Point-Point Distance--1-D

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 of the points being a (Positive) distance $d$ apart (i.e., without regard to ordering) is given by

$$P(d) = {\int_0^1\int_0^1 \delta(d-\vert x_2-x_1\vert)\,dx_1\,dx_2\over \int_0^1\int_0^1 dx_1\,dx_2}$$

$$= (1-d)[H(1-d)-H(d-1)+H(d)-H(-d)]$$

$$= \left\{\begin{array}{ll} 2(1-d) & \mbox{for $0\leq d\leq 1$}\\ 0 & \mbox{otherwise,}\end{array}\right.$$ (1)

where $\delta$ is the Dirac Delta Function and $H$ is the Heaviside Step Function. The Moments are then

$$\mu'_m = \int_0^1 d^m P(d)\,dd = 2\int_0^1 d^m(1-d)\,dd$$

$$= 2\left[{{d^{m+1}\over m+1}-{d^{m+2}\over m+2}}\right]_0^1$$

$$= 2\left({{1\over m+1}-{1\over m+2}}\right)=2\left[{(m+2)-(m+1)\over (m+1)(m+2)}\right]$$

$$= {2\over (m+1)(m+2)}$$

$$= \left\{\begin{array}{ll} {1\over (n+1)(2n+1)} & \mbox{for $m=2n$}\\ {1\over (n+1)(2n+3)} & \mbox{for $m=2n+1$,}\end{array}\right.$$ (2)

giving Moments about 0

$$\mu'_1 = {\textstyle{1\over 3}}$$ (3)

$$\mu'_2 = {\textstyle{1\over 6}}$$ (4)

$$\mu'_3 = {\textstyle{1\over 10}}$$ (5)

$$\mu'_4 = {\textstyle{1\over 15}}.$$ (6)

The Moments can also be computed directly without explicit knowledge of the distribution

$$\mu'_1 = {\int_0^1 \int_0^1 \vert x_2-x_1\vert\,dx_1\,dx_2 \over \int_0^1 \int_0^1 dx_1\,dx_2}$$

$$= \int_0^1 \int_0^1 \vert x_2-x_1\vert\,dx_1\,dx_2$$

$$= {\int_0^1 \int_0^1\atop\scriptstyle x_2-x_1>0} (x_2-x_1)\,dx_1\,dx_2+ {\int_0^1 \int_0^1\atop\scriptstyle x_2-x_1<0} (x_1-x_2)\,dx_1\,dx_2$$

$$= \int_0^1 \int_{x_1}^1 (x_2-x_1)\,dx_1\,dx_2+ \int_0^1 \int_0^{x_1} (x_2-x_1)\,dx_1\,dx_2$$

$$= \int_0^1 \left[{{\textstyle{1\over 2}}{x_2}^2-x_1x_2}\right]_{x_1... ...dx_1+\int_0^1 \left[{x_1x_2-{\textstyle{1\over 2}}{x_2}^2}\right]^{x_1}_0\,dx_1$$

$$= \int_0^1 \left[{({\textstyle{1\over 2}}-x_1)-({\textstyle{1\over ... ..._1+ \int_0^1 \left[{({x_1}^2-{\textstyle{1\over 2}}{x_1}^2)-(0-0)}\right]\,dx_1$$

$$= \int_0^1 ({\textstyle{1\over 2}}-x_1+{x_1}^2)\,dx_1 = [{\textstyle{1\over 2}}x_1-{\textstyle{1\over 2}}{x_1}^2+{\textstyle{1\over 3}}{x_1}^3]^1_0$$

$$= ({\textstyle{1\over 2}}-{\textstyle{1\over 2}}+{\textstyle{1\over 3}})-(0-0+0) = {\textstyle{1\over 3}}$$ (7)

$$\mu'_2 = \int_0^1\int_0^1 (\vert x_2-x_1\vert)^2\,dx_2\,dx_1$$

$$= \int_0^1\int_0^1 (x_2-x_1)^2\,dx_1\,dx_2$$

$$= \int_0^1\int_0^1 ({x_2}^2-2x_1x_2+{x_1}^2)\,dx_1\,dx_2$$

$$= \int_0^1 [{\textstyle{1\over 3}}{x_2}^3-x_1{x_2}^2+{x_1}^2x_2]^1_0\,dx_1$$

$$= \int_0^1 ({\textstyle{1\over 3}}-x_1+{x_1}^2)\,dx_1 = [{\textstyle{1\over 3}}{x_1}^3-{\textstyle{1\over 2}}{x_1}^2+{\textstyle{1\over 3}}x_1]^1_0$$

$$= {\textstyle{1\over 3}}-{\textstyle{1\over 2}}+{\textstyle{1\over 3}} = {\textstyle{1\over 6}}.$$ (8)

The Moments about the Mean are therefore

$$\mu_2 = \mu_2'-{\mu_1'}^2={\textstyle{1\over 6}}-({\textstyle{1\over 3}})^2={\textstyle{1\over 18}}$$ (9)

$$\mu_3 = \mu_3'-3\mu_2'\mu_1'+2(\mu_1')^3={\textstyle{1\over 135}}$$ (10)

$$\mu_4 = \mu_4'-4\mu_3'\mu_1'+6\mu_2'(\mu_1')^2-3(\mu_1')^4={\textstyle{1\over 135}},$$ (11)

so the Mean, Variance, Skewness, and Kurtosis are

$$\mu = \mu_1'={\textstyle{1\over 3}}$$ (12)

$$\sigma^2 = \mu_2={\textstyle{1\over 18}}$$ (13)

$$\gamma_1 = {\mu_3\over\sigma^3}={\textstyle{2\over 5}}\sqrt{2}$$ (14)

$$\gamma_2 = {\mu_4\over\sigma^4}-3=-{\textstyle{3\over 5}}.$$ (15)

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).

See also Point-Point Distance--2-D, Point-Point Distance--3-D, Point-Quadratic Distance, Tetrahedron Inscribing, Triangle Inscribing in a Circle

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.