 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-4/28k/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Take the proposal density

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

\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} .

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