The Poisson distribution and Stirling numbers

While working on an assignment for my machine learning class, I rediscovered the fact that if X is a random variable from a Poisson distribution with parameter \(\lambda\), then

\(\displaystyle E[X^n] = \sum_{k=1}^n S(n,k) \lambda^k,\)

where \(S(n,k)\) denotes a Stirling number of the second kind. (I actually prefer Knuth’s curly bracket notation, but I can’t seem to get it to work on this blog.) In particular, if \(\lambda = 1\), then \(E[X^n]\) is the nth Bell number \(B_n\), the number of ways of partitioning a set of size n into subsets!

As it turned out, this didn’t help me at all with my assignment, I just thought it was nifty.