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  "body": "\n## Definition\n\nLet $X_1, \\dots, X_n$ be real [[random variables|0DKRR22AEQ]] on a probability space $(\\Omega, \\mathcal{F}, P)$.\nFor $\\omega \\in \\Omega$, list the numbers $X_1(\\omega), \\dots, X_n(\\omega)$ in increasing order, with repetitions, as $X_{(1)}(\\omega) \\le X_{(2)}(\\omega) \\le \\dots \\le X_{(n)}(\\omega)$.\nFor $1 \\le k \\le n$, the random variable $X_{(k)}$ is the *$k$-th order statistic* of the sample.\n\n## Remarks\n\nEach $X_{(k)}$ is a [[statistic|036756NFD6]], since sorting is a continuous map from $\\R^n$ to $\\R^n$.\nExample: if $X_1(\\omega) = 4$, $X_2(\\omega) = 1$, $X_3(\\omega) = 4$, then $X_{(1)}(\\omega) = 1$ and $X_{(2)}(\\omega) = X_{(3)}(\\omega) = 4$.\nNon-example: $X_2$ itself is not the second order statistic in general; in the example $X_2(\\omega) = 1 \\ne 4 = X_{(2)}(\\omega)$.\nBy definition, $X_{(1)} = \\min_i X_i$ and $X_{(n)} = \\max_i X_i$.\nIf the $X_i$ are i.i.d. with cumulative distribution function $F$, then $P(X_{(n)} \\le t) = F(t)^n$ and $P(X_{(1)} \u003e t) = (1 - F(t))^n$.\nIf the $X_i$ are i.i.d. uniform on $[0, 1]$, then $X_{(k)}$ has the beta distribution $\\mathrm{Beta}(k, n - k + 1)$.\n",
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