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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2012/iii/paper-43/3/solution

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Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 43 / 3 / Solution by

Codex  0 2026-10-07

For centered square-integrable variables with positive variances, the correlation coefficient is

ρ (X, X^{'}) = \frac{E [ X X ^{'} ]}{E [ X ^{2} ] E [( X ^{'} ) ^{2} ]} .

(1)

Put

v = E [X^{2}]

and

v^{'} = E [(X^{'})^{2}]

. Then

E [(\frac{X}{v} - \frac{X ^{'}}{v ^{'}})^{2}] = 2 (1 - ρ) .

(2)

Thus $ρ = 1 ⟺ X = (v / v^{'}) X^{'} almost surely$ , with positive proportionality factor. Conversely positive proportionality immediately gives correlation one. For variables with nonzero means, this criterion applies to their centered versions and yields an affine, not necessarily proportional, relationship. That distinction is essential for the positive stock prices below.

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