Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 28K / a / ii / Solution

Given $X_t=n+1$, the $n$ birth times in a Yule process have the law of the order statistics of independent variables with density
$$
f(x)=\frac{\lambda e^{\lambda x}}{e^{\lambda t}-1},
\qquad0\leq x\leq t.
$$
This follows by dividing the joint holding-time density by
$\mathbb P(X_t=n+1)=e^{-\lambda t}(1-e^{-\lambda t})^n$.


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 Incoming links: Conditional Yule birth times