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Past exam of the mathematics course of the University of Oxford

2023

Prelims

Mathematics III

{c}

Put $p_*=p/[p+(1-p)r]$. Then
$$
G_Z(s)=\frac{p_*}{1-(1-p_*)s}.
$$
Uniqueness of the <probability generating function> therefore gives
$$
\boxed{Z\sim\operatorname{Geom}_0\!\left(\frac p{p+(1-p)r}\right).}
$$
This is <Bernoulli thinning of a geometric count>, using the convention of counting failures before success.

Solution

Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 5 / b / iii / Solution

Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 5 / b / iii / Solution

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Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 5 / b / iii / Solution

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