Expected Values Let X be the number appearing on the first die when two
fair dice are rolled, and let Y be the sum of the numbers appearing
on the two dice. Show that E(X)E(Y) does not equal to E(XY).
I found E(X) and E(Y) but I don't know how to find E(XY). 
The answer in the textbook says that E(XY)= 329/12. How is this the case?

Here is a reasonably efficient way to find $E(XY)$. Let $W$ be the number on the second die. Then $Y=X+W$ so $XY=X^2+XW$. It follows that $E(XY)=E(X^2)+E(XW)$.
By independence, $E(XW)=E(X)E(W)$. Finally, to find $E(X^2)$, we can compute directly. We have $E(X^2)=\frac{1}{6}(1^2+2^2+3^2+4^2+5^2+6^2)$.
Now put the pieces together. If we wish, we can compute $1^2+2^2+\cdots+6^2$  by the formula for the sum of the first $n$ squares, but direct calculation is really no harder.
Remark: Alternately, we can find $E(XY)$ by finding $\Pr(XY)=k$ for all possible values of $k$, and using the ordinary formula for expectation. In principle this is straightforward, in practice excessively tedious.
