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.


  • 1
    $\begingroup$ This question appears in many forms. Answers can be found by a search for skellam. $\endgroup$ – 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 ?

enter image description here

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.