Ignoring terms independent of $r_0$, the <log-likelihood> is
$$
\ell(r_0)=-3N\log r_0-\frac1{r_0}\sum_{s=1}^Nr_s.
$$
Its score vanishes at
$$
\widehat r_0=\frac1{3N}\sum_{s=1}^Nr_s.
$$
The expected <Fisher information> is
$$
\mathcal I_N(r_0)=-\mathbb E\ell''(r_0)=\frac{3N}{r_0^2}.
$$
Because a shape-three gamma variable has mean $3r_0$ and variance $3r_0^2$,
$$
\mathbb E\widehat r_0=r_0,
\qquad
\operatorname{Var}(\widehat r_0)=\frac{r_0^2}{3N}.
$$
The estimator is unbiased and attains the <Cramér-Rao lower bound> $\mathcal I_N(r_0)^{-1}$.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-219/1/b/solution