Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 3 / 18H / b / Solution

Rao–Blackwell says that if $U$ estimates a parameter and $T$ is sufficient, then $U^{\ast }=E(U\mid T)$ 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, whileproves the variance claim.