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.