 Poisson process conditional arrival times

Conditioned on $N_t=n$, the arrival times are distributed as the order statistics of $n$ independent uniform points on $[0,t]$. Their conditional joint density onis $n!/t^n$, obtained by dividing the Poisson-process density ${\lambda }^n{e}^{-\lambda t}$ by $\mathbb{P}(N_t=n)$.