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Let $X_1,\ldots,X_n$ be random variables with realizations $x_i=X_i(\omega)\in\mathbb{R}$ , $i=1,\ldots,n$ . The empirical distribution function $F_n(x,\omega)$ based on $x_1,\ldots,x_n$ is $$F_n(x,\omega)=\frac{1}{n}\sum_{i=1}^{n}\chi_{A_i}(x,\omega),$$ where $\chi_{A_i}$ is the indicator function (or characteristic function) and $A_i=\lbrace(x,\omega)\mid x_i\leq x \rbrace$ . Note that each indicator function is itself a random
variable.
The empirical function can be alternatively and equivalently defined by using the order statistics $X_{(i)}$ of $X_i$ as:
where $x_{(i)}$ is the realization of the random variable $X_{(i)}$ with outcome $\omega$ .
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