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

where $\chi_{{A_{i}}}$ is the indicator function (or characteristic function) and $A_{i}=\{(x,\omega)\mid x_{i}\leq x\}$. 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 $\omega$.