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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?

Kind regards,

Michel

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  • $\begingroup$ I have reformulated my problem and I have posted it again: link $\endgroup$
    – Michel
    Feb 9 '13 at 19:34