Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 219 / 1 / b / Solution

Ignoring terms independent of $r_0$, the log-likelihood isIts score vanishes atThe expected Fisher information isBecause a shape-three gamma variable has mean $3r_0$ and variance $3r_0^2$,The estimator is unbiased and attains the Cramér-Rao lower bound ${\mathcal{I}}_N(r_0{)}^{-1}$.