 Solution

A rate-$\lambda$ Poisson process has $N_0=0$, right-continuous paths with upward jumps of size one, stationary independent increments, and $N_t-N_s\sim \mathrm{Poisson}(\lambda (t-s))$ for $t\ge s$. Assume $\lambda >0$. Its successive waiting times are independent exponential variables of rate $\lambda$. For $0<t_1<\cdots <t_n<t$, the product of their densities and the probability of no further jump before $t$ isDivide by $\mathbb{P}(N_t=n)=e^{-\lambda t}(\lambda t{)}^n/n!$ to obtain the conditional joint densityBoundary inequalities do not affect a density. Equivalently these are the ordered values of $n$ independent uniform points of $[0,t]$. For $n=0$, the conditional tuple is empty.