Section 8.4

Order Statistics

Distributions of the minimum, maximum, median or range of i.i.d. samples, probabilities about spacings, guessing ranks.

Toolkit

For i.i.d. $X_1,\dots,X_n$ with common CDF $F$ and density $f$ , order statistics $X_{(1)}\le\cdots\le X_{(n)}$ satisfy:

F_{\max}(x)=F(x)^n,\qquad F_{\min}(x)=1-(1-F(x))^n,

f_{(k)}(x)=\frac{n!}{(k-1)!(n-k)!}F(x)^{k-1}(1-F(x))^{n-k}f(x).

More generally $\mathbb{P}(X_{(k)}\le x)=\mathbb{P}(\text{at least }k\text{ of the }X_i\le x)$ , a Binomial $(n,F(x))$ tail — useful for medians and ranks. For spacings/symmetric events, exploit distributional symmetries (e.g. $x\mapsto 1-x$ for $\mathrm{Unif}(0,1)$ ) to pair or cancel events. For extremes of exponentials/large $n$ , rescale (e.g. $x=\ln n+c$ ) to get Gumbel-type limits $\exp(-e^{-c})$ . For ranking problems with a revealed value $a$ , compute posterior probabilities of each rank via Bayes’ rule using the binomial-type weights $\binom{n-1}{j}a^j(1-a)^{n-1-j}$ , then optimise the decision pointwise before integrating over $a$ .

Problems