Let $X_1,\ldots,X_n$ be random variables with realizations in $\mathbb{R}$ . Given an outcome $\omega$ , order $x_i=X_i(\omega)$ in non-decreasing order so that $$x_{(1)}\leq x_{(2)}\leq\cdots\leq x_{(n)}.$$ Note that $x_{(1)}=\operatorname{min}(x_1,\ldots,x_n)$ and $x_{(n)}=\operatorname{max}(x_1,\ldots,x_n)$ . Then each $X_{(i)}$ , such that $X_{(i)}(\omega)=x_{(i)}$ , is a random variable. Statistics defined by $X_{(1)},\ldots,X_{(n)}$ are called order statistics of $X_1,\ldots,X_n$ . If all the orderings are strict, then $X_{(1)},\ldots,X_{(n)}$ are the order statistics of $X_1,\ldots,X_n$ . Furthermore, each $X_{(i)}$ is called the $i$ th order statistic of $X_1,\ldots,X_n$ .

Remark. If $X_1,\ldots,X_n$ are iid as $X$ with probability density function $f_X$ (assuming $X$ is a continuous random variable), Let ${Z}$ be the vector of the order statistics $(X_{(1)},\ldots,X_{(n)})$ (with strict orderings), then one can show that the joint probability density function $f_{{Z}}$ of the order statistics is: $$f_{\textbf{Z}}(\boldsymbol{z})=n!\prod_{i=1}^{n}f_X(z_i),$$ where $\boldsymbol{z}=(z_1,\ldots,z_n)$ and $z_1<\cdots<z_n$ .