Chapter 5.3, Problem 1E

Let $X_1$ and $X_2$ be independent Poisson random variables with respective means ${\lambda }_1=2$ and ${\lambda }_2=3$.

Find

(a) $P\left(X_1=3,X_2=5\right)$.

(b) $P\left(X_1+X_2=1\right)$.

HINT: Note that this event can occur if and only if $\left\{X_1=1,X_2=0\right\}$ or $\left\{X_1=0,X_2=1\right\}$.