Suppose $X$ and $Y$ take the values ${0,1,...,N}$ with probability $\frac{1}{N+1}$. What is the PMF of $|X-Y|$?
I started it like so: Let $|X-Y| = Z$
$P(Z<z) = P(-z < X-Y<z)$
In the continuous case, I'd just integrate the joint over the domain. But I'm having trouble figuring out how to translate that strategy to the discrete case.