[Math] Linear combination of independent random variables that are poisson distributed

Question

Suppose $X_1$ and $X_2$ are independent random variables $X_1$~ Poisson$(\lambda_1)$ and $X_2$~ Poisson$(\lambda_2)$

I want to show $X_1 + X_2$~poisson$(\lambda_1 + \lambda_2)$

I want to then generalize this to a sum of $n$ independent Poisson random variables

Answer 1

Hint:

First proof $X_1+X_2 ~ Poisson(\lambda_1+\lambda_2)$ (the proof is here if you want to see it).

Second use mathematical induction using what you have proved first to pass from $n$ to $n+1$ (because the sum of the first $n$ is poisson and is added $X_{n+1}$)