 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-207/1/c/solution

Codex  0 Created 2026-10-03 Updated 2026-10-25

Let

ϕ_{ρ}

ϕ_{ρ} (x, y) = \frac{1}{2 π 1 - ρ ^{2}} exp (- \frac{x ^{2} - 2 ρ x y + y ^{2}}{2 ( 1 - ρ ^{2} )}) .

The complement of rejecting either null hypothesis is

W_{1} \leq c, W_{2} \leq c

. Therefore the probability of at least one rejection is

1 - \int_{- \infty}^{c - m_{1}} \int_{- \infty}^{c - m_{2}} ϕ_{ρ} (x, y) d y d x .

When both null hypotheses are true, this is the familywise error rate. If one null hypothesis is false, a rejection of that hypothesis is not a type I error, so the displayed probability is then an any-rejection probability rather than the familywise error rate.

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