Two points are uniformly distributed around the circumference of a unit circle. What is the expected value of the distance between them?

Free worked solution · Quant Blueprint

4/π for independent uniform points on a unit circle.

1. By rotational symmetry, fix the first point at angle zero. The angular difference θ to the second point is uniform from zero to 2π.
2. The chord length is 2 sin(θ/2). It is nonnegative throughout this interval.
3. Average over the angle: (1/(2π)) × ∫₀²π 2 sin(θ/2) dθ = 8/(2π) = 4/π, approximately 1.273.
4. The result is between zero and the diameter two, as required.

Common mistake

Do not use arc length in place of straight-line distance. Also specify uniform points on the circumference; uniform points inside the disk define a different problem.

Follow-up

For radius R, distances scale by R, so the answer is 4R/π. The expected squared distance is 2R².

Original educational explanation for this practice question. Reviewed 2026-10-13. Firm-style practice; the label describes the collection’s preparation context.