 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-2/5j/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

ℓ^{'} (θ) = \sum_{i} Y_{i} - n A^{'} (θ) .

Differentiating the normalizing identity for the family gives

A^{'} (θ) = E_{θ} Y_{1} = μ (θ), A^{''} (θ) = Var_{θ} (Y_{1}) \geq 0.

Hence an interior maximum satisfies, and under nondegeneracy uniquely satisfies,

μ (θ_{MLE}) = \overline{Y} .

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