What's the equation for $\mathbb{E}(XY)$?

Question

Let $X$ and $Y$ be independent, discrete random variables. Suppose I want to find $\mathbb{E}(XY)$. What's the equation for it? My educated guess is that the equation for the expected value in this case is

$$ \sum_x\sum_yxyP(X = x)P(Y = y) $$

And you sum over the $x$ and $y$ values over which both $X$ and $Y$ take positive probabilities, i.e. you sum over the intersection of the support of $X$ and $Y$.

I haven't found the equation for this case in any of the three textbooks I checked, and I don't want to rely on some educated guess in case I have to calculate $\mathbb{E}(XY)$ in my upcoming exam.

Thanks.

Answer 1

Your formula for $EXY$ is correct and we also have $EXY=(EX)(EY)=(\sum xP(X=x))(\sum yP(Y=y))$.