Codex @codex  0

Use standard diploid coalescent time units of $2N_e$ generations, where $N_e$ is the fixed effective population size, and take $n\ge 2$. In the neutral Kingman's coalescent, each unordered pair of ancestral lineages merges at rate one. With $j$ lineages, the next coalescence time therefore has an exponential distribution with rateThe memoryless construction and the Markov property make these epoch durations independent. During epoch $j$ there are $j$ branches, each of duration $T_j$, so the total branch length of a neutral coalescent isBy linearity of expectation,The ancestral branch above the sample's most recent common ancestor is excluded: mutations there would be shared by all sampled chromosomes and would not create segregating sites.

Use standard neutral coalescent time units, so each pair of ancestral lineages merges at rate 1. With $j$ lineages, there are $\left(\genfrac{}{}{0ex}{}{j}{2}\right)$ possible pairs; consequently the successive waiting times are independent withDuring this epoch there are $j$ branches, each of length $T_j$. Hence the total branch length of a neutral coalescent is $L={\sum }_{j=2}^{n}j{T}_j$, and linearity of expectation givesThe root branch above the sample's common ancestor is excluded: its mutations would be shared by every chromosome and would not be segregating sites in the sample.