 Solution

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

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

In one proposal of rejection sampling, the probability of both proposing a value in

d x

and accepting it is

h (x) d x \frac{f ( x )}{M h ( x )} = \frac{1}{M} f (x) d x .

Integrating shows that the acceptance probability is

1/ M

. Conditional on acceptance, the density of the proposed value is therefore

\frac{f ( x ) / M}{1/ M} = f (x) .

Each unsuccessful proposal restarts independently, so the eventual output

Y

has the same conditional density:

Y \sim f .

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