Given a unit circle, pick two points at random on its circumference, forming a chord. Without loss of generality,
the first point can be taken as , and the second by , with (by symmetry, the range can be limited to instead of ). The distance between the two points is then

Sheng, T. .K. "The Distance between Two Random Points in Plane Regions." Adv. Appl. Prob.17, 748-773, 1985.Sloane,
N. J. A. Sequences A000984/M1645,
A001803/M2986, A061549,
and A093581 in "The On-Line Encyclopedia
of Integer Sequences."