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Definition D5236
Empirical probability distribution measure

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $M = (\Xi, \mathcal{S})$ is a D1108: Measurable space (ii) $X_1, \ldots, X_N : \Omega \to \Xi$ are each a D202: Random variable from $P$ to $M$
The empirical probability distribution measure with respect to $X_1, \ldots, X_N$ is the D3650: Random probability measure $$\Omega \times \mathcal{S} \to [0, 1], \quad (\omega, S) \mapsto \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \in S \}} (\omega)$$
Children
 ▶ Random real number empirical probability distribution measure