---
tag: 6Y451VYFV9
kind: definition
name: Order statistic
msc: [62G30]
defines: 58NHGZDSRE
uses: [0DKRR22AEQ, 036756NFD6]
sources:
  - url: https://en.wikipedia.org/wiki/Order_statistic
---

## Definition

Let $X_1, \dots, X_n$ be real [[random variables|0DKRR22AEQ]] on a probability space $(\Omega, \mathcal{F}, P)$.
For $\omega \in \Omega$, list the numbers $X_1(\omega), \dots, X_n(\omega)$ in increasing order, with repetitions, as $X_{(1)}(\omega) \le X_{(2)}(\omega) \le \dots \le X_{(n)}(\omega)$.
For $1 \le k \le n$, the random variable $X_{(k)}$ is the *$k$-th order statistic* of the sample.

## Remarks

Each $X_{(k)}$ is a [[statistic|036756NFD6]], since sorting is a continuous map from $\R^n$ to $\R^n$.
Example: if $X_1(\omega) = 4$, $X_2(\omega) = 1$, $X_3(\omega) = 4$, then $X_{(1)}(\omega) = 1$ and $X_{(2)}(\omega) = X_{(3)}(\omega) = 4$.
Non-example: $X_2$ itself is not the second order statistic in general; in the example $X_2(\omega) = 1 \ne 4 = X_{(2)}(\omega)$.
By definition, $X_{(1)} = \min_i X_i$ and $X_{(n)} = \max_i X_i$.
If the $X_i$ are i.i.d. with cumulative distribution function $F$, then $P(X_{(n)} \le t) = F(t)^n$ and $P(X_{(1)} > t) = (1 - F(t))^n$.
If the $X_i$ are i.i.d. uniform on $[0, 1]$, then $X_{(k)}$ has the beta distribution $\mathrm{Beta}(k, n - k + 1)$.