Suppose $x_1$ and $x_2$ are two uniformly distributed points from unit cube $(0,1)^3$, what's the distribution of the distance between $x_1$ and $x_2$?
I did a quick simulation and find that this distribution's kurtosis is 2.5, smaller than 3. can anyone help me with a closed form pdf? thanks.
[Math] the distribution of distance between two random points from $U(0,1)^3$
probabilityprobability distributions
Best Answer
This is such a natural problem to study that I would check to see if it has been done before, and it seems it has:
"On the distribution of the distance between two points in a cube" by Antanas Žilinskas, Random Operators and Stochastic Equations, Volume 11, Issue 1, Pages 21–24, March 2003. Abstract: We are interested in the distribution of distance between two random points in a cube. It is well known, that the derivation of the formulae of the distribution function of interest implies integration problems which are almost intractable. We show, that the problem may be successfully solved using a symbolic computation tool.
I was unable to find the paper for free online.
I also found what looks like a proof of the exact same thing at http://www.degruyter.com/view/j/rose.2000.8.issue-4/rose.2000.8.4.339/rose.2000.8.4.339.xml . It also is behind a paywall. Tantalizingly, you can see part of the answer on the first page of the paper, which is displayed at that URL.
This paper was free online: "THE PROBABILITY DISTRIBUTION OF THE DISTANCE BETWEEN TWO RANDOM POINTS IN A BOX", posted at http://www.math.kth.se/~johanph/habc.pdf , which actually answers the more difficult problem with a box with not necessarily equal sides. This would be harder to use, but easier to get for free right away.