 Solution

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

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

Let

m = 1/ σ^{2}

. Within block

i

, the

m

replicated observations have dispersion

a_{i}

and common natural parameter

θ_{i}

. Evaluating their joint MGF at the blockwise argument

σ^{2} t_{i}

gives

[exp {\frac{b ( θ _{i} + a _{i} σ ^{2} t _{i} ) - b ( θ _{i} )}{a _{i}}}]^{m} = exp {\frac{b ( θ _{i} + a _{i} σ ^{2} t _{i} ) - b ( θ _{i} )}{a _{i} σ ^{2}}} .

Multiplying over blocks reproduces the MGF of

Y

from part (a). Since moment-generating functions uniquely determine distributions in this family,

\overset{ˉ}{Y} = d Y

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