 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/discrete-mathematics-probability-and-continuous-mathematics/8/b/iii/solution

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

p_{*} / [1 - (1 - p_{*}) s]

, identifying

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

Equivalently, for

z \in N

P (Z = z) = p_{*} [(1 - p_{*})]^{z}

. Since

0 < p, r < 1

, also

0 < p_{*} < 1

. This is the existing geometric count under Bernoulli thinning, with the zero-based convention specified.

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