 Solution

In the Yule process, each of $k$ branches gives birth independently at rate $\lambda$ per generation. The next birth therefore has total rate $k\lambda$. Its waiting time has an exponential distribution, so the mean phylogenetic epoch duration isFor the ensuing calculation use $E_2=1/(2\lambda )$, $E_3=1/(3\lambda )$ and $E_4=1/(4\lambda )$. This is the inter-event waiting-time convention requested in the question, including an observation interval with four branches. Stopping the forward process immediately when the fourth branch is born would instead leave a zero terminal interval and describe a different sampling convention.