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

Set $p_{\ast }=p/[p+(1-p)r]$. The preceding probability generating function becomes $p_{\ast }/[1-(1-p_{\ast })s]$, identifyingEquivalently, for $z\in \mathbb{N}$, $\mathbb{P}(Z=z)=p_{\ast }[(1-p_{\ast }){]}^z$. Since $0<p,r<1$, also $0<p_{\ast }<1$. This is the existing geometric count under Bernoulli thinning, with the zero-based convention specified.