 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iii/8/b/ii/solution

Each residual has mean zero and variance

σ^{2} (1 - 1/ n)

. By linearity of expectation, without needing independence of residuals,

E [S^{2}] = \frac{1}{n - 1} \sum_{i = 1}^{n} E [D_{i}^{2}] = \frac{n}{n - 1} σ^{2} (1 - 1/ n) = σ^{2} .

Hence

S^{2}

σ^{2}

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