Empirical Distribution of a Dataset
Why
A natural distribution to associate with a dataset is to assign to each outcome a probability which reflects the number of times it appears in the dataset.
Definition
Given a dataset $x_1, \dots , x_n$ is a finite
set $X$, the empirical
distribution is the function $q: X \to
\R $ which associates each outcome with the
proportion of times it appears in the dataset.
In other words, $q$ is defined by
\[
q(a) = \frac{1}{n} \num{\Set*{k \in \set{1, \dots , n}}{a^k =
a}}.
\]