Solution

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

{title2=$S=\sum_{j=2}^nY_j$}

In <Kingman's coalescent>, the epoch with $j$ ancestral lineages lasts $T_j\sim\operatorname{Exp}(\binom j2)$, independently across epochs. If each lineage carries an independent <Poisson process> of <mutations> with rate $\theta/2$, its epoch mutation count satisfies $Y_j\mid T_j\sim\operatorname{Poisson}(j\theta T_j/2)$. The total mutation count is $S=\sum_{j=2}^nY_j$. Under the <infinite sites mutation model>, this also counts <segregating sites>; without that assumption it still counts mutation events, which need not correspond one-to-one to observed polymorphisms.

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 Incoming links: Mutation count on a neutral coalescent tree