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.

About Brent

Assistant Professor of Computer Science at Hendrix College. Functional programmer, mathematician, teacher, pianist, follower of Jesus.
This entry was posted in combinatorics, grad school, learning, math and tagged , , , . Bookmark the permalink.

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!

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