Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 33 / 1 / a / ii

Solution

p_{acc} = E_{g} [\frac{f ( Y )}{M g ( Y )}] = \frac{C}{M} .

The inequality

f \leq M g

also gives

C \leq M

, so this is a valid probability. It is the same at every independent trial.

J_{i}

indicates acceptance of the

i

th proposal, then

J_{1}, \dots, J_{n}

C / M

. Hence the accepted count

K = \sum_{i} J_{i}

satisfies

K \sim Bin (n, \frac{C}{M}), E K = \frac{n C}{M} = \frac{n}{M} \int_{R} f (y) d y .

n p_{acc} (1 - p_{acc})

. Conditional on any acceptance pattern, the accepted values have independent target probability density function

h

; mixing over patterns retains this product law conditional on their count.

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