Hello! I am trying to understand an example from my book that deals with two independent Poisson random variables X1 and X2 with parameters λ1 and λ2. The problem is to find the probability distribution of Y = X1 + X2. I am aware this can be done with the moment-generating function technique, but the author is using this problem to illustrate the transformation technique. He starts by obtaining the joint probability distribution of the two variables: f(x1, x2) = p1(x1)p2(x2) for x1 = 0, 1, 2,... and x1 = 0, 1, 2,... Then he proceeds onto saying: "Since y = x1 + x2 and hence x1 = y - x2, we can substitute y - x2 for x1, getting: g(y, x2) = f(y - x2, x2) for y = 0, 1, 2,... and x2 = 0, 1,..., y for the joint distribution of Y and X2." Then he goes ahead and obtains the marginal distribution of Y by summing over all x2. My question is this. How did he obtain the region of support (y = 0, 1, 2,... and x2 = 0, 1,..., y) for g(y, x2). I can't for the life of me understand this. Thank you for your help!