ThmDex – An index of mathematical definitions, results, and conjectures.
F9074
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space.
Let $X_j$ be a D202: Random variable on $P$ for each $j \in J$ such that
(i) $\sigma_{\text{pullback}} \langle X_j \rangle$ is the D1730: Pullback sigma-algebra of $X_j$ on $P$ for each $j \in J$
Let $\mathcal{P}_{\text{finite}}(J)$ be the D2337: Set of finite subsets of $J$.
The D1721: Random collection $X = \{ X_j \}_{j \in J}$ is independent if and only if \begin{equation} \forall \, I \in \mathcal{P}_{\text{finite}}(J) \left[ \forall \, i \in I : E_i \in \sigma_{\text{pullback}} \langle X_i \rangle \quad \implies \quad \mathbb{P} \left( \bigcap_{i \in I} E_i \right) = \prod_{i \in I} \mathbb{P}(E_i) \right] \end{equation}