Conditional Distribution of arrival times

Just got my final back, and this was the only problem I couldn't figure out. Could someone show me how to do this? Thanks!

A Poisson process N(t) has rate = 1. Let be the time between the first and second arrivals of the process. Suppose there are exactly 2 arrivals in the interval , so N(1) = 2. Find the conditional pdf of given that N(1) = 2. Use your result to calculate the mean and standard deviation of given that N(1) = 2.

We were given the hint to use the theorem of conditional arrival times, namely that given N(t) = n, the n arrival times have the same distribution as the order statistics corresponding to n independent random variables uniformly distributed on (0,t).

The only headway I was able to make was that , so the conditional pdf of would be the pdf of minus the pdf of . But when I used that to find the expected value, I got a ln(t), which is undefined at 0 so I figured I was wrong.

Re: Conditional Distribution of arrival times

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)

http://www.sv-luka.org/ikone/ikone180a.jpg

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