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).