 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/13k/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Conditionally on

X

, the log likelihood differs from

- \frac{1}{2 σ ^{2}} (Y - Xβ)^{T} Σ^{- 1} (Y - Xβ)

only by terms independent of

β

X^{T} Σ^{- 1} X β = X^{T} Σ^{- 1} Y,

and therefore

β (Σ) = (X^{T} Σ^{- 1} X)^{- 1} X^{T} Σ^{- 1} Y .

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