the order statistics property of arrival times

Here is a small miracle of the Poisson process. Suppose you are told only that exactly n events happened during a fixed window, say [0, T] — but not when. Where, inside that window, should you expect them to be? The answer is wonderfully simple: completely at random, as if you had thrown n darts uniformly at the interval and then sorted them left to right. Knowing the count tells you nothing about clustering — the positions are as uniform as can be.

Stated carefully: condition on N(T) = n. Then the n arrival times, viewed as an unordered set, are distributed exactly like n independent Uniform(0, T) points; and their sorted versions S1 < S2 < ... < Sn are distributed like the order statistics of those n uniform points. So to simulate the arrival times of a Poisson process on [0, T] you can first draw n from a Poisson(lambda*T), then drop n uniform points on [0, T] and sort them — the rate lambda only sets how many, not where. The intuition is that the memoryless, no-clustering nature of the process leaves each instant as good as any other once the total is fixed.

This is more than a curiosity: it is a workhorse. It gives a fast way to simulate Poisson processes, it underlies the construction of non-homogeneous and marked processes, and it makes many expectations easy — for instance the expected sum of arrival times given n arrivals is just n*(T/2), because each uniform point averages T/2. The one caveat: the property holds for the HOMOGENEOUS (constant-rate) Poisson process. For a time-varying rate the conditional positions are still independent but distributed according to the normalized rate function, not uniformly.