 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-iii/5/b/iii/solution

Put

p_{*} = p / [p + (1 - p) r]

. Then

G_{Z} (s) = \frac{p _{*}}{1 - ( 1 - p _{*} ) s} .

Z \sim Geom_{0} (\frac{p}{p + ( 1 - p ) r}) .

This is Bernoulli thinning of a geometric count, using the convention of counting failures before success.

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