September « 2008 « blog :: Brent -> [String]

The Poisson distribution and Stirling numbers

September 16, 2008

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!