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 ▼ Measure space ▼ Probability space
Definition D1720
Independent event collection

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 $$\forall \, I \in \mathcal{P}_{\mathsf{finite}}(J) : \mathbb{P} \left( \bigcap_{i \in I} E_i \right) = \prod_{i \in I} \mathbb{P}(E_i)$$

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 $$\forall \, N \in \{ 1, 2, 3, \ldots \} : \forall \, j_1, \ldots, j_N \in J : \left[ j_1, \ldots, j_N \text{ distinct} \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^N E_{j_n} \right) = \prod_{n = 1}^N \mathbb{P}(E_{j_n}) \right]$$
Children
 ▶ D3831: Independent collection of event collections
Results
 ▶ R2374: Borel-Cantelli zero-one law ▶ R4348: Two disjoint events are not necessarily independent ▶ R4349: Two disjoint events independent iff one is of probability zero