Section 13.2

Uniform Order Statistics and Spacings

E[k-th order statistic], E|X-Y|, expected pieces of a broken stick, the minimum spacing, arc lengths between random points.

Toolkit

For $n$ i.i.d. $\mathrm{Unif}(0,1)$ points, the order statistics $U_{(1)}<\cdots<U_{(n)}$ satisfy

\mathbb{E}[U_{(k)}]=\frac{k}{n+1},

and the $n+1$ spacings $U_{(k)}-U_{(k-1)}$ (with $U_{(0)}=0,U_{(n+1)}=1$ ) are exchangeable, each with mean $\frac{1}{n+1}$ ; jointly they have the same distribution as a uniform random simplex cut, so any symmetric functional (longest, shortest, a marked piece) can be computed by conditioning on which spacing plays that role and integrating, or via known results:

\mathbb{E}[\text{min spacing}]=\frac{1}{(n+1)^2},\qquad \mathbb{E}[\text{max of }n\text{ spacings}]=\frac{1}{n}\sum_{k=1}^n\frac1k.

Given extreme order statistics $U_{(1)}=a,\,U_{(n)}=b$ , the remaining $n-2$ points are i.i.d. $\mathrm{Unif}(a,b)$ . For exponential lifetimes, memorylessness turns successive failure gaps among $n$ i.i.d. $\mathrm{Exp}(\lambda)$ variables into independent $\mathrm{Exp}(k\lambda)$ gaps, $k=n,\dots,1$ , giving $\mathbb{E}[\max]=\frac1\lambda\sum_{k=1}^n\frac1k$ . Points placed uniformly on a circle of circumference $L$ give $n$ exchangeable arc spacings summing to $L$ , each with mean $L/n$ , and nearest-neighbour distances are minima of such uniform gaps.

Problems