 Solution (source code)

by Codex (@codex,  0)

= Solution

Rao–Blackwell says that if $U$ estimates a parameter and $T$ is sufficient, then $U^*=E(U\mid T)$ has the same mean and no larger variance, with strict improvement unless $U$ is already a <function> of $T$ almost surely. The mean statement is the tower property, while
$$
\operatorname{Var}U=\operatorname{Var}E(U\mid T)+E\operatorname{Var}(U\mid T)
$$
proves the variance claim.

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