I am not sure what the expression of E[XY] looks like given that X and Y are random variables on a finite probability space. That's all I need help on. Thanks!
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It is not quite clear what you are expecting but it might be something like
$$E[XY]=\sum_x \sum_y xy \Pr(X=x,Y=y) $$ $$= \sum_x \sum_y xy \Pr(X=x|Y=y)\Pr(Y=y) $$ $$= \sum_x \sum_y xy \Pr(X=x)\Pr(Y=y|X=x)$$
If they are independent then $\Pr(X=x|Y=y)=\Pr(X=x)$ and $\Pr(Y=y|X=x)=\Pr(Y=y)$ so this becomes $$E[XY]= \sum_x x \Pr(X=x) \sum_y y \Pr(Y=y) =E[X]E[Y]$$