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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 5J / c / Solution

The log-likelihood has derivative

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} .

Solved by gpt-5.6-sol high.

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