 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-3/28j/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Multiplying the likelihood by the gamma distribution prior density gives

π (θ ∣ x_{1}, \dots, x_{n}) \propto θ^{n} e^{- θ S_{n}} θ^{α - 1} e^{- βθ} = θ^{α + n - 1} e^{- (β + S_{n}) θ} .

θ ∣ X_{1}, \dots, X_{n} \sim Gamma (α + n, β + S_{n}),

θ_{π} = \frac{α + n}{β + S _{n}} .

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