State and prove the Rao-Blackwell Theorem.

Suppose that $X_{1}, X_{2}, \ldots, X_{n}$ are independent, identically-distributed random variables with distribution

$$P\left(X_{1}=r\right)=p^{r-1}(1-p), \quad r=1,2, \ldots$$

where $p, 0<p<1$, is an unknown parameter. Determine a one-dimensional sufficient statistic, $T$, for $p$.

By first finding a simple unbiased estimate for $p$, or otherwise, determine an unbiased estimate for $p$ which is a function of $T$.