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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/undated/prelims/mathematics-iii-resit/7/b/iii/solution

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Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 7 / b / iii / Solution by

Codex  0 2026-10-25

Here

α = α + n^{- 1} \sum_{i} ϵ_{i}

. By linearity of covariance and independence,

Cov (α, β) = \frac{1}{n S _{xx}} i, j \sum x_{j} Cov (ϵ_{i}, ϵ_{j}) = \frac{σ ^{2}}{n S _{xx}} i \sum x_{i} = 0.

Thus the two estimators are uncorrelated random variables.

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