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 expectation 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 $\frac{1}{k-1}$.

Update 10/26/2010: In general, $S_N$ can only take a finite number of values, so the expectation is finite, as noted in the comments below. It seems that the distribution of $S_N$ should be computable using some simple algebra, but I was unable to do this. The finite-set of values property allows one however to compute $E(S_N)$ to a reasonable acurracy for a given $p$. For $p=\frac{1}{3}$, the expectation is larger than 1 and does not seem to be rational.

`$p$`

is of the form`$1−1/k$`

where`$k$`

is an integer, then`$S_N$`

is equal to`$1/(k−1)$`

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