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

Table of contents
- Solution b

Solution

Ignoring terms independent of

r_{0}

, the log-likelihood is

ℓ (r_{0}) = - 3 N lo g r_{0} - \frac{1}{r _{0}} \sum_{s = 1}^{N} r_{s} .

Its score vanishes at

r_{0} = \frac{1}{3 N} \sum_{s = 1}^{N} r_{s} .

The expected Fisher information is

I_{N} (r_{0}) = - E ℓ^{''} (r_{0}) = \frac{3 N}{r _{0}^{2}} .

Because a shape-three gamma variable has mean

3 r_{0}

3 r_{0}^{2}

,

E r_{0} = r_{0}, Var (r_{0}) = \frac{r _{0}^{2}}{3 N} .

The estimator is unbiased and attains the Cramér-Rao lower bound

I_{N} (r_{0})^{- 1}

.

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