X and Y are independent Poisson random variables, each with rate lambda.

Question

I tried using the poisson formula to solve this but came to nothing. The answer: ln 4.

Answer 1

Hint

You should know that if $X\sim \mathcal P(\lambda_1)$ and $Y\sim \mathcal P(\lambda _2)$ are independents, then $X+Y\sim\mathcal P(\lambda _1+\lambda _2)$. If not, prove it ! Given this information, the result is straightforward.

Answer 2

Because of independence we have the equality:

$$P(X=1)=2P(X=0)P(Y=1)+2P(X=1)P(Y=0)$$

Work it out.