yego.me
💡 Stop wasting time. Read Youtube instead of watch. Download Chrome Extension

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


3m read
·Nov 11, 2024

  • [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 0.1 is 0.4.

And so, we just have to add up these numbers. So, we get 0.15, plus 0.8, plus 0.75, plus 0.4, 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 0.1 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.

More Articles

View All
Envy Can Be Useful, or It Can Eat You Alive
Do you want to tell us about some of the jobs that you had as a youth and the specific job that kicked off your fanatical obsession with creating wealth? This gets a little personal, and I don’t want to do the humble brag thing. There was some thread goin…
Game theory worked example from A P Microeconomics
What we have here is a free response question that you might see on an AP Microeconomics type exam that deals with game theory. It tells us Bread Basket and Quick Lunch are the only two sandwich shops serving a small town, so we’re in an oligopoly situati…
Weak base–strong acid titrations | Acids and bases | AP Chemistry | Khan Academy
Ammonia is an example of a weak base, and hydrochloric acid is an example of a strong acid. If we’re doing a weak base-strong acid titration, that means that ammonia is the analyte, the substance we’re analyzing, and we’re titrating ammonia with hydrochlo…
I can't keep doing this to myself
Guys, I’m making this video out of necessity. There’s a large part of me—it’s been the prevailing part of me as of late—that makes me not want to make this video because it’s unfiltered, because it’s not ready. What I want to say isn’t totally polished. I…
Zeros of polynomials introduction | Polynomial graphs | Algebra 2 | Khan Academy
Let’s say that we have a polynomial ( p ) of ( x ) and we can factor it. We can put it in the form ( (x - 1)(x + 2)(x - 3)(x + 4) ). What we are concerned with are the zeros of this polynomial. You might say, “What is a zero of a polynomial?” Well, those …
The elements of a poem | Reading | Khan Academy
Hello readers! Let’s talk about poems. Poetry is a special kind of writing. If ordinary writing is like talking, then poetry is like singing. Poetry is a way of making art with language. Poems can express huge ideas or feelings. They can be about the soun…