Order statistic

Remarks

Each $X_{(k)}$ is a statistic, 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 $\mathrm{Beta}(k, n - k + 1)$.