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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ii/paper-4/27i/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2005 / ii / Paper 4 / 27I / i / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

Complete the square in the independent normal prior and likelihood:

τ θ_{i}^{2} + τ_{ε} (X_{i} - θ_{i})^{2} = (τ + τ_{ε}) (θ_{i} - \frac{τ _{ε} X _{i}}{τ + τ _{ε}})^{2} + \frac{τ τ _{ε}}{τ + τ _{ε}} X_{i}^{2} .

(1)

Thus, conditional on the full observations, the hospital parameters remain independent and

θ_{i} ∣ X \sim N (\frac{τ _{ε}}{τ + τ _{ε}} X_{i}, \frac{1}{τ + τ _{ε}}) .

(2)

j

is the observed minimum index and

X_{j} = x

, its conditional distribution is exactly this formula with

X_{j} = x

. No extra truncation is introduced because the selection index is already a function of the conditioned data. Its posterior mean is shrunk toward zero, so the worst measured performance is not simply identified with the worst true performance.

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