Math 2.0

Probability theory and stochastic processes Foundations: probability spaces and events Concept

Probability axioms

Builds on Sample spaceEventSigma-algebra

Summary

The probability axioms of Kolmogorov are the three rules that a probability function PP on the events of a sample space Ω\Omega must obey. First, P(A)≥0P(A) \ge 0 for each event AA. Second, P(Ω)=1P(\Omega) = 1. Third, for each sequence A1,A2,…A_1, A_2, \dots of pairwise disjoint events, P(⋃nAn)=∑nP(An)P\left(\bigcup_{n} A_n\right) = \sum_{n} P(A_n). This is countable additivity. A function that obeys these rules is a probability measure.

Remarks

Andrey Kolmogorov stated these axioms in 1933. The other rules of probability follow from the axioms. Examples are P(∅)=0P(\varnothing) = 0, P(A)≤1P(A) \le 1, the rule for the complement and the addition rule for two events.

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Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptRandom experiment, concept, no evidence levelRandom experimentconceptSet, definition, no evidence levelSetdefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionUnion, definition, no evidence levelUniondefinitionSample space, definition, no evidence levelSample spacedefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionSubset, definition, no evidence levelSubsetdefinitionPower set, definition, no evidence levelPower setdefinitionUniversal set, definition, no evidence levelUniversal setdefinitionComplement, definition, no evidence levelComplementdefinitionSigma-algebra, definition, no evidence levelSigma-algebradefinitionEvent, definition, no evidence levelEventdefinitionProbability axioms, concept, no evidence levelProbability axiomsconcept
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  1. Element definition
  2. Proposition concept
  3. Random experiment concept
  4. Set definition
  5. Disjunction definition
  6. Implication definition
  7. Predicate definition
  8. Union definition
  9. Sample space definition
  10. Universal quantifier definition
  11. Subset definition
  12. Power set definition
  13. Universal set definition
  14. Complement definition
  15. Sigma-algebra definition
  16. Event definition
  17. Probability axioms concept

Where this leads

374 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Probability_axioms, Kolmogorov axioms
About this page
Tag0DAX8PMAJ4
URIm2://0DAX8PMAJ4
Aliaseskolmogorov-axioms
Kindconcept
WritingW1 stated
MSC60A05
UsesSample space, Event, Sigma-algebra
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