 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-4/13j/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

f_{i} (y_{i}; θ_{i}, σ_{i}^{2}) = exp {\frac{y _{i} θ _{i} - b ( θ _{i} )}{σ _{i}^{2}} + c (y_{i}, σ_{i}^{2})},

where

σ_{i}^{2} = a_{i} σ^{2}

μ_{i} = E [Y_{i}] = b^{'} (θ_{i})

g (μ) = Xβ

. It follows from normalization of the density after shifting the natural parameter that

E [e^{t_{i} Y_{i}}] = exp {\frac{b ( θ _{i} + σ _{i}^{2} t _{i} ) - b ( θ _{i} )}{σ _{i}^{2}}} .

Independence therefore gives the joint moment-generating function

E [e^{t^{T} Y}] = exp {\sum_{i = 1}^{n} \frac{b ( θ _{i} + a _{i} σ ^{2} t _{i} ) - b ( θ _{i} )}{a _{i} σ ^{2}}},

whenever every shifted natural parameter lies in its natural-parameter domain.

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