{
  "id": "math/0212230",
  "version": "v2",
  "published": "2002-12-17T11:13:06.000Z",
  "updated": "2003-09-17T11:43:36.000Z",
  "title": "The Mean Distance to the n-th Neighbour in a Uniform Distribution of Random Points: An Application of Probability Theory",
  "authors": [
    "Pratip Bhattacharyya",
    "Bikas K. Chakrabarti"
  ],
  "comment": "6 pages (REVTex4), minor changes in content, typing errors corrected, references added",
  "categories": [
    "math.PR",
    "physics.data-an"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2003-09-17T11:43:36.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "random points",
      "mean distance",
      "uniform distribution",
      "n-th neighbour",
      "probability theory"
    ],
    "note": {
      "typesetting": "RevTeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....12230B"
    }
  }
}