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

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- Solution b

Solution

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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