# Fast way to compute central moments of a Poisson random variable?

I am looking for a way to quickly compute the central moments of a Poisson random variable. I've found a couple of resources on how to compute the central moments, but I'm still trying to figure out if there are any fast algorithms out there. Of course I'd prefer an exact answer, but I'd be happy to trade off accuracy for speed.

Any ideas? Any papers I should look into?

Edit: To be more specific, given a $\lambda$ and $n$, I need to calculate the $n^{th}$ central moment. In other words, for a Poisson random variable $X$ with parameter $\lambda$, I need to calculate $$E[(X-\lambda)^n]$$

• What do you know? The moment generating function? Stirling numbers of the second kind? Moving from moments about $0$ to moments about the mean? – Henry Nov 19 '14 at 20:13
• You mean population moments, given the parameter? Or something else? What's known here? Do you need all central moments or just some? – Glen_b -Reinstate Monica Nov 19 '14 at 20:13
• The problem is given a $\lambda$, I need to compute the $n^{th}$ central moment. $n$ could be quite large. I edited the question to give more detail. – Kurt Nov 19 '14 at 20:21

Use the recurrence relation

$$\mu_{r+1}=\lambda\left(\frac{d\mu_r}{d\lambda}+r\mu_{r-1}\right).$$

The initial conditions are

• 1st central moment = $0$
• 2nd central moment = $\lambda$

Using these in the equation you will find the 3rd central moment is $\lambda.$ (Bear in mind that all central moments are zero when $\lambda=0,$ implying the differential equation has a unique solution.)
Again use the 2nd and 3rd central moments to obtain the 4th and so on.

• Could you provide a reference for this result? Thanks. – Xi'an Jun 4 '18 at 15:30
• Here is a reference of the result. Similar recursions for other popular distributions are well known. – StubbornAtom Jun 4 '18 at 16:35