Let $X_{1}, \ldots, X_{n}$ be i.i.d. Bernoulli $(p)$ random variables, where $n \geqslant 3$ and $p \in(0,1)$ is unknown.

(a) What does it mean for a statistic $T$ to be sufficient for $p$ ? Find such a sufficient statistic $T$.

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

(c) By considering the estimator $X_{1} X_{2}$ of $p^{2}$, find an unbiased estimator of $p^{2}$ that is a function of the statistic $T$ found in part (a), and has variance strictly smaller than that of $X_{1} X_{2}$.