Hi, can someone help me with this question:

Let X~Poisson(1), Y~Poisson(2), define T = X + Y.

a) Show that T~Poisson(3)

b) Find the joint probability function of X and T

c) Find the conditional distribution of X given T = n

d) Compute Cov(X,T) and the correlation coefficient.

I have no problems with a). But I got stuck at b) because I get that the jpf of X and T, f(X,T) = P ( X = x , T = t) = f(T) ...?! Which makes no sense. Any help would be appreciated!