Timeline for Why is this distribution exponential?
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| when toggle format | what | by | license | comment | |
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| Jun 22, 2015 at 22:38 | comment | added | Henry | 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. | |
| Jun 22, 2015 at 21:51 | comment | added | Ganesh | Reference: David & Nagaraja, Order Statistics. This is a fairly standard result in order statistics. | |
| Jun 22, 2015 at 21:49 | comment | added | Alex R. | Try working this out with just two points. | |
| Jun 22, 2015 at 21:43 | comment | added | wdlang | 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. | |
| Jun 22, 2015 at 21:39 | comment | added | Alex R. | 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. | |
| Jun 22, 2015 at 21:27 | comment | added | wdlang | actually 1000 points are sufficient to obtain a very good curve. | |
| Jun 22, 2015 at 21:26 | history | asked | wdlang | CC BY-SA 3.0 |