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
(1)
| |||
(2)
|
where is the delta function. The distribution function is then given by
(3)
|
Both are plotted above.
The raw moments are then
(4)
| |||
(5)
| |||
(6)
| |||
(7)
|
(Uspensky 1937, p. 257), giving raw moments
(8)
| |||
(9)
| |||
(10)
| |||
(11)
|
(OEIS A000217), which are simply one over the triangular numbers.
The raw moments can also be computed directly without explicit knowledge of the distribution
(12)
| |||
(13)
| |||
(14)
| |||
(15)
| |||
(16)
| |||
(17)
| |||
(18)
| |||
(19)
| |||
(20)
| |||
(21)
| |||
(22)
| |||
(23)
| |||
(24)
| |||
(25)
| |||
(26)
| |||
(27)
|
The th central moment is given by
(28)
|
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
(29)
| |||
(30)
| |||
(31)
| |||
(32)
|
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).