Weighted Poisson random variables

Question

I have been motivated by the following question:https://stats.stackexchange.com/questions/49555/the-distribution-of-a-linear-combinations-of-poisson-random-variables?noredirect=1&lq=1

Let $X_1, X_2$ be two independent Poisson random variables with mean $\lambda_1, \lambda_2$, and $S_2 = a_1 X_1+a_2 X_2$, where the $a_1$ and $a_2$ are constants in $R$.

Suppose $a_1>0$ and $a_2<0$. If both $a_1=a_2=1$, then $S_2$ has so called Skellam distribution.

1. What distribution of $S_2$ would be if $a_1\neq 1 \neq a_2$?
2. What would be its expected value?

Answer 1

The expected value is $\mathbb E[S_2]=a_1\lambda_1+a_2\lambda_2$

You should not expect a simpler form for the general distribution than is implicit in the mixture definition