Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 3 / 20H / a

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- Solution a

Solution

The sample mean is

\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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