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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 1

Conditionally on $X$, the log likelihood differs from
$$
-\frac1{2\sigma^2}(Y-X\beta)^T\Sigma^{-1}(Y-X\beta)
$$
only by terms independent of $\beta$. <Differentiation> gives the weighted normal equations
$$
X^T\Sigma^{-1}X\widehat\beta=X^T\Sigma^{-1}Y,
$$
and therefore
$$
\boxed{\widehat\beta(\Sigma)=(X^T\Sigma^{-1}X)^{-1}X^T\Sigma^{-1}Y.}
$$

Solution

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

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

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

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