 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/iii/paper-37/3/a/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

Pr (Y_{i} = y_{i}) = p_{i}^{y_{i}} (1 - p_{i})^{1 - y_{i}} = exp {y_{i} lo g \frac{p _{i}}{1 - p _{i}} + lo g (1 - p_{i})} .

θ_{i} = lo g [p_{i} / (1 - p_{i})]

gives

p_{i} = e^{θ_{i}} / (1 + e^{θ_{i}})

and

lo g (1 - p_{i}) = - lo g (1 + e^{θ_{i}})

. Therefore the exponential family representation has

θ_{i} = logit (p_{i}) = β^{T} x_{i}, b (θ) = lo g (1 + e^{θ}), ϕ = 1, c (y, ϕ) = 0.

The logit link is the canonical link.

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