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Prove that if $X_i \sim \text{Poi}(λ_i)$, $i = 1, 2$, are independent, the sum $X_1 + X_2$ has the Poisson distribution as well.

Prove that if $X_i \sim \text{Gamma}(\alpha_i,\beta)$, $i = 1, 2$, are independent, the sum $X_1 + X_2$ has the gamma distribution as well ($i$'s are meant to be subscript).

I am not sure how to go about solving these problems, help would be greatly appreciated!

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  • $\begingroup$ These questions have already been asked and answered before on this website. Try searching them. $\endgroup$ – Shubham Johri Dec 10 '19 at 22:34
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The respective characteristic functions are $\exp[\lambda(\exp it-1)]$, $(1-it/\beta)^{-\alpha}$.

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