Hi. I need some help verifying something.
Let X, Y be discrete random variables such that 1<=X,Y<=10. Then:
$\displaystyle \mathbb{E}[XY]=\sum_{1\leq x,y\leq10}xyP(X=x\cap Y=y)$
Thanks in advance.
This is correct; it follows from $\displaystyle \displaystyle \mathbb{E}[X]=\sum xP(X=x)$ and the fact that XY is a discrete random variable.
Note that $\displaystyle P(XY=2) = P(X=1\cap Y=2) + P(X=2\cap Y=1)$ and that $\displaystyle 2P(XY=2) = 2P(X=1\cap Y=2) + 2P(X=2\cap Y=1)$, illustrating that the sum you wrote down is in fact equal to $\displaystyle \sum zP(XY=z)$ where I wrote z to avoid confusion; z ranges over the values that XY can take on.
Coincidentally, I just came across the formal statement of this recently in another thread on interestingly named math concepts. It is true, and sometimes known as the law of the unconscious statistician.
This isn't your question, but a word of caution on something semi-related: It is not correct to do things like E(|X-Y|) = |E(X)-E(Y)|.