Solution
= Solution
For <independent random variables>, the <probability generating function of an independent sum> is the product of the individual <probability generating functions>. Thus
$$
G_{X+Y}(s)=e^{\lambda(s-1)}e^{\mu(s-1)}=e^{(\lambda+\mu)(s-1)}.
$$
Expanding this last expression as $e^{-(\lambda+\mu)}\sum_{r\geq0}(\lambda+\mu)^rs^r/r!$ identifies each probability coefficient. Therefore
$$
\boxed{X+Y\sim\operatorname{Poisson}(\lambda+\mu)}.
$$