Re: Sum of two correlated Poisson distribution
- From: David Winsemius <doe_snot@xxxxxxxxxxx>
- Date: Wed, 22 Nov 2006 08:42:44 -0600
"David Jones" <dajxxx@xxxxxxxxx> wrote in news:456439b9$1@xxxxxxxxxxxxxx:
Jacqueline wrote:
I am trying to develop the Sum of two correlated Poissondistribution.
Given that W1, W2 and W3 are independent Poisson distribtuion with
parameter s1,s2 and s3 respectly.
Let X=W1+W3 and Y=W2+W3.
How to compute the desity funtion of Z=X+Y (distribution of the sum
of those variables ? )
You are possibly better off trying this a different way. You want
Z=X+Y=W1+W2+2*W3
But the probabilities for (W1+W2) are well known, since this is just
Poisson, and the probabilities for (2*W3) are easy to get, since they
are a version of the Poisson probabilities where only even values are
possible outcomes. You then have to find the probabilities for the sum
of two independent variables (W1+W2) and (2*W3).
It may be that the problem you gave is only a special case of
something more general for which the above would not work. It might be
that using probability generating functions would provide a better
approach in both the above case, and more generally.
A more general formulation might involve copulas. Poisson models are common
in insurance applications. Embrechts,Lindskog and McNeil have written a
chapter on modeling correlated risks with copulas:
http://www.math.ethz.ch/~baltes/ftp/copchapter.pdf
Perhaps the OP may get useful approaches there.
--
David Winsemius
.
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