ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D1720
Independent event collection
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E_j \in \mathcal{F}$ is an D1716: Event in $P$ for each $j \in J$
Let $\mathcal{P}_{\mathsf{finite}}(J)$ be the D2337: Set of finite subsets of $J$.
Then $\{ E_j \}_{j \in J}$ is an independent event collection in $P$ if and only if \begin{equation} \forall \, I \in \mathcal{P}_{\mathsf{finite}}(J) : \mathbb{P} \left( \bigcap_{i \in I} E_i \right) = \prod_{i \in I} \mathbb{P}(E_i) \end{equation}
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $E_j \in \mathcal{F}$ is an D1716: Event in $P$ for each $j \in J$
Then $\{ E_j \}_{j \in J}$ is an independent event collection in $P$ if and only if \begin{equation} \forall \, N \in \{ 1, 2, 3, \ldots \} : \forall \, j_1, \ldots, j_N \in J : \left[ j_1, \ldots, j_N \text{ distinct} \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^N E_{j_n} \right) = \prod_{n = 1}^N \mathbb{P}(E_{j_n}) \right] \end{equation}
Children
Independent collection of event collections
Results
Borel-Cantelli zero-one law
Two disjoint events are not necessarily independent
Two disjoint events independent iff one is of probability zero