First two moments and cross-moments of some coalescent times of the pure birth tree

TL;DR

This paper provides a detailed analysis of the moments and cross-moments of coalescent times and tree heights in pure birth trees, enhancing understanding of their probabilistic structure.

Contribution

It introduces explicit calculations for moments and cross-moments of coalescent times and tree heights in pure birth trees, including conditional expectations.

Findings

Explicit formulas for first two moments of tree height.

Explicit formulas for first two moments of coalescent times.

Analysis of cross-moments between tree height and coalescent times.

Abstract

We present here a thorough study of the first two moments and cross-moments for the pure birth tree's height and for the coalescent time of a randomly sampled pair of tips. We consider also the first two moments of the conditional, on the tree, expectation of this coalescent time.