# Proof involving Poisson and Gamma distributions for two random variables [duplicate]

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!

• These questions have already been asked and answered before on this website. Try searching them. – Shubham Johri Dec 10 '19 at 22:34

The respective characteristic functions are $$\exp[\lambda(\exp it-1)]$$, $$(1-it/\beta)^{-\alpha}$$.