# Conditional distribution of arrival times in Poisson process

Suppose I know over a window $$[0, T)$$ that I have observed $$n$$ samples from a poisson process $$N_t \sim p(n|\lambda t) = \frac{1}{n!}(\lambda t)^{n}\exp(-\lambda t)$$.

What is the conditional distribution of the arrival times $$t_1 < t_2 < ... < t_n$$ on $$[0, T)$$? In particular, what is the marginal distribution of $$t_1$$?

An interesting property of Poisson processes is that each event can be considered as "placed" independently and uniformly at a given time $$t$$ in $$[0,T]$$ (just like rain drops falling uniformly over the length of a board of length $$T$$). In other words, if there are $$n$$ events $$\{\tau_i\}_{i=1}^n$$, we have $$\tau_i \sim \mathrm{Unif}(0,T)$$ for all $$i$$. What you are looking for is the distribution of $$t_1=\min \{\tau_i: i=1,\dots,n\}$$. Please see this answer for distribution of the minimum of uniform IID variables. In your case: $$p(t_1|n) = \frac{n}{T^n}(T-t_1)^{n-1} \quad\text{for t_1\in[0,T]}.$$
Since the inter-arrival times are independent exponentially distributed, the joint pdf of the $$n$$ first arrival times is \begin{align} f(t_1,\dots,t_n) &=f(t_1)f(t_2|t_1)\dots f(t_n|t_{n-1}) \\&=\lambda e^{-\lambda t_1}\lambda e^{-\lambda(t_2-t_1)}\cdots \lambda e^{-\lambda(t_n-t_{n-1})} \\&=\lambda^n e^{-\lambda t_n} \end{align} for $$0 and 0 elsewhere. Conditional on $$n$$ arrivals in $$(0,T)$$, the conditional pdf is \begin{align} f(t_1,\dots,t_n|N(T)=n) &=\frac{f(t_1,\dots,t_n,N(T)=n)}{P(N(T)=n)} \\&=\frac{f(t_1,\dots,t_n)P(N(T)=n|t_1,\dots,t_n)}{P(N(T)=n)} \\&=\frac{\lambda^n e^{-\lambda t_n}e^{-\lambda(T-t_n)}}{e^{-\lambda T}(\lambda T)^n/n!} \\&=\frac{n!}{T^n}, \end{align} for $$0 and 0 elsewhere. This is the joint pdf of the order statistics of $$n$$ iid uniform random variables on $$(0,T)$$.