Variables with infinite expected value Consider the following:
Give examples of two discrete random variables $X$ and $Y$, whose expected value is infinite (i.e. don't have expected value). 
Also:
1) $X$ only gets positive values.
2) $Y$ ~ $-Y$, so $Y$ and $-Y$ are similarly distributed.
I think that one way of finding a random variable with no expected value, is to find a variable with value $x$ and probability $p$, so that infinite sum their product is infinite.
 A: No, not quite; that variable $X$ is extremely easy to compute the expectation of: it is $0$.  What you need to look for here is more technical: remember, expectations are just integrals (or might look like sums, depending on the associated measure).  What you want is to create variables whose expectations are improper integrals (or infinite series) that don't converge.
To that end, here are a couple of hints:
(1) Consider a random variable taking values in $\{1,2,3,\ldots\}$. If $p_n=P(X=n)$, then you want
$$
\sum_{n=1}^{\infty}p_n=1\qquad\text{and}\qquad\sum_{n=1}^{\infty}np_n=\infty.
$$
What if $p_n=C\frac{1}{n^k}$ for some constants $C$ and $k$?  How could they be chosen to make the above results hold?
(2) This is a similar thought, but we can cheat and use part (1).  Remember: an integral on $(-\infty,\infty)$ (or a sum for $n=-\infty$ to $n=\infty$) can't converge if the piece to the left of $0$ tends to $-\infty$ and the piece to the right tends to $\infty$.  Could you combine a couple independent copies of the variable from (1) to get a variable that is symmetric and satisfies this property?
