 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ib/paper-3/20h/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

\overline{X} \sim N (μ, \frac{σ ^{2}}{n}) .

Write the Gaussian sample vector as its orthogonal projection onto the span of

(1, \dots, 1)

plus its projection onto the orthogonal complement. These two Gaussian projections are independent. Their squared standardized lengths give Cochran's theorem:

\frac{S _{XX}}{σ ^{2}} \sim χ_{n - 1}^{2}, \overline{X} is independent of S_{XX} .

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