Math 2.0

Probability theory and stochastic processes Definition

Independent sigma-algebras

Builds on Sigma-algebraMutual independence of events

Definition

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space, let II be an index set, and for each i∈Ii \in I let ℱi⊆ℱ\mathcal{F}_i \subseteq \mathcal{F} be a σ-algebra on Ω\Omega. The family (ℱi)i∈I(\mathcal{F}_i)_{i \in I} is independent if for each finite nonempty subset J⊆IJ \subseteq I and each choice of events Aj∈ℱjA_j \in \mathcal{F}_j for j∈Jj \in J, P(⋂j∈JAj)=∏j∈JP(Aj).P\left(\bigcap_{j \in J} A_j\right) = \prod_{j \in J} P(A_j).

Remarks

Events (Ai)i∈I(A_i)_{i \in I} are mutually independent if and only if the σ-algebras {∅,Ai,Aic,Ω}\{\varnothing, A_i, A_i^c, \Omega\} are independent. Random variables are independent when the σ-algebras that they generate are independent. To check independence, it is enough to check the product formula on a generating family of each ℱi\mathcal{F}_i that is closed under finite intersections. This follows from Dynkin's π-λ theorem.

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To understand this page from the ground up, read these 32 pages in order. The foundations come first.

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Construction of the reals from Cauchy sequences, concept, no evidence levelConstruction of the r…conceptElement, definition, no evidence levelElementdefinitionIntegers, object, no evidence levelIntegersobjectOrdered field, definition, no evidence levelOrdered fielddefinitionProposition, concept, no evidence levelPropositionconceptRandom experiment, concept, no evidence levelRandom experimentconceptRational numbers, object, no evidence levelRational numbersobjectSet, definition, no evidence levelSetdefinitionRing of integers, statement, no evidence levelRing of integersstatementDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionUnion, definition, no evidence levelUniondefinitionSample space, definition, no evidence levelSample spacedefinitionThe rationals form a field, statement, no evidence levelThe rationals form a …statementUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSubset, definition, no evidence levelSubsetdefinitionThe rationals form an ordered field, statement, no evidence levelThe rationals form an…statementPower set, definition, no evidence levelPower setdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionOrder on the reals, definition, no evidence levelOrder on the realsdefinitionComplement, definition, no evidence levelComplementdefinitionInterval, definition, no evidence levelIntervaldefinitionSigma-algebra, definition, no evidence levelSigma-algebradefinitionEvent, definition, no evidence levelEventdefinitionProbability axioms, concept, no evidence levelProbability axiomsconceptProbability measure, definition, no evidence levelProbability measuredefinitionProbability space, definition, no evidence levelProbability spacedefinitionConditional probability, definition, no evidence levelConditional probabili…definitionIndependence of two events, definition, no evidence levelIndependence of two e…definitionMutual independence of events, definition, no evidence levelMutual independence o…definitionIndependent sigma-algebras, definition, no evidence levelIndependent sigma-alg…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Set definition
  9. Ring of integers statement
  10. Disjunction definition
  11. Implication definition
  12. Predicate definition
  13. Union definition
  14. Sample space definition
  15. The rationals form a field statement
  16. Universal quantifier definition
  17. Subset definition
  18. The rationals form an ordered field statement
  19. Power set definition
  20. Universal set definition
  21. Order on the reals definition
  22. Complement definition
  23. Interval definition
  24. Sigma-algebra definition
  25. Event definition
  26. Probability axioms concept
  27. Probability measure definition
  28. Probability space definition
  29. Conditional probability definition
  30. Independence of two events definition
  31. Mutual independence of events definition
  32. Independent sigma-algebras definition

Where this leads

149 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Independence_(probability_theory), Independent σ-algebras
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WritingW1 stated
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DefinesIndependent sigma-algebras
UsesSigma-algebra, Mutual independence of events

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