## 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.

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### 2 Responses to The Poisson distribution and Stirling numbers

1. keyholecontrol says:

I think \left\{k \atop b\right\} worked for me to for bracket notation.
I hope that shows up as code…

$\left\{k \atop b\right\}$

2. Brent says:

Aha, thanks!