Let $p$ be a parameter in $]0,1[$. Let $(X_k)_{k\geq 0}$ be an independent, identically distributed sequence of random variables, such that each $X_k$ takes values only in $\lbrace -1, \frac{1-p}{p} \rbrace$ and $P(X_k=-1)=1-p$ (so that $X_k$ has mean $0$). Let $S_n=X_1+X_2+ \ldots +X_n$ for $n\geq 1$ and let $N$ denote the smallest integer such that $S_{N} > 0$ (it is well known that $N$ exists almost surely). What is the expectancy of $S_{N}$ ?

If $p$ is of the form $1-\frac{1}{k}$ where $k$ is an integer, it is easily seen that $S_{N}$ is constant and equal to $\frac{1}{k}$.