= Geometric mutation count in a coalescent epoch
{title2=$Y_j\sim\operatorname{Geom}_0\left(\frac{j-1}{j-1+\theta}\right)$}
Integrating the <Poisson distribution> <probability generating function> over the <exponential distribution> of the epoch length gives
$$
\mathbb E[s^{Y_j}]=\frac{j-1}{j-1+\theta(1-s)}.
$$
Thus $Y_j$ has a <geometric distribution> on $\{0,1,\ldots\}$, with <expectation> $\theta/(j-1)$ and <variance> $\theta/(j-1)+\theta^2/(j-1)^2$. The support convention matters: the count includes zero mutations.
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