The average distance across a circle
Choose two points independently and uniformly on a circle of radius \(1\). What is their expected straight-line distance?
Hint
Fix one point by rotational symmetry and average over the angle of the other.
Solution
By rotational symmetry, fix the first point. The angle \(\theta\) of the second point relative to it is uniform on \([0,2\pi)\). The cosine rule gives the chord length
\[D=\sqrt{2-2\cos\theta}=2\sin\frac{\theta}{2},\]where the sine is nonnegative on this interval. Therefore,
\[\begin{aligned} \mathbb E[D] &=\frac{1}{2\pi}\int_0^{2\pi}2\sin\frac{\theta}{2}\,d\theta\\ &=\frac{4}{\pi}. \end{aligned}\]