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I am trying to understand an example from my book that deals with two independent Poisson random variables X_{1}and X_{2}with parameters λ_{1}and λ_{2}. The problem is to find the probability distribution of Y = X_{1}+ X_{2}. 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(x_{1}, x_{2}) = p_{1}(x_{1})p_{2}(x_{2})

for x_{1}= 0, 1, 2,... and x_{1}= 0, 1, 2,...

Then he proceeds onto saying: "Since y = x_{1}+ x_{2}and hence x_{1}= y - x_{2}, we can substitute y - x_{2}for x_{1}, getting:

g(y, x_{2}) = f(y - x_{2}, x_{2})

for y = 0, 1, 2,... and x_{2}= 0, 1,..., y for the joint distribution of Y and X_{2}."

Then he goes ahead and obtains the marginal distribution of Y by summing over all x_{2}.

My question is this. How did he obtain the region of support (y = 0, 1, 2,... and x_{2}= 0, 1,..., y) for g(y, x_{2}). I can't for the life of me understand this.

Thank you for your help!

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# Sum of two independent Poisson random variables

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