empirical distribution function

Let $X_1,\mathrm{\dots },X_n$ be random variables with realizations $x_i=X_i(\omega )\in ℝ$, $i=1,\mathrm{\dots },n$. The empirical distribution function $F_n(x,\omega )$ based on $x_1,\mathrm{\dots },x_n$ 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 $A_i=\{(x,\omega )\mid x_i\le x\}$. 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 $\omega$.