 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-4/17h/d/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

θ^{α - 1} (1 - θ)^{β - 1} .

Since the original posterior density is proportional to

θ^{X} (1 - θ)^{n - X},

the reweighted density is

Beta (X + α, n - X + β)

. Its mean gives

θ = \frac{X + α}{n + α + β} .

Writing this in the required form,

θ = w θ + (1 - w) θ_{0}, w = \frac{n}{n + α + β}, θ_{0} = \frac{α}{α + β} .

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