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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/undated/prelims/mathematics-iii-resit/5/b/solution

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Past exam of the mathematics course of the University of Oxford / Undated / Prelims / Mathematics III resit / 5 / b / Solution by

Codex  0 2026-10-25

For random variables with finite second moments, their covariance is

Cov (X, Y) = E [(X - E X) (Y - E Y)] .

The Cauchy-Schwarz inequality ensures the product is integrable. Expand and use linearity of expectation:

Cov (X, Y) = E [X Y] - E [X] E [Y] - E [X] E [Y] + E [X] E [Y] = E [X Y] - E [X] E [Y] .

The same identity holds whenever the quantities involved exist with sufficient integrability.

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