Prove that

(a) If \(y \sim \operatorname{Poisson}(\lambda)\), then both the mean and variance of \(y\) are \(\lambda\).

(b) If \(y_{1}\) and \(y_{2}\) are independent and \(y_{j} \sim \operatorname{Poisson}\left(\lambda_{j}\right)(j=1,2)\), then the sum \(y_{1}+y_{2} \sim\operatorname{Poisson}\left(\lambda_{1}+\lambda_{2}\right)\).