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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 28K / c / Solution

Take the proposal density

h (x) = e^{- x} 1_{x \geq 0}

. For the target half-normal distribution,

\frac{f ( x )}{h ( x )} = \frac{2}{π} exp (- \frac{x ^{2}}{2} + x) = \frac{2 e}{π} exp (- \frac{( x - 1 ) ^{2}}{2}) .

The ratio is maximized at

x = 1

, so the sharp envelope constant is

M = \frac{2 e}{π} .

Generate

X = - lo g U_{1}

and an independent uniform

U_{2}

, and accept

X

when

U_{2} \leq \frac{f ( X )}{M h ( X )} = e^{- (X - 1)^{2} /2} .

Solved by gpt-5.6-sol high.

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