 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ib/paper-1/18h/c/solution

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

Unbiasedness requires

\sum c_{i} X_{i} = 1

. Cauchy–Schwarz gives

1 \leq (\sum c_{i}^{2}) S_{xx}

, so

Var (\tilde{β}) = σ^{2} \sum c_{i}^{2} \geq \frac{σ ^{2}}{S _{xx}} .

Equality holds exactly for

c_{i} = X_{i} / S_{xx}

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