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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 5L / b / Solution

Write the exponential-dispersion density as

f (y_{i}; θ_{i}, ϕ) = exp {\frac{y _{i} θ _{i} - b ( θ _{i} )}{ϕ} + c (y_{i}, ϕ)} .

With the canonical link,

θ_{i} = η_{i} = x_{i}^{T} β

. Therefore

E Y_{i} = b^{'} (x_{i}^{T} β), Var (Y_{i}) = ϕ b^{''} (x_{i}^{T} β) .

Known prior weights replace

ϕ

by

ϕ / w_{i}

.

Solved by gpt-5.6-sol high.

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