The bivariate poisson distribution.

Suppose that are independent Poisson random variables with respective means . Let and [TEXY=X_2+X_3[/TEX]. The random vector is said to have a bivariate Poisson distribution. Find its joint probability mass function. That is, find

I tried:

We know

However I get stuck as these two are not independent and cannot therefore just be multiplied

I also tried to create a new variable Z and use the equation for functions of random variables, but the Jacobian is zero due to the zero row.