Conditional Distribution of Arrival Times

Theorem

Let $\{N(t),t\ge 0\}$ be a Poisson process of rate $\lambda$.

Given $N(t)=n$, the $n$ arrival times $S_1,S_2,\dots ,S_n$ have the same distribution as the order statistics of $n$ i.i.d. $\text{Uniform}(0,t)$ random variables.

The joint density is:

$f(s_1,s_2,\dots ,s_n\mid N(t)=n)=\frac{n!}{t^n},0<s_1<s_2<\cdots <s_n<t$

Interpretation

If you know exactly $n$ events happened in $[0,t]$, the times at which they occurred are just like $n$ points dropped uniformly at random in the interval — there’s no clustering or pattern.

Proof

The event $\{S_1=s_1,\dots ,S_n=s_n,N(t)=n\}$ is equivalent to $\{T_1=s_1,T_2=s_2-s_1,\dots ,T_n=s_n-s_{n-1},T_{n+1}>t-s_n\}$.

Using inter-arrival times are i.i.d. $\text{Exp}(\lambda )$:

$$\begin{array}{rl}f(s_1,\dots ,s_n\mid N(t)=n)& =\frac{\lambda e^{-\lambda s_1}\lambda e^{-\lambda (s_2-s_1)}\cdots \lambda e^{-\lambda (s_n-s_{n-1})}{e}^{-\lambda (t-s_n)}}{e^{-\lambda t}(\lambda t{)}^n/n!}\\ & =\frac{n!}{t^n}\end{array}$$

which is the joint density of order statistics of $n$ i.i.d. $\text{Uniform}(0,t)$ variables.

Remark

This is often paraphrased as: given $n$ events in $(0,t)$, the event times considered as unordered variables are distributed independently and uniformly over $(0,t)$.

Related

• Poisson Process
• Waiting Times