 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2004/ia/paper-2/3f/b/solution

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

The covariance is symmetric, because multiplication of real random variables is commutative. Expanding with the bilinearity just proved gives

cov (X + Y, X - Y) = var (X) - cov (X, Y) + cov (Y, X) - var (Y) = var (X) - var (Y) .

Identical probability distributions imply equal second moments and equal means, hence equal variances. Therefore the assertion is true, with

cov (X + Y, X - Y) = 0.

The cancellation shows exactly why no assumption of independence is needed.

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