= Solution
Let $\mu$ be the <mutation> rate per chromosomal segment per generation, and use the population-scaled parameter $\theta=4N_e\mu$. In $2N_e$-generation coalescent units, <mutations> occur on each ancestral lineage at rate $\theta/2$. Independent <mutations> on all branches of a given <phylogenetic tree> superpose to give a <Poisson distribution> with <mean> $\theta L/2$.
Under the <infinite sites mutation model>, each <mutation> occurs at a distinct position, and every <mutation> below the sample's common ancestor separates a nonempty proper subset of sampled <chromosomes> from the remainder. Thus the <mutation> count is exactly the <segregating site> count, giving
$$
\boxed{S\mid L,\theta\sim\operatorname{Poisson}\!\left(\frac{\theta L}{2}\right).}
$$
By the <law of total expectation> and the previous branch-length formula,
$$
\boxed{E[S\mid\theta]=E\!\left[\frac{\theta L}{2}\right]=\theta a_n.}
$$
This also gives the <Watterson estimator> $S/a_n$ as an unbiased moment estimator of $\theta$. Here all sample <segregating sites> are counted, including singletons; filtering sites by a minimum sample frequency or by external <SNP> discovery would introduce <genetic-study ascertainment> and change this calculation.
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