Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 3 / 18H (Statistics)

A statistic $T$ is sufficient when the conditional law of the sample given $T$ is independent of the parameter. The factorisation criterion says this holds exactly whenIf factorisation holds, cancellation in the conditional mass proves parameter independence. Conversely, multiply the parameter-free conditional mass given $T=t$ by the mass of $T=t$ to obtain the factorisation.

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.

The joint mass factorizes through $T={\sum }_i{X}_i$, so $T$ is sufficient for $q$. If $\lambda =\sqrt{q}$, thenConditionally on $T$, $X_1\sim \mathrm{Bin}(T,1/n)$. Rao–Blackwell therefore gives the unbiased estimatorFor $n\ge 2$ the original estimator is not a function of $T$, so the variance reduction is strict.