Given a unit Line Segment , pick two points at random on it. Call the first point and the second
point . Find the distribution of distances between points. The probability of the points being a
(Positive) distance apart (i.e., without regard to ordering) is given by

(1) |

where is the Dirac Delta Function and is the Heaviside Step Function. The Moments are then

(2) |

giving Moments about 0

(3) | |||

(4) | |||

(5) | |||

(6) |

The Moments can also be computed directly without explicit knowledge of the distribution

(7) | |||

(8) |

The Moments about the Mean are therefore

(9) | |||

(10) | |||

(11) |

so the Mean, Variance, Skewness, and Kurtosis are

(12) | |||

(13) | |||

(14) | |||

(15) |

The probability distribution of the distance between two points randomly picked on a Line Segment is germane to the problem of determining the access time of computer hard drives. In fact, the average access time for a hard drive is precisely the time required to seek across 1/3 of the tracks (Benedict 1995).

**References**

Arfken, G. *Mathematical Methods for Physicists, 3rd ed.* Orlando, FL: Academic Press, pp. 930-931, 1985.

Benedict, B. *Using Norton Utilities for the Macintosh.* Indianapolis, IN: Que, pp. B-8-B-9, 1995.

© 1996-9

1999-05-25