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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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This gives a definition when u have the density function of X and Y – Bhargav Sep 29 '11 at 6:13
I already looked at that. I am still a bit uncertain. Given {w_n} in the probability space, I define E[X}=\sum (X(w_n))p_n, but does E[XY] have P_n^2? – john brown Sep 29 '11 at 6:16
Perhaps you should look at the relevant section and replace the integration with a summation. – anon Sep 29 '11 at 6:20
Can you please stop being so hostile? The density function is not making sense to me. I already looked at the section... – john brown Sep 29 '11 at 6:23
Let $W=XY$. Make a list of all the possible values taken on by $W$. With any luck you can find, from the information about $X$ and $Y$, the probability $P(W=w)$ for all possible values of $w$. After that, expectation is the standard one variable stuff. If you need further detail, please supply the specific question you have in mind. When $X$ and $Y$ are independent life becomes simpler, since then $E(XY)=E(X)E(Y)$. – André Nicolas Sep 29 '11 at 6:23

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

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