Average Distance Between Random Points on a Line Segment

Suppose I have a line segment of length L. I now select two points at random along the segment. What is the expected value of the distance between the two points, and why?

Answer

Byron has already answered your question, but I will attempt to provide a detailed solution…

Let X be a random variable uniformly distributed over [0,L], i.e., the probability density function of X is the following

fX(x)={1Lifx[0,L]0otherwise

Let us randomly pick two points in [0,L] independently. Let us denote those by X1 and X2, which are random variables distributed according to fX. The distance between the two points is a new random variable

Y=|X1X2|

Hence, we would like to find the expected value E(Y)=E(|X1X2|). Let us introduce function g

g(x1,x2)=|x1x2|={x1x2ifx1x2x2x1ifx2x1

Since the two points are picked independently, the joint probability density function is the product of the pdf’s of X1 and X2, i.e., fX1X2(x1,x2)=fX1(x1)fX2(x2)=1/L2 in [0,L]×[0,L]. Therefore, the expected value E(Y)=E(g(X1,X2)) is given by

E(Y)=L0L0g(x1,x2)fX1X2(x1,x2)dx1dx2=1L2L0L0|x1x2|dx1dx2=1L2L0x10(x1x2)dx2dx1+1L2L0Lx1(x2x1)dx2dx1=L36L2+L36L2=L3

Attribution
Source : Link , Question Author : Kenshin , Answer Author : Michael Hardy

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