Expected chord between two random points

HardProbability Expected Value~7m

Two points are chosen independently and uniformly on the circumference of a unit circle. Find the expected length of the chord joining them. The answer has the form $\dfrac{x}{\pi}$ for a rational $x$ ; find $x$ .

Show hints (2)+

By symmetry, only the angular separation $\Theta \sim \text{Uniform}(0,2\pi)$ matters.
A chord subtending angle $\Theta$ on a unit circle has length $2\sin(\Theta/2)$ . Integrate against the uniform density.

Answer

Reveal answer →

Want the full step-by-step worked solution? It's part of Premium - along with a worked solution for every question in the bank.

Practise free Unlock solutions

Asked at: Jane Street, SIG

Expected chord between two random points

Answer

Related questions