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 , then
where 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 , then is the nth Bell number , 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.