Yule process

The Yule process is the pure birth process with rate $n\lambda$ in population state $n$. Starting from one individual, it is nonexplosive because the independent holding times have divergent total almost surely.

For the Yule process starting from one individual, the generator applied to $f(n)=n$ givesand hence $\mathbb{E}{X}_t=e^{\lambda t}$.

Given $X_t=n+1$, the $n$ birth times in a Yule process have the law of the order statistics of independent variables with densityThis follows by dividing the joint holding-time density by
$\mathbb{P}(X_t=n+1)=e^{-\lambda t}(1-e^{-\lambda t}{)}^n$.