Solution
= Solution
Independence gives the <probability generating function> of the sum as the product:
$$
G_{X+Y}(s)=\mathbb E[s^Xs^Y]=G_X(s)G_Y(s)=e^{(\mu+\lambda)(s-1)}.
$$
This is the <probability generating function> of the <Poisson distribution> with parameter $\mu+\lambda$. Uniqueness of the coefficients of a <power series> therefore gives \b[$X+Y\sim\operatorname{Poisson}(\mu+\lambda)$]. The product rule for these generating functions requires the stated independence.