4, plus two times 0.4. Plus, the consequence three has a probability of 0.25, plus three times 0.25. And then last but not least, we have the result four workouts in a week, that has a probability of 0.1, plus four times 0.1. well, we can simplify this a little spot. Zero times anything is merely 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 sol, we good have to add up these numbers. then, 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 in concert. And so, let ‘s see, five plus five is 10. And then this is two asset 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. therefore, one way to think about it is the expected prize of x, the expect number of workouts for me in a week, given this probability distribution, is 2.1. now you might be saying, delay, hold on a second. All of the outcomes hera are whole numbers. How can you have 2.1 workouts in a week ? What is .1 of a exercise ? Well, this is n’t saying that in a given week, you would expect me to work out precisely 2.1 times. But this is valuable because you could say, well, in 10 weeks, you would expect me to do approximately 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. then, evening for a random variable that can only take on integer values, you can silent have a non-integer expected value, and it is still useful.

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