Question

# Compute the distribution of X+Y in the following cases:X and Y are independent Poisson random variables with means respective \lambda_{1} and \lambda_{2}.

Random variables

Compute the distribution of $$X+Y$$ in the following cases:
X and Y are independent Poisson random variables with means respective $$\lambda_{1} and \lambda_{2}$$.

2021-06-04

$$X\sim P(\lambda_{1})$$
$$X\sim P(\lambda_{2})$$
X and Y are independent
$$P(\lambda_{1})+P(\lambda_{2})\sim P(\lambda_{1}+\lambda_{2})$$
So,
$$X+Y$$ follows poisson distribution.
$$(X+Y)\sim P(\lambda_{1}+\lambda_{2})$$