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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.

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