# Empirical distribution function

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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.