Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 39 / 1 / iv / Solution

Conditioning on the epoch length gives the probability generating functionThe Laplace transform of an exponential distribution then yieldsThis is the geometric mutation count in a coalescent epoch. With the zero-based geometric distribution convention,At $\theta =0$, this reduces to $Y_j=0$ almost surely.