In my work I've come across the following problem.
For $Y_k\sim\text{Binomial}(X_k,p)$ and $X_0=1$, compute the probability distribution for the following recursion,
$$X_k=X_{k-1}+Y_{k-1}$$
This problem is the same one discussed here.
This distribution has expected value $(1+p)^k$, and if one looks at its histogram, it sort of looks like a log-normal distribution, except over-dispersed. However, I was wondering if anything else was known about it, e.g. a closed form, or an efficient way to compute probabilities that does not require running the full recursion each time.
For those of you who are interested, this is used to model PCR amplification bias, reference.
