# Proper name for the generalized binomial distribution with two trials?

Two Bernoulli trials are performed, first with success probability $p_1$ and second with success probability $p_2 \not=p_1$. The resulting distribution for the number of positive outcomes is (obviously),

$P_2=p_1 p_2$, $P_1=(1-p_1)(1-p_2)$ and $P_2=1-P_0-P_2$.

I need a name for this distribution which is as informative as possible. Is there a better alternative than a generalized binomial distribution? This question is part of an earlier SE question here but it is not in the focus and didn't receive an answer.

(Sorry if the questions sounds boring, but I'm writing a paper and want to be right with terminology).

## 2 Answers

It's called a Poisson binomial distribution. You can find useful information about it on Wikipedia.

• +1. Never heard of this one... I guess one learns new things every day... :) – user76844 Mar 14 '14 at 16:50
• It comes up occasionally on both the stats and math SE sites. It's a handy one to know. – soakley Mar 15 '14 at 1:55

Edit Based on OP Comments

Actually, I realised that the sum of two bernoulli rvs with different ps will result in *under*dispersion, so perhaps underdispersed?

If you don't want to be associated with dispersion models, then why not "Heterogeneous Binomial Sum", its clearer than generalized binomial, as there are several ways you could generalize it.

• Thanks, but this probably won't help. The variability of $p_i$ in my case is deterministically tuned, and some data points for $P_n$'s even go beyond the generalized/overdispersed binomial constraint. Here is the physics context: physics.stackexchange.com/questions/61835 – Slava Kashcheyevs Mar 7 '14 at 21:04
• @Slaviks the p's don't need to be random, and they don't need to have a particular distribution. All that matters is that each Bernoulli trial does not have the same p. In your case, you have a simple prior distribution. – user76844 Mar 7 '14 at 21:17
• @Slaviks see revised response. – user76844 Mar 7 '14 at 21:25