 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ib/paper-1/18h/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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.

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