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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ii/paper-4/29k/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 4 / 29K / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

Write

X = μ + σ Z

, where

Z

has the standard normal distribution. Apply Gaussian integration by parts to

g (z) = U^{'} (μ + σ z)

:

E [Z U^{'} (X)] = E [g^{'} (Z)] = σ E [U^{''} (X)] .

Since

X - μ = σ Z

,

E [U^{'} (X) (X - μ)] = σ^{2} E [U^{''} (X)] .

The polynomial growth assumption and decay of the normal distribution density justify the integration by parts.

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