Needs:
Cardinality
Empirical Distribution of a Dataset
Probability Measures
Needed by:
None.
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Empirical Measure

Why

There is a natural probability measure on a measurable space to associate with a dataset from the base set of that space.

Definition

The empirical measure for a dataset in some measurable space is the measure which associates to each event the proportion of the records which are elements of that event.

Notation

Let (a1,,an) be a dataset in a measurable space (A,A). Let P:P(A)[0,1] be the probability measure that assigns to each set BA the number

P(B)=1n|{k{1,,n}|akB}|.

Then P is the empirical measure.

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