Math 2.0

Probability theory and stochastic processes Definition

Mutual independence of events

Builds on Independence of two eventsProbability spaceEvent

Definition

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space, let II be an index set, and let Ai∈ℱA_i \in \mathcal{F} be an event for each i∈Ii \in I. The events (Ai)i∈I(A_i)_{i \in I} are mutually independent if for each finite nonempty subset J⊆IJ \subseteq I, P(⋂j∈JAj)=∏j∈JP(Aj).P\left(\bigcap_{j \in J} A_j\right) = \prod_{j \in J} P(A_j).

Remarks

For two events this is the independence of two events. For three events A,B,CA, B, C the definition asks for four equations: one for each of the pairs {A,B}\{A, B\}, {A,C}\{A, C\}, {B,C}\{B, C\} and one for the triple {A,B,C}\{A, B, C\}. Mutual independence implies pairwise independence. The converse is false: see pairwise independent events that are not mutually independent. If we replace some of the events AiA_i by their complements, the family stays mutually independent.

Reading path

To understand this page from the ground up, read these 31 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…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.
  • no evidence level
  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

Where this leads

152 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Independence_(probability_theory), More than two events
About this page
Tag0GCVEV6BJA
URIm2://0GCVEV6BJA
Kinddefinition
WritingW0 stub
MSC60A05
DefinesMutual independence of events
UsesIndependence of two events, Probability space, Event

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