 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/ib/paper-1/6h/c/solution

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

Since

E X_{(1)} = θ + (1 - θ) / (n + 1)

, take

\hat{θ} = ((n + 1) X_{(1)} - 1) / n

. It is unbiased, and the supplied variance formula gives

var \hat{θ} = (1 - θ)^{2} / [n (n + 2)]

, which is its MSE.

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