 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2008/iii/paper-44/2/c/solution

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

Integrate the latent variable out to obtain the hospital's marginal likelihood:

p (y_{i} ∣ ψ) = \frac{ψ}{y _{i} !} \int_{0}^{\infty} θ^{y_{i}} e^{- (1 + ψ) θ} d θ = \frac{ψ}{( 1 + ψ ) ^{y_{i} + 1}} .

For

ϕ = (1 + ψ)^{- 1}

this becomes

p (y_{i} ∣ ϕ) = (1 - ϕ) ϕ^{y_{i}}

1 - ϕ

. The transformation concerns the likelihood, so no Jacobian is inserted here.

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