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

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.