I would like to know what the distribution is of linear combinations of Poisson random variables.
I know that a linear combinations of Poisson random variables is not always a Poisson random variable, namely
$Y : = \sum_k a_k X_k$
with $X_k \sim Pois(\lambda_k)$. Now, $Y$ is only Poisson if all $a_k$ are 1 and else $Y$ is a not Poisson random variable but follows a 'compound Poisson distribution'.
My question is can I still find an expression for this distribution or can I generate it using for instance Matlab?