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Take the interval $[0, 1]$.

Now sample 10000 points in this interval randomly according to the uniform distribution.

The fact is that the distribution of the distances between adjacent points on this segment is an exponential one.

This is a fact pointed out to me by a friend years ago. Can anyone prove it? The problem is that two successively sampled points are generally not adjacent.

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  • $\begingroup$ actually 1000 points are sufficient to obtain a very good curve. $\endgroup$
    – wdlang
    Commented Jun 22, 2015 at 21:27
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    $\begingroup$ Basically by rescaling the interval as a function of $n$ the waiting time between successive points becomes a poisson process. This is because the probability of seeing the next point is proportional to the interval you are looking at. $\endgroup$
    – Alex R.
    Commented Jun 22, 2015 at 21:39
  • $\begingroup$ But the points are not generated from left to right. Numerically, one generates them first, and then order them, and then get the gaps, and then analyse the gaps. I cannot see how to turn this numerical process to the 'left-to-right' process. $\endgroup$
    – wdlang
    Commented Jun 22, 2015 at 21:43
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    $\begingroup$ Reference: David & Nagaraja, Order Statistics. This is a fairly standard result in order statistics. $\endgroup$
    – Ganesh
    Commented Jun 22, 2015 at 21:51
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    $\begingroup$ Including the gap at the beginning and the end, you have $10001$ identically distributed gaps (to see this, add an extra point, distribute the points randomly along a circle of circumference $1$ and then split the circle at the extra point to recreate the interval). So you may as well consider the distribution of the initial gap. The probability this is less than or equal to $x$ is $1-(1-x)^{10000}$. This is not quite exponentially distributed, but for small $x$ it is close to $1-\exp(-10000x)$ which is the cumulative distribution function of an exponential distribution. $\endgroup$
    – Henry
    Commented Jun 22, 2015 at 22:38

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