Empirical distribution function

In statistics, an empirical distribution function is a cumulative probability distribution function that concentrates probability 1/n at each of the n numbers in a sample.

Let $$X_1,\ldots,X_n$$ be random variables with realizations $$ x_i\in\mathbb{R}, i=1,\ldots,n\in\mathbb{N}$$.

The empirical distribution function $$ F_n(x) $$ based on sample $$ x_1,\ldots,x_n$$ is a step function defined by


 * $$F_n(x) = \frac{ \mbox{number of elements in the sample} \leq x}n =

\frac{1}{n} \sum_{i=1}^n I(x_i \le x),$$

where I(A) is an indicator function.