Line Line Picking

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LineLinePicking

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) = [画像:(int_0^1int_0^1delta(d-|x_2-x_1|)dx_1dx_2)/(int_0^1int_0^1dx_1dx_2)]

(1)

= 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^' = [画像:int_0^1d^mP(d)dd]

(4)

= [画像:2int_0^1d^m(1-d)dd]

(5)

= [画像:2/((m+1)(m+2))]

(6)

= [画像:{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^' = [画像:(int_0^1int_0^1|x_2-x_1|dx_1dx_2)/(int_0^1int_0^1dx_1dx_2)]

(12)

= [画像:int_0^1int_0^1|x_2-x_1|dx_1dx_2]

(13)

= [画像: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)

= [画像: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)

= [画像: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)

= [画像: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)

= [画像:int_0^1(1/2-x_1+x_1^2)dx_1]

(18)

= [1/2x_1-1/2x_1^2+1/3x_1^3]_0^1

(19)

= 1/3

(20)

mu_2^' = [画像:int_0^1int_0^1(|x_2-x_1|)^2dx_2dx_1]

(21)

= [画像:int_0^1int_0^1(x_2-x_1)^2dx_1dx_2]

(22)

= [画像:int_0^1int_0^1(x_2^2-2x_1x_2+x_1^2)dx_1dx_2]

(23)

= [画像:int_0^1[1/3x_2^3-x_1x_2^2+x_1^2x_2]_0^1dx_1]

(24)

= [画像:int_0^1(1/3-x_1+x_1^2)dx_1]

(25)

= [1/3x_1^3-1/2x_1^2+1/3x_1]_0^1

(26)

= 1/6.

(27)

The nth central moment is given by

[画像: 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

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

Line Line Picking

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