Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 223 / 1 / a

Solution

Let

m = F^{- 1} (1/2)

IF (x; T, F) = \frac{1 _{{x > m}} - 1 _{{x < m}}}{2 f ( m )}

away from

m

A (T, F) = E_{F} [IF (X; T, F)^{2}] = \frac{1}{4 f ( F ^{- 1} ( 1/2 ) ) ^{2}} .

No symmetry assumption is used: the density is evaluated at the actual median of

F

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