Suppose we have a random walk starting at the origin on $\mathbb{Z}_2$. Let $X$ denote the $\ell_1$ distance from the origin (i.e. $\vert x \vert + \vert y \vert$, where $(x,y)$ is the position of the walk). What is $\mathbb{E}X$?
I'm sure this is very well-known, but I didn't find anything.