TITLE: Addition of two Binomial Distribution
QUESTION [15 upvotes]: What is the distribution of the variable $X$ given $$ X = Y + Z, $$where $Y \sim $ Binomial($n$, $P_Y$) and $Z\sim$ Binomial($n$, $P_Z$)?

For the special case, when $P_Y = P_Z = P$, I think that X~Binomial($2n$, $P$) is correct. If $P_A ≠ P_B$, the distribution might eventually just be Binomial$\left(2n, \frac{P_A + P_B}{2}\right)$ but I can't prove it.
If the problem is more complicated than I expect and we can't derive the whole distribution, can we tell something about the mean and the variance of $X$?

REPLY [9 votes]: It will be a special case of the Poisson Binomial Distribution.