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

Mean of sum and difference of random variables | Random variables | AP Statistics | Khan Academy


2m read
·Nov 11, 2024

Let's say that I have a random variable X, which is equal to the number of dogs that I see in a day. Random variable Y is equal to the number of cats that I see in a day. Let's say I also know what the mean of each of these random variables are, the expected value.

So, the expected value of X, which I could also denote as the mean of our random variable X, let's say I expect to see three dogs a day. Similarly, for the cats, the expected value of Y is equal to, I could also denote that as the mean of Y, is going to be equal to, and this is just for the sake of argument, let's say I expect to see four cats a day.

In previous videos, we defined how do you take the mean of a random variable or the expected value of a random variable. What we're going to think about now is what would be the expected value of X plus Y, or another way of saying that, the mean of the sum of these two random variables.

Well, it turns out—and I'm not proving it just yet—that the mean of the sum of random variables is equal to the sum of the means. So, this is going to be equal to the mean of random variable X plus the mean of random variable Y.

In this particular case, if I were to say, well, what's the expected number of dogs and cats that I would see in a given day? I would add these two means: it would be 3 + 4, and it would be equal to 7. So, in this particular case, it is equal to 3 + 4, which is equal to 7.

Similarly, if I were to ask you the difference, if I were to say, well, how many more cats in a given day would I expect to see than dogs? The expected value of Y minus X, what would that be? Well, intuitively, you might say, well, hey, if we can add random variables, if the expected value of the sum is the sum of the expected values, then the expected value—or the mean—of the difference will be the difference of the means, and that is absolutely true.

So, this is the same thing as the mean of Y minus X, which is equal to the mean of Y, is going to be equal to the mean of Y minus the mean of X. In this particular case, it would be equal to 4 - 3, which is equal to 1.

So, another way of thinking about this intuitively is I would expect to see, on a given day, one more cat than dogs. Now, the example that I've just used—this is discrete random variables. On a given day, I wouldn't see 2.2 dogs or pi dogs. The expected value itself does not have to be a whole number because you could, of course, average it over many days.

But this same idea—that the mean of a sum is the same thing as the sum of means, and that the mean of a difference of random variables is the same as the difference of the means—in a future video, I'll do a proof of this.

More Articles

View All
It Looks Like a Velociraptor Foot | Photographer | National Geographic
Oh, you can see it! Heart starting to beat right there. Oh, that’s crazy, look at that! Oh my God, beyond that, of course, like that turning into a chicken. There’s a lot that has to happen, but like, this is such a… it looks like a river Delta, and it’s …
Water Technology in Architecture | National Geographic
[Music] Here on the snowy slopes of Mount Hood, Oregon, it seems impossible that the U.S. could ever run low on water. But government-backed research says we could in little more than 50 years. [Music] Oregon relies heavily on snowmelt for its fresh water…
The Most Important Things That Make or Break a Good Life
Hello Elixers and welcome back to our channel! This video is for everybody, regardless of where you are in your life, sort of a back to basics. You know, it’s good to have a refresher once in a while. We know you’ll love this one. Welcome to Alux! Now, …
The aftermath... Tenants Stopped Paying Rent
What’s big, eyes? It’s Graham here. So last month, I addressed a highly publicized article which found that nearly 1⁄3 of Americans were unable to pay their rent for the month of April. At the time, that was a very alarming statistic. For someone who has…
Male Polar Bear Fight Club - Ep. 2 | Wildlife: The Big Freeze
(Polar bear whining) - It’s been four months since your last bite. (Snow crunching) You may fool yourself into digging out some frozen kelp, but you know that dinner actually sits comfortably (seal purring) 200 miles away. It’s so frustrating. (Polar bear…
Chris Hemsworth sends his best mates in search of the secret elixir of Bali | Azza & Zoc Do Earth
[cheering] - Hello. Chris Hemsworth here. I’ve decided to create a new series about unlocking health and wellness secrets around the world. Here’s the catch. [announcer] Chris Hemsworth! [Chris] I’m too busy to travel to all these countries and get the go…