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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2010/ia/paper-2/4f/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2010 / ia / Paper 2 / 4F / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-14

Put

U = X - EX

and

V = Y - E Y

. Their second moments are the finite positive variances of

X, Y

. The Cauchy-Schwarz inequality for square-integrable random variables gives

∣ E (U V) ∣^{2} \leq E (U^{2}) E (V^{2}) = Var X Var Y .

(1)

Divide by the positive product of variances and take square roots. The correlation bound is $- 1 \leq ρ (X, Y) \leq 1$ . As usual, non-constant here means not almost surely constant, so that the denominator is nonzero.

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