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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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    – jbowman
    Aug 20, 2023 at 22:58

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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 $G_S(z) = G_X(z)G_Y(z)$.

With these two properties you should be able to deduce the PGF of the sum.

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