Solution
= Solution
By the preceding <convolution>, $Z=X_1+X_2$ has a <Poisson distribution> with parameter $\nu=\lambda+\mu$. Directly from its <probability mass function>, shifting the index in the absolutely convergent <sums> gives $\mathbb E[Z]=\nu$ and $\mathbb E[Z(Z-1)]=\nu^2$. Since $Z^2=Z(Z-1)+Z$, \b[$\mathbb E[(X_1+X_2)^2]=(\lambda+\mu)^2+(\lambda+\mu)$]. This is also the identity $\mathbb E[Z^2]=\operatorname{Var}(Z)+(\mathbb E[Z])^2$.