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 
, then
![\displaystyle E[X^n] = \sum_{k=1}^n S(n,k) \lambda^k,](http://l.wordpress.com/latex.php?latex=%5Cdisplaystyle+E%5BX%5En%5D+%3D+%5Csum_%7Bk%3D1%7D%5En+S%28n%2Ck%29+%5Clambda%5Ek%2C&bg=ffffff&fg=333333&s=0 "\displaystyle E[X^n] = \sum_{k=1}^n S(n,k) \lambda^k,")
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 ![E[X^n]](http://l.wordpress.com/latex.php?latex=E%5BX%5En%5D&bg=ffffff&fg=333333&s=0 "E[X^n]")
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.
2 Comments | 
combinatorics, grad school, learning, math | Tagged: Bell numbers, moments, Poisson, Stirling numbers | 
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Posted by Brent
