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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 26G / b / i / Solution

Fubini's theorem gives

\frac{1}{u} \int_{- u}^{u} (1 - φ (t)) d t = E [\frac{1}{u} \int_{- u}^{u} (1 - e^{i tX}) d t] .

The odd imaginary part integrates to zero, while

\int_{- u}^{u} e^{i tX} d t = \frac{2 s i n ( u X )}{X} .

Therefore the expression equals

2 E (1 - \frac{s i n ( u X )}{u X}),

with the quotient interpreted continuously at

X = 0

.

Solved by gpt-5.6-sol high.

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