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The independent mutation counts in coalescent epochs have the geometric mutation count in a coalescent epoch laws. Summing their expectations and variances gives the displayed formulas, where $H_m={\sum }_{i=1}^{m}1/i$ and $H_m^{(2)}={\sum }_{i=1}^{m}1/i^2$. The extra quadratic term records randomness of the ancestral tree, beyond the conditional Poisson distribution variance.

Apply the law of total variance to each epoch. The Poisson distribution has equal conditional mean and variance, while $\mathrm{Var}(T_j)={\left(\genfrac{}{}{0ex}{}{j}{2}\right)}^{-2}$. HenceThe independent mutation counts in coalescent epochs permit variance additivity for independent random variables. ThusThe first term is conditional Poisson distribution variation; the second comes from the random lengths of the ancestral tree. These are the moments of the mutation count on a neutral coalescent tree.