Geometric mutation count in a coalescent epoch ($Y_j\sim {\mathrm{Geom}}_0\left(\frac{j-1}{j-1+\theta }\right)$)

Integrating the Poisson distribution probability generating function over the exponential distribution of the epoch length givesThus $Y_j$ has a geometric distribution on $\{0,1,\dots \}$, with expectation $\theta /(j-1)$ and variance $\theta /(j-1)+{\theta }^2/(j-1{)}^2$. The support convention matters: the count includes zero mutations.