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.