State and prove the Rao-Blackwell theorem.

Individuals in a population are independently of three types $\{0,1,2\}$, with unknown probabilities $p_{0}, p_{1}, p_{2}$ where $p_{0}+p_{1}+p_{2}=1$. In a random sample of $n$ people the $i$ th person is found to be of type $x_{i} \in\{0,1,2\}$.

Show that an unbiased estimator of $\theta=p_{0} p_{1} p_{2}$ is

$$\hat{\theta}= \begin{cases}1, & \text { if }\left(x_{1}, x_{2}, x_{3}\right)=(0,1,2) \\ 0, & \text { otherwise. }\end{cases}$$

Suppose that $n_{i}$ of the individuals are of type $i$. Find an unbiased estimator of $\theta$, say $\theta^{*}$, such that $\operatorname{var}\left(\theta^{*}\right)<\theta(1-\theta)$.