# Math Help - A question on Poisson distribution

1. ## A question on Poisson distribution

Hi,

Let X1 and X2 be Poisson distributed variables with the same mean m and Y be defined as multiplication of X1 and X2 (Y=X1X2). How can i obtain the probability function (pf) of Y

I tried to find it such that

I define the following Y1=X1X2 and Y2=X2 so X1=Y1/Y2 and X2=Y2. When i put Y1/Y2 in place of X1 in the joint pf of X1 and X2, there is an unidentifiability for Y2=0 because denominator has such a term: (Y1/Y2)!.

Thanks in advance...

2. For fixed y1, calculate P(Y1 = y1, Y2 = 0) y referring back to the original random variables. If y1 = 0, this is just the probability that Y2 = 0, while if y1 != 0, the probability is 0. That will take care of the case Y2 = 0, and you can sum Y2 out in the usual way.

(I would be more clear but Latex is acting up, sorry )

3. Take the characteristic or moment generating functions for the two distributions, then use the appropriate form of the convolution theorem to get the corresponding characteristic or moment generator for the product, and find the inverse transform of the result.

CB