Line Line Picking

Line line picking is the selection of two independent, uniformly distributed points on a unit line segment . Call the first point and the second point . The distance between these points is denoted . The probability density function for the points being a (positive) distance apart (i.e., without regard to ordering) is given by

			(1)
			(2)

where , the density is zero otherwise, and is the delta function. The distribution function is then given by

(3)

Here for and for . Both are plotted above.

The raw moments, for nonnegative integers , 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 statistical 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. If seek time is proportional to track separation, the average seek time is the time required to traverse 1/3 of the tracks in this model (Benedict 1995).