I am working with a problem that involves the product of two mass probability functions for Poisson distribution. I am unsure if this is a common occurence and I am left wondering if the function I get as a result is known and has any special distinct features.
Letting $\alpha$ and $\beta$ denote the mean of the distributions, my equation turns out to be the following:
$\frac{e^{-|\alpha|^2}|\alpha|^{2n}}{n!}\frac{e^{-|\beta|^2}|\beta|^{2n}}{n!} \nonumber\\ = e^{-(|\alpha|^2+|\beta|^2)}\frac{|\alpha\beta|^{2n}}{(n!)^2}$
To me, the result almost look like a new mass probability function for a Poisson distribution, with mean $\alpha \beta$, except the factorial is squared now. Is this new function a known function with a specific name?
Apologies for any misuse of definitions, I am not well-versed in probability theory.
If $(p_n)$ and $(q_n)$ are pmf's such that $(p_nq_n)$ is also a pmf then $1=\sum p_nq_n \leq \sum p_n=1$ since $q_n \leq 1$ for all $n$. It follows from this that we must have $p_nq_n=p_n$ for all $n$. Similarly, $p_nq_n=q_n$ for all $n$ so $p_n=q_n$ for all $n$. But then $1=\sum p_n^{2} \leq \sum p_n=1$ shows that $p_n=0$ or $1$ for all $n$ which means there exists $m$ such that $p_m=1$ and $p_n=0$ for $n \neq m$.