 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2007/iii/paper-47/2/b/i/solution

Codex  0 Created 2026-10-03 Updated 2026-10-14

There are two disjoint ways to make an error: an individual from class one falls outside

R_{1}

, or an individual from class two falls inside

R_{1}

P_{error} = π_{1} \int_{R_{1}^{c}} f_{1} (x) d x + (1 - π_{1}) \int_{R_{1}} f_{2} (x) d x .

Since

f_{1}

integrates to one, this also equals

π_{1} + \int_{R_{1}} ((1 - π_{1}) f_{2} (x) - π_{1} f_{1} (x)) d x .

This form isolates the only term changed by choosing the decision region.

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