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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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    I seriously wasn't trying to be hostile - I should've said 'could' instead of 'should' - and I really thought you didn't see the section. I apologize. Do you know what a [joint probability mass function](http://en.wikipedia.org/wiki/Joint_probability_distribution#Discrete_case) is? For a concrete example, let $X$ be the outcome of a coin flip ($0$ or $1$ with probability $1/2$ for each outcome) and define $Y=1-X$. Then $\mathbb{E}(XY)=1\cdot1\cdot(0)+1\cdot0\cdot(1/2)+0\cdot1\cdot(1/2)+0\cdot0 \cdot (0)=0$ because $P(X=1,Y=1)=0, P(X=1,Y=0)=1/2,$ $P(X=0,Y=1)=1/2,P(X=0,Y=0)=0.$2011-09-29

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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 $E[XY]= \sum_x x \Pr(X=x) \sum_y y \Pr(Y=y) =E[X]E[Y]$