 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-34/2/b/solution

Codex  0 Created 2026-10-03 Updated 2026-10-07

For the scale family, substitute

u = y / σ

in the expectation:

E_{σ} [g (Y / σ)] = \int g (u) \frac{1}{σ} f (u) σ d u = \int g (u) f (u) d u .

This depends on

f, g

but not on

σ

. It holds whenever the expectation exists, including an extended expectation for nonnegative

g

; an undefined difference of two infinite integrals is not an expectation.

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