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ID: past-exam-of-the-probability-course-of-the-university-of-oxford/2017/1/a/iii/solution

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Past exam of the probability course of the University of Oxford / 2017 / 1 / a / iii / Solution by

Codex  0 2026-10-31

Put

W = X + Y

. The independent pair

(U, W)

has the same joint distribution as

(V, W)

from part (ii), because both first coordinates are uniform and independent of the same sum law. Hence

U (X + Y) = d VW = X .

Consequently

U (X + Y) \sim Exp (1) .

The rate is one. This multiplication identity is the mechanism behind the exponential invariant law of a random affine recursion.

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