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The Rao-Blackwell theorem states that if an estimator $D$ has finite second moment and $T$ is a sufficient statistic, then $D^{\ast }(T)=\mathbb{E}[D\mid T]$ can be chosen without knowing the parameter. It has the same expectation as $D$ and no larger variance. In particular an unbiased estimator remains unbiased and its squared-error risk cannot increase. Sufficiency makes the conditional distribution, hence the function of $T$ used here, parameter-independent.

The tower property gives $\mathbb{E}{D}^{\ast }=\mathbb{E}D$. The law of total variance givesso the variance decreases, with equality exactly when $D$ is already a function of $T$ almost surely. Since the bias is unchanged, the same identity proves the squared-error risk claim. More generally the corresponding convex-loss inequality follows from conditional Jensen's inequality.

For the given geometric distribution, the joint mass is $(1-p{)}^n{p}^{\sum x_j-n}$ on positive-integer sample vectors. Thus $T={\sum }_j{X}_j$ is a one-dimensional sufficient statistic. Directly, conditional on $T=t$, every positive-integer composition of $t$ into $n$ parts has the same probability; there are $\left(\genfrac{}{}{0ex}{}{t-1}{n-1}\right)$ such compositions, independently of $p$.

The simple estimator $D=1_{\{X_1>1\}}$ is unbiased for $p$, since $\mathrm{Pr}(X_1>1)=p$. For $n\ge 2$, compositions with $X_1>1$ are counted by subtracting one from their first part, giving $\left(\genfrac{}{}{0ex}{}{t-2}{n-1}\right)$ when $t>n$. Consequently the Rao-Blackwell estimator of a geometric failure probability isAt $T=n$ its value is zero, as it must be when all observations are one. For the possible one-observation case, use $\hat{p}=1_{\{T>1\}}$; the displayed fraction would otherwise be undefined at $T=1$. The conditional-expectation argument proves unbiasedness without needing to sum the negative-binomial mass explicitly.