# Sum of squared Poisson probability masses

Let $(p_k)_{k=0, \dots, \infty}$ denote the probability masses of a Poisson distribution with parameter $\lambda$. I'm looking for the sum of their squares, $$\sum_{k=0}^\infty p_k^2,$$ as a function of $\lambda$. In other words I am interested in (the exponential of) the second-order Renyi entropy of a Poisson distribution.

Background:

• This question appears in many forms. Answers can be found by a search for skellam. – whuber Jul 21 '15 at 10:48

Don't hesitate to use WolframAlpha to get the sum of a series. Or do you need a mathematical proof ?

This gives $\exp(-2\lambda)I_0(2\lambda)$. The link to the documentation of the Bessel function $I_0$ is this one.

Actually the proof here would just mean the series representation of $I_0$.

If you want to use R to evaluate this Bessel function, you can do it with the help of the gsl package:

> library(gsl)
> lambda <- 1
> exp(-2*lambda)*bessel_I0(2*lambda)
[1] 0.3085083
> sum(dpois(0:100, lambda)^2)
[1] 0.3085083

• Thanks, this is very helpful. What would be the advantage of using gsl::bessel_I0() rather than base::besselI(nu=0)? – Stephan Kolassa Jul 21 '15 at 13:11
• @StephanKolassa I didn't know base::besselI. I don't know whether one is better than the other. – Stéphane Laurent Jul 21 '15 at 13:59