What is the probability $X+Y=0$ for two independent Poisson random variables?

Question

For two independent Poisson random variables, $X$ and $Y$, with parameters $\lambda_1 > 0$ and $\lambda_2>0$ respectively, how do I find P$\{X+Y=0\}$ in terms of $\lambda_1$ and $\lambda_2$?

Answer 1

Hint: a sum of independent Poisson variables with parameters $\lambda_1$ and $\lambda_2$ is a Poisson variable with parameter $\lambda_1 + \lambda_2$.

Answer 2

Hint: $X+Y = 0$ if and only if $X = 0$ and $Y = 0$.