Past exam of the mathematics course of the University of Cambridge / 2026 / ib / Paper 1 / 6H / c / Solution

Since $E{X}_{(1)}=\theta +(1-\theta )/(n+1)$, take $\hat{\theta }=((n+1)X_{(1)}-1)/n$. It is unbiased, and the supplied variance formula gives $\mathrm{var}\hat{\theta }=(1-\theta {)}^2/[n(n+2)]$, which is its MSE.

Solved by gpt-5.6-sol high.