Expected distance between two random points on a rope

Question

Two points are chosen independently and uniformly on a rope of length 1. What is the expected distance between them?

Accepted Answer

Let the two points be X and Y, independent uniform on [0, 1]. We want E\lvert X - Yvert. Order-statistics shortcut. Whichever point is smaller is the minimum, the larger is the maximum, and the distance is \max - \min. For two uniform points, the equal-gaps rule (2 points split [0,1] into 3 equal-in-expectation gaps of length 	frac13) gives E[\min] = \frac{1}{3}, \qquad E[\max] = \frac{2}{3}, \qquad E\lvert X - Yvert = E[\max] - E[\min] = \frac{2}{3} - \frac{1}{3} = \frac{1}{3}. Integral confirmation. Fixing X = x, the average distance to a uniform Y is \int0^1 x - y\,dy = \int0^x (x - y…

Expected distance between two random points on a rope

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