Suppose two fair dice are tossed find the expected value of the product of the faces showing

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1) Suppose two fair dice are tossed. Find the expected value of the product of the faces showing. 2) Suppose that fx,y(x,y) = 2/3(x+2y), x and y are in [0,1]. Find Var(X+Y)

3) A gambler plays n hands of poker. If he wins the kth hand, he collects k dollars; if he loses the kth hand, he collects nothing. Let T denote his total winnings in n hands. Assuming that his chances of winning each hand are constant and are independent of his success or failure at any other hand, find E(T) and Var(T).

1) Just form the 6 by 6 grid and in each place put the product.
From that get the distribution of the product, where all 36 sample points have a probability of 1/36.

Ok, thanks. I figured it was something pretty simple like that. And for #3, is it like a geometric answer or something? Seems like the expectation would be infinite, though.

Expectation is infinite right? Because if you win on like the 100th hand, you get 100 dollars, etc.... and the values keep going up as you go. Don't know what variance would be though.

And #2 is still confusing me I just can't figure out what I'm supposed to do.

There's only n hands, hence this is a bounded random variable. So the mean and variance are finite. Let p be the chance of winning any one hand.

$\displaystyle E(T)=(1)(p)+(2)(p)+....+(n)(p)=p\sum_{k=1}^nk={pn(n+1)\over 2}$

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Question: Given two fair dice, what is the expected value of their product?

My attempt:

Let $X_1$ and $X_2$ be scores by first and second die respectively. Note that $X_1$ and $X_2$ are independent. Then $$E(X_1X_2) = E(X_1)E(X_2) = 3.5^2 = 12.25.$$

Is my calculation correct?