The Mean Distance to the n-th Neighbour in a Uniform Distribution of Random Points: An Application of Probability Theory

Pratip Bhattacharyya; Bikas K. Chakrabarti

paper

The Mean Distance to the n-th Neighbour in a Uniform Distribution of Random Points: An Application of Probability Theory

arXiv:math/0212230

Abstract

We study different ways of determining the mean distance $< r_{n} >$ between a reference point and its $n$ -th neighbour among random points distributed with uniform density in a $D$ -dimensional Euclidean space. First we present a heuristic method; though this method provides only a crude mathematical result, it shows a simple way of estimating $< r_{n} >$ . Next we describe two alternative means of deriving the exact expression of $< r_{n} >$ : we review the method using absolute probability and develop an alternative method using conditional probability. Finally we obtain an approximation to $< r_{n} >$ from the mean volume between the reference point and its $n$ -th neighbour and compare it with the heuristic and exact results.

6 pages (REVTex4), minor changes in content, typing errors corrected, references added