Conditional Yule birth times
= Conditional Yule birth times
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
$\mathbb P(X_t=n+1)=e^{-\lambda t}(1-e^{-\lambda t})^n$.