Combining Random Variables

Means behave simply: the mean of X + Y is mx + my, and the mean of X - Y is mx - my. Averages just combine the way arithmetic suggests, because expected value is linear no matter what.

Spread works differently. Think about subtracting two independent random variables, say your commute time minus your friend's commute time. You might guess the uncertainty should shrink, since subtraction feels like it should cancel things out. But it doesn't: both variables are still bouncing around independently, and both sources of randomness add uncertainty to the result. There's no cancellation, because the two variables never coordinate their wobbles to offset each other. So for independent random variables, variances add whether you're finding X + Y or X - Y: var(X+Y) = var(X) + var(Y), and var(X-Y) = var(X) + var(Y). Only after adding the variances do you take a square root to get the standard deviation of the combination. This only works when X and Y are independent; if they're correlated, a covariance term enters and changes everything.

This rule is the bridge from single random variables into sampling distributions. Once you can combine two variables, you can build the distribution of a sample mean or a sample proportion, which is really just combining many independent random variables at once. Every variance-adds argument you'll meet later, from the standard error of a sample mean to the pooled variance in a two-sample test, traces back to this same idea.