There is a line which can be considered as an interval [0,L], here we drop N points randomly on the line which obey the uniform distribution, namely, the probability of the location of any points in the line is equal. Now the distance between any two neighboring points will obey an distribution, how to derive the distribution theoretically. (I think the distribution is very similar to the poisson distribution and the time gap for any neighboring possion points obeying the exponential distribution, but there is also some difference between them. Therefore, I do some experiment in the matlab, the result is that the distribution for the distance is very close to exponential distribution)
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Before determining distances this is the $n$ order statistics from uniform $[0,L]$ which is a known distribution, Then there are either $n-1$ or $n+1$ distances determined, depending on whether you want to include distances between $0$ and least, and between greatest and $L.$ So this in turn will be a multiple variable distribution, How did you get from the various distances to a single number to compare it with an exponential distribution? Did you average the distances? or sum them? – coffeemath May 10 '16 at 06:56
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Yes, you are right. And when I do an experiment on MATLAB, I just give N random numbers which are in [0,L] to describe this question, these random numbers can be considered as the coordinates of the points, and the distance between any neighboring points can be obtained, this is an observed experiment result, and calculate the pdf we can find it very similar to the exponential distribution. We can repeat doing the same experiments, and the result is the same. If we average all the experiment result, the distance obeys the exponential distribution. – user330187 May 10 '16 at 07:18
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Maybe you can find some hint here – N74 May 10 '16 at 08:21