 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-1/18h/c/solution

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

δ

R (δ) = 2 \int_{γ \leq δ} (δ - γ) d F (γ) + \int_{γ > δ} (γ - δ) d F (γ) .

At continuity points its derivative is

R^{'} (δ) = 2 F (δ) - [1 - F (δ)] = 3 F (δ) - 1.

Thus the risk is minimized at a posterior one-third quantile:

γ_{B} = F^{- 1} (\frac{1}{3}) .

More generally, any

δ

satisfying

F (δ -) \leq \frac{1}{3} \leq F (δ)

is optimal. The quantile lies below the posterior median, reflecting the smaller penalty assigned to underestimation.

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