# Empirical Distribution¶

Often we are working directly with data and we don’t know the parent distribution that generated the data.

We often denote a dataset with $$N$$ data points indexed by $$i$$ as $$\{x_i\}_{i=1}^N$$.

Sometimes this dataset is thought of a samples or realizatiosn from some parent distribution. For instance, we often assume that we have independent and identically distributed (iid) data $$x_i \sim p_X$$ for $$i=1\dots N$$.

In other cases one thinks of this data set as an emperical distribution

$p_\textrm{emp, X} = \frac{1}{N} \sum_{i=1}^N \delta(x-x_i)$