Section 2.5 - Product Expected Value

Given independent random variables $X$ and $Y$,^?That is, knowning $X = x$ provides no help in determining what value $Y$ might be, and vice versa (knowing $Y = y$ says nothing about $X$’s value. what is the expected value of the product $XY$?

From the definition of expected vlaue, $$E(XY) = \sum_x\sum_y xy\,p(XY = xy)$$ Because $X$ and $Y$ are independent, the probability of their product is the product of their probabilities $$p(XY = xy) = p_X(X = x)\,p_Y(Y = y)$$ and the expected value becomes $$E(XY) = \sum_x\sum_y xy\,p_X(X = x)\,p_Y(Y = y)$$ where $p_X$ is the probability distribution for $X$ and similarly for $p_Y$.^?Independence includes not only values, but probability distributions too.

Evaluating the expected value as given takes work proportional to $n^2$. Can the sum be evaluated with less work?