Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 219 / 1 / c / Solution

Writing $r_s=1/{\alpha }_s$, the log likelihood isHenceThe score variance and negative expected Hessian give Fisher informationSince $r_s$ is Gamma with mean $3r_0$ and variance $3r_0^2$,The estimator is unbiased and attains the Cramér-Rao lower bound.