Assume that Xi, i = 1; 2; 3 are independent Poisson random variables with respective means ?i, i = 1; 2; 3. Let X = X1 + X2 and Y = X2 + X3. The random vector X; Y is said to have a bivariate Poisson distribution. Find its joint probability mass function. That is, find P{X = n;

Y = m} for all possible n and m. (Hint: Express the event

{X = n; Y = m} in terms X1; X2 ;X3 and then use the independence.)