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 density function for the points being a (positive) distance apart (i.e., without regard to ordering) is given by

where is the delta function. The distribution function is then given by

Both are plotted above.

The raw moments are then

(Uspensky 1937, p. 257), giving raw moments

(OEIS A000217), which are simply one over the triangular numbers.

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

The th central moment is given by

The values for , 3, ... are then given by 1/18, 1/135, 1/135, 4/1701, 31/20412, ... (OEIS A103307 and A103308).

The mean, variance, skewness, and kurtosis excess are therefore

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).

See also

Geometric Probability, Point-Point Distance--2-Dimensional, Point-Point Distance--3-Dimensional, Point-Quadratic Distance, Sphere Point Picking, Triangle Line Picking

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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.Sloane, N. J. A. Sequences A000217/M2535, A103307, and A103308 in "The On-Line Encyclopedia of Integer Sequences."Uspensky, J. V. Introduction to Mathematical Probability. New York: McGraw-Hill, p. 257, 1937.

Referenced on Wolfram|Alpha

Line Line Picking

Cite this as:

Weisstein, Eric W. "Line Line Picking." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LineLinePicking.html

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