Line Line Picking

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Given a unit line segment 👁 [0,1]
, pick two points at random on it. Call the first point 👁 x_1
and the second point 👁 x_2
. Find the distribution of distances 👁 d
between points. The probability density function for the points being a (positive) distance 👁 d
apart (i.e., without regard to ordering) is given by

👁 P(d)	👁 =	👁 (int_0^1int_0^1delta(d-\|x_2-x_1\|)dx_1dx_2)/(int_0^1int_0^1dx_1dx_2)	(1)
👁 Image	👁 =	👁 2(1-d),	(2)

where 👁 delta(x)
is the delta function. The distribution function is then given by

👁 D(d)=d(2-d).

(3)

Both are plotted above.

The raw moments are then

👁 mu_m^'	👁 =	👁 int_0^1d^mP(d)dd	(4)
👁 Image	👁 =	👁 2int_0^1d^m(1-d)dd	(5)
👁 Image	👁 =	👁 2/((m+1)(m+2))	(6)
👁 Image	👁 =	👁 {1/((n+1)(2n+1)) for m=2n; 1/((n+1)(2n+3)) for m=2n+1	(7)

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

👁 mu_1^'	👁 =	👁 1/3	(8)
👁 mu_2^'	👁 =	👁 1/6	(9)
👁 mu_3^'	👁 =	👁 1/(10)	(10)
👁 mu_4^'	👁 =	👁 1/(15)	(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

👁 mu_1^'	👁 =	👁 (int_0^1int_0^1\|x_2-x_1\|dx_1dx_2)/(int_0^1int_0^1dx_1dx_2)	(12)
👁 Image	👁 =	👁 int_0^1int_0^1\|x_2-x_1\|dx_1dx_2	(13)
👁 Image	👁 =	👁 int_0^1int_0^1; x_2-x_1>0(x_2-x_1)dx_1dx_2+int_0^1int_0^1; x_2-x_1<0(x_1-x_2)dx_1dx_2	(14)
👁 Image	👁 =	👁 int_0^1int_(x_1)^1(x_2-x_1)dx_1dx_2+int_0^1int_0^(x_1)(x_2-x_1)dx_1dx_2	(15)
👁 Image	👁 =	👁 int_0^1[1/2x_2^2-x_1x_2]_(x_1)^1dx_1+int_0^1[x_1x_2-1/2x_2^2]_0^(x_1)dx_1	(16)
👁 Image	👁 =	👁 int_0^1[(1/2-x_1)-(1/2x_1^2-x_1^2)]dx_1+int_0^1[(x_1^2-1/2x_1^2)-(0-0)]dx_1	(17)
👁 Image	👁 =	👁 int_0^1(1/2-x_1+x_1^2)dx_1	(18)
👁 Image	👁 =	👁 [1/2x_1-1/2x_1^2+1/3x_1^3]_0^1	(19)
👁 Image	👁 =	👁 1/3	(20)
👁 mu_2^'	👁 =	👁 int_0^1int_0^1(\|x_2-x_1\|)^2dx_2dx_1	(21)
👁 Image	👁 =	👁 int_0^1int_0^1(x_2-x_1)^2dx_1dx_2	(22)
👁 Image	👁 =	👁 int_0^1int_0^1(x_2^2-2x_1x_2+x_1^2)dx_1dx_2	(23)
👁 Image	👁 =	👁 int_0^1[1/3x_2^3-x_1x_2^2+x_1^2x_2]_0^1dx_1	(24)
👁 Image	👁 =	👁 int_0^1(1/3-x_1+x_1^2)dx_1	(25)
👁 Image	👁 =	👁 [1/3x_1^3-1/2x_1^2+1/3x_1]_0^1	(26)
👁 Image	👁 =	👁 1/6.	(27)

The 👁 n
th central moment is given by

👁 mu_n=(3^(-(n+2))[2(-1)^n(3n+5)+2^(n+3)])/((n+1)(n+2)).

(28)

The values for 👁 n=2
, 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

👁 mu	👁 =	👁 1/3	(29)
👁 sigma^2	👁 =	👁 1/(18)	(30)
👁 gamma_1	👁 =	👁 2/5sqrt(2)	(31)
👁 gamma_2	👁 =	👁 -3/5.	(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).

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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.

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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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