 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ib/paper-3/18h/iii/solution

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

Since

β

Y

β \sim N_{p} (β, σ^{2} (X^{T} X)^{- 1}) .

The residual projection

I - H

has rank

n - p

. Cochran's theorem therefore gives

\frac{n σ ^{2}}{σ ^{2}} = \frac{∥ ( I - H ) Y ∥ ^{2}}{σ ^{2}} \sim χ_{n - p}^{2} .

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