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Line Point Picking


Consider a line segment of length 1, and pick a point x at random between [0,1]. This point x divides the line into line segments of length x and 1-x. If a set of points are thus picked at random, the resulting distribution of lengths has a uniform distribution on [0,1]. Similarly, separating the two pieces after each break, the larger piece has uniform distribution on [1/2,1] (with mean 3/4), and the smaller piece has uniform distribution on [0,1/2] (with mean 1/4).

The probability that the line segments resulting from cutting at two points picked at random on a unit line segment determine a triangle is given by 1/4.

LinePointPickingSmall

The probability and distribution functions for the ratio of small to large pieces are given by

P(x)=2/((1+x)^2)
(1)
D(x)=(2x)/(1+x)
(2)

for x in [0,1]. The raw moments are therefore

 mu_n^'=n[psi_0(1/2(n+1))-psi_0(1/2n)]-1,
(3)

where psi_0(x) is the digamma function. The first few are therefore

mu_1^'=2ln2-1
(4)
mu_2^'=-4ln2+3
(5)
mu_3^'=6ln2-4
(6)
mu_4^'=-8ln2+(17)/3
(7)

(OEIS A115388 and A115389). The central moments are therefore

 mu_n=((-1)^(n-1)(n-1)(2ln2)^n(1-n)_(n-1)(-n)_(n-1))/(Gamma(n)Gamma(n+1)) 
 +sum_(k=0)^(n-2)((ln2)^k(2^n-2^(k+1))(1-n)_k(-n)_k)/((n-1)k!(2-n)_k),
(8)

where (x)_n is a Pochhammer symbol. The first few are therefore

mu_2=2-4ln^22
(9)
mu_3=3-12ln2+16ln^32
(10)
mu_4=(14)/3-24ln2+48ln^22-48ln^42.
(11)

This therefore gives mean, variance, skewness, and kurtosis excess of

mu=2ln2-1
(12)
sigma^2=2-4ln^22
(13)
beta=(3-12ln2+16ln^32)/((2-4ln^22)^(3/2))
(14)
gamma=((25)/6-6ln2)/((1-2ln^22)^2)-6.
(15)

The mean can be computed directly from

mu=int_0^1(min(x,1-x))/(max(x,1-x))dx
(16)
=2int_0^(1/2)x/(1-x)dx
(17)
=2ln2-1.
(18)
LinePointPickingLarge

The probability and distribution functions for the ratio of large to small pieces are given by

P(x)=2/((1+x)^2)
(19)
D(x)=(x-1)/(x+1)
(20)

for x in [1,infty). Paradoxical though it may be, this distribution has infinite mean and other moments. The reason for this is that a theoretical bone can be cut extremely close to one end, thus giving huge ratio of largest to smallest pieces, whereas there is some limit for a real physical bone. Taking epsilon to be the smallest possible piece in which is bone cen be cut, the mean is then given by

 x^_=2ln(1/(2epsilon))+2epsilon-1.
(21)

See also

Cube Point Picking, Hypercube Point Picking, Line Line Picking, Square Point Picking

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References

Pickover, C. A. "The Problem of the Bones." Ch. 8 in The Mathematics of Oz: Mental Gymnastics from Beyond the Edge. New York: Cambridge University Press, pp. 21-23 and 243-249, 2002.Sloane, N. J. A. Sequences A115388 and A115389 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Line Point Picking

Cite this as:

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

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