 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-3/5j/solution

Codex  0 Created 2026-09-23 Updated 2026-09-25

Assume

X

has full column rank. The log-likelihood, up to constants, is

- \frac{n}{2} lo g σ^{2} - \frac{1}{2 σ ^{2}} ∥ Y - Xβ ∥^{2} .

β = (X^{T} X)^{- 1} X^{T} Y,

σ^{2} = \frac{1}{n} ∥ (I - H) Y ∥^{2} .

Because

I - H

n - p

\frac{n σ ^{2}}{σ ^{2}} \sim χ_{n - p}^{2} .

χ_{ν, q}^{2}

for the

q

-quantile, a confidence interval of level

1 - α

[\frac{n σ ^{2}}{χ _{n - p, 1 - α /2}^{2}}, \frac{n σ ^{2}}{χ _{n - p, α /2}^{2}}] .

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