Definition
Independence
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space (Definition~Probability Space).
- Events $A, B \in \mathcal{F}$ are independent if $\mathbb{P}(A \cap B) = \mathbb{P}(A)\mathbb{P}(B)$.
- Random variables $X, Y$ (Definition~Random Variable) are independent if $\sigma(X)$ and $\sigma(Y)$ are independent sub-$\sigma$-algebras, i.e.\ $\mathbb{P}(X \in A, Y \in B) = \mathbb{P}(X \in A)\mathbb{P}(Y \in B)$ for all Borel $A, B$.
- A collection $\{X_i\}_{i \in I}$ is independent if for every finite $J \subseteq I$, $\mathbb{P}(\bigcap_{j \in J} \{X_j \in B_j\}) = \prod_{j \in J} \mathbb{P}(X_j \in B_j)$.
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