ThmDex – An index of mathematical definitions, results, and conjectures.
 ▼ Set of symbols ▼ Alphabet ▼ Deduction system ▼ Theory ▼ Zermelo-Fraenkel set theory ▼ Set ▼ Subset ▼ Power set ▼ Hyperpower set sequence ▼ Hyperpower set ▼ Hypersubset ▼ Subset algebra ▼ Subset structure ▼ Measurable space ▼ Measurable map ▼ Measurable function ▼ Pre-kernel ▼ Measure kernel ▼ Random unsigned basic measure ▼ Random probability measure ▼ Empirical probability distribution measure
Definition D5680
Random real number empirical probability distribution measure

Let $X_1, \ldots, X_N \in \text{Random}(\Omega \to \mathbb{R})$ each be a D3161: Random real number such that
 (i) $X_1, \ldots, X_N$ is an D3357: Identically distributed random collection
The empirical probability distribution measure with respect to $X_1, \ldots, X_N$ is the D3650: Random probability measure $$\Omega \times \mathcal{B}(\mathbb{R}) \to [0, 1], \quad (\omega, B) \mapsto \frac{1}{N} \sum_{n = 1}^N I_{\{ X_n \in B \}} (\omega)$$
Results
 ▶ I.I.D. real empirical distribution measure converges to a probability for a fixed Borel set