(a) State and prove the Rao-Blackwell theorem.

(b) Let $X_{1}, \ldots, X_{n}$ be an independent sample from $\operatorname{Poisson}(\lambda)$ with $\theta=e^{-\lambda}$ to be estimated. Show that $Y=1_{\{0\}}\left(X_{1}\right)$ is an unbiased estimator of $\theta$ and that $T=\sum_{i} X_{i}$ is a sufficient statistic.

What is $\mathbb{E}[Y \mid T] ?$