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

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

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

Interpret the random choice as an independent selector

B

with

P (B = 1) = r

, and write

Y = BX + (1 - B) X^{'}

. Its distribution is again uniform on

{- 1, 1}

, so

EX = E Y = 0

and

Var X = Var Y = 1

. By independence,

E (X Y) = r E (X^{2}) + (1 - r) E (X X^{'}) = r .

Consequently $ρ (X, Y) = r$ . The independent selector is the standard meaning of the stated mixture; a choice depending on the signs would describe a different model.

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