What is the probability that a given $ n $ event trains match the beginning of a Poisson process? Here is my question with which I'm confusing myself:
Assume that some event times $ \{\tau_i\}_{i \in \mathbb{N}} $ are a point process with rate $ \mu $ such that number of events that occurred before $ t $, denoted $ N = \# \{ \tau_i \in [0,t] \} $, is fixed to $ n $. Consider one event train $ T = \{t_1<\cdots<t_n\} $ such that $ 0 \leq t_1 $ and $ t_n \leq t $.
What is the probability that $ T $ match the beginning of $ \{ \tau_i \}_{i \in \mathbb{N}} $ (i.e. what is the probability density function $ f(t_1,\cdots,t_n;t,\mu) $) ?
 A: So, you have a Poisson point process and under the condition that $n$ events have happened in the time interval, $(0;t]$, you want the joint probability density function of times for the sequence of times of these events. $\{\tau_i\}_{i\in\{1,..,n\}}$.
Under this condition, the time of each event will be independently and uniformly distributed over the interval.   The conditional probability density function, and CDF, are:
$$\begin{align}f_{\tau}(t_i)~=~& 1/t~\big[0< t_i\leq t\big] \\ F_{\tau}(t_i) ~=~& t_i/t~\big[0< t_i\leq t\big]~+~ 1~\big[t< t_i\big]\end{align}$$
However, the distribution of the time that $i$-th events occurs is distribution of the $i$-th order statistic for the sequence, and we want the joint distribution for all $n$ times:
$$\begin{align}
f_{\tau_{(1)},..,\tau_{(n)}}(t_1,..,t_n) ~=~& n!\prod_{i=1}^{n}f_\tau(t_i)
\\ =~ & \dfrac{n!}{t^n}~~\Big[0 < t_1< t_2<\cdots<t_n\leq t\Big]
\end{align}$$

PS: This answer assumes that the length of the interval is also fixed.   If $t$ is the time of the $n$-th or $(n+1)$-st occurrence, or something, then the answer will need to be adjusted.
