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Independence

Events

Given a probability space

\[\big(\Omega, \mathcal{F}, \mathbb{P}\big)\]
  • Two Events: \(A, B\) are independent under \(\mathbb{P}\) if
\[\mathbb{P}( A \cap B) = \mathbb{P}(A) \mathbb{P}(B)\]
  • Finite Collection of Events: \(A_1, A_2, \dots, A_n\) are independent if
\[\forall I_0 \subset \{1,2, \dots, n\}, \quad \mathbb{P}\big(\cap _{i \in I_0} A_i \big) = \prod _{i \in I_0} \mathbb{P}(A_i)\]

  • Arbitrary Collection of Events: \(\{A_i, i \in I\}\)

    • Independent if for any finite subset, the independent condition defined above holds

Sub-\(\sigma\)-algebras

  • Two sub-\(\sigma\)-algebras: \(\mathcal{F}_1, \mathcal{F}_2\)
\[\forall A_1 \in \mathcal{F}_1, \forall A_2 \in \mathcal{F}_2, \quad \mathbb{P}( A_1 \cap A_2) = \mathbb{P}(A_1) \mathbb{P}(A_2)\]
  • Arbitrary Collection of Sub-\(\sigma\)-algebras: \(\{\mathcal{F}_i, i \in I\}\)

Are independent if for any subset of \(\Omega\) selected from any sub-\(\sigma\)-algebras, that collection of subsets is indepdent as defined above

Random Variables

\(X, Y\) are independent if the \(\sigma\)-algebras generated by these random variables are independent as defined above

  • \(\sigma\)-algebra generated by the random variable \(X\)
\[\sigma(X) := \{A \in \mathcal{F} \mid A = X^{-1}(B) \ \textrm{for} \ B \in \mathcal{B}(\mathcal{R})\}\]