Section 13.4

Expected Lengths and Distances

Expected chord lengths, distances from a centre, perimeters of random triangles.

Toolkit

For expected lengths or distances built from points placed uniformly on a disc or circle, exploit rotational/scaling symmetry to reduce to a single random variable, then use either direct integration of the density or the tail-sum formula

\mathbb{E}[R]=\int_0^\infty \mathbb{P}(R\ge r)\,dr .

Key facts:

Point uniform in a disc of radius $1$ : $\mathbb{P}(R\ge r)=1-r^2$ , giving moments $\mathbb{E}[R^k]=\frac{2}{k+2}$ .
Chord from two uniform points on a circle (or one fixed by symmetry): length $=2\sin(\theta/2)$ , with $\theta\sim\mathrm{Unif}(0,2\pi)$ (or $\mathrm{Unif}(0,\pi)$ by symmetry of the sine), so $\mathbb{E}[\text{length}]=\frac{1}{2\pi}\int_0^{2\pi}2\sin(\theta/2)\,d\theta$ .
For triangle perimeters, decompose into pairwise chord lengths and use linearity of expectation with the single-chord result.

Problems

13.17 Dart Distance From the Centre+1 variation Easy Free
13.18 Expected Chord Length Medium