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# Mean of sum and difference of random variables

AP.STATS:
VAR‑5 (EU)
,
VAR‑5.E (LO)
,
VAR‑5.E.1 (EK)

## Video transcript

let's say that I have a random variable X which is equal to the number of dogs that I see in a day and random variable Y is equal to the number of cats that I see in a day and 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 and similarly for the cat the expected value of y is equal to I could also note 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 and in previous videos we define how do you take the mean of an amine 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 this 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 a random variable Y and so 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 well I would add these two means it would be three plus four it would be equal to seven so in this particular case it'd be it is equal to three plus four which is equal to which is equal to seven and similarly if I were to ask you the difference if I were to say well what's the how many more cats in a given day would I expect to see then dogs so the expected value of y minus X what would that be well intuitively you might say well okay if we can add random 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 is 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 minus the mean of X and in this particular case it would be equal to four minus three minus three is equal to one 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 variable and a given day I wouldn't see two point two 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 under the 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
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