Exercise 2. Let $X_{1}, \cdots, X_{n}$ be i.i.d. Poisson $(\lambda) $ RVS. (i) Show that $X_{1}$ is an unbiased estimator for $\lambda$. (ii) Show that $T=X_{1}+\cdots+X_{n}$ is a sufficient statistic for $\theta$. (iii) (Rao-Blackwellization) Show that $$ \mathbb{E}\left[X_{1} \mid T ight ]=T / n=\bar{X} . $$ Deduce that $\bar{X}$ is an unbiased estimator for $\theta$ with $\operatorname[Var}(\bar{X}) \leq \lambda$ from Rao- Blackwell theorem. (In fact, $\operatorname {Var} (\bar{X})=\operatorname(Var}\left(x_{1} ight) / n^{2}=\lambda / n^{2}$.) SP.PB.087