# Thread: linear combination of Poisson random variables

1. ## linear combination of Poisson random variables

hi!
does anybody know the probability distribution of a linear combination of Poisson variables?
If it is a simple sum, then i know that it is a Poisson variable with a parameter that is the sum of the parameters of the Poisson variables of the sum.
But I have no clues about a linear combination...

2. Hello,
Originally Posted by ravari
hi!
does anybody know the probability distribution of a linear combination of Poisson variables?
If it is a simple sum, then i know that it is a Poisson variable with a parameter that is the sum of the parameters of the Poisson variables of the sum.
But I have no clues about a linear combination...
Let's assume they're independent.

And as far as I can see, there is no particular distribution for a general linear combination... (you can have a general idea by studying the mgf)

3. yes, independent sorry.

but, do you know of any approximate distributions for the linear combination?

4. Originally Posted by ravari
yes, independent sorry.

but, do you know of any approximate distributions for the linear combination?
Read this: Sum of two independent Poisson random variables

5. Thanks,but that's the result for the sum. I'm interested in the probability distribution of:

X = a1 x1 + a2 x2 + ... + aN xN

with ai positive real numbers and xi independent poisson variables of parameters gi.

6. Originally Posted by ravari
Thanks,but that's the result for the sum. I'm interested in the probability distribution of:

X = a1 x1 + a2 x2 + ... + aN xN

with ai positive real numbers and xi independent poisson variables of parameters gi.
Well, what does the approach taken to get the sum tell you ....? And what does this link suggest: Skellam distribution - Wikipedia, the free encyclopedia

Perhaps you should state the problem that has motivated your interest.