Consider a Poisson counting process with arrival rate, λ. Suppose it is observed that there have been exactly arrivals in [0, t] and let S1, S2… Sn be the times of those arrivals. Next, define X1, X2… Xn to be a sequence of IID random variables uniformly distributed over [0, t] and let Y1, Y2… Yn be the order statistics associated with the Xi. Show that the joint PDF of the order statistics, fY (y), is identical to the joint PDF of the Poisson arrival times, fs (s). Hence, the order statistics are statistically identical to the arrival times.
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