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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 13K / a / Solution

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

. Differentiation gives the weighted normal equations

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 .

Solved by gpt-5.6-sol high.

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