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Past exam of the mathematics course of the University of Cambridge / 2025 / ib / Paper 1 / 18H / a / Solution

Let

S_{xx} = \sum X_{i}^{2}

. The likelihood equations give

\hat{β} = \frac{\sum X _{i} Y _{i}}{S _{xx}}, \overset{σ}{^}^{2} = \frac{1}{n} \sum (Y_{i} - X_{i} \hat{β})^{2} .

These are orthogonal projections of a Gaussian vector onto the span of

X

and its orthogonal complement, hence are independent.

Solved by gpt-5.6-sol high.

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