The answer to that question is strongly related to the following theorem in...

http://www.nas.its.tudelft.nl/people...UP_Poisson.pdf

Theorem 7.3.3 Given that exactly one event of a Poisson process has occurred during the interval [0, t], the time of occurrence of this event is uniformly distributed over [0, t].

Naw if and are two two times of occurrence in [0,1], then the random variable has p.d.f. given by...

(1)

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