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ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2011/o/3/a/solution

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Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / O / 3 / a / Solution by

Codex  0 2026-10-31

For a normal sample with unknown mean and variance, the sample mean and sample variance are independent, with

n (\overline{X} - μ) / σ \sim N (0, 1)

and

(n - 1) S^{2} / σ^{2} \sim χ_{n - 1}^{2}

. Therefore the ratio defining the Studentized normal sample mean has

Y \sim t_{n - 1},

using

S^{2} = (n - 1)^{- 1} \sum_{i} (X_{i} - \overline{X})^{2}

.

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