I am looking for a reference (perhaps an example in a text or an article) for the following stopping-time problem. I suspect the problem is well known.
Random variables in an i.i.d. sequence $\langle X_n: n\ge1\rangle$ are summed until the sum first exceeds a given value $\alpha$. I'm interested in two random variables: the number $N$ of summands needed for the sum to first exceed $\alpha$, as well as the value of this sum $S=\sum_{n=1}^N X_n\ge\alpha$, but $\sum_{n=1}^{N-1} X_n<\alpha$.
I've done some numerical experiments. It looks like $N$ has something like a Poisson distribution with parameter $\alpha/E[X_n]$, and $S$ looks like the portion of a normal distribution above the mean. Perhaps there's a distribution for the $X_n$'s where $N$ and $S$ can be simply described in terms of well-known distributions, but my formal calculations for these make me pessimistic.