# Probability generating function [closed]

X and Y are independent random variables having poisson distribution with parameters a and b respectively. By using probability generating function, prove that X + Y have a poisson distribution and give it's parameter

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So the PGF of a discrete random variable X is defined as $$E[z^X]$$. By the law of the unconscious statistician, the expectation of a function of $$X$$ is the sum (over the support of the random variable) of the function times the PDF (PMF in this case). For the Poisson distribution this works out to
$$\sum_{x=0}^\infty z^x \frac{\lambda^x e^{-\lambda}}{x!} = e^{-\lambda} \sum_{x=0}^\infty z^x \frac{\lambda^x}{x!}$$
Now remember that the Taylor expansion of $$e^x = \sum_{k=0}^\infty \frac{x^k}{k!}$$
Also, much like the MGF, the PGF $$G(z)$$ has the property that if 2 random variables are $$X$$ and $$Y$$ independent, and $$S = X+Y$$, then $$G_S(z) = G_X(z)G_Y(z)$$.