 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-36/2/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

E_{i}

as known and positive. The Poisson distribution log-likelihood, up to a parameter-independent constant, is

ℓ_{i} (λ_{i}) = y_{i} lo g λ_{i} - E_{i} λ_{i}

ℓ_{i}^{'} (λ_{i}) = \frac{y _{i}}{λ _{i}} - E_{i}, ℓ_{i}^{''} (λ_{i}) = - \frac{y _{i}}{λ _{i}^{2}} .

Since

E (Y_{i}) = E_{i} λ_{i}

I_{i} (λ_{i}) = - E ℓ_{i}^{''} (λ_{i}) = \frac{E _{i}}{λ _{i}} .

The Jeffreys prior is proportional to its square root. Removing the constant

E_{i}

gives

π_{J} (λ_{i}) \propto λ_{i}^{- 1/2}, λ_{i} > 0.

This is an improper prior: its integral diverges at infinity.

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