Math 2.0

Probability theory and stochastic processes Definition

Probability space

Builds on Sample spaceSigma-algebraProbability measure

Definition

A probability space is a triple (Ω,ℱ,P)(\Omega, \mathcal{F}, P) with these parts. Ω\Omega is a nonempty set, the sample space. ℱ\mathcal{F} is a σ-algebra on Ω\Omega. Its elements are the events. PP is a probability measure on (Ω,ℱ)(\Omega, \mathcal{F}).

Remarks

For one roll of a fair die: Ω={1,…,6}\Omega = \{1, \dots, 6\}, ℱ\mathcal{F} is the power set of Ω\Omega, and P(A)=∣A∣/6P(A) = |A| / 6. A discrete probability space has a countable Ω\Omega, the power set as ℱ\mathcal{F}, and P(A)=∑ω∈Ap(ω)P(A) = \sum_{\omega \in A} p(\omega) for a function p≥0p \ge 0 with total sum 11. The interval [0,1][0, 1] with its Borel σ-algebra and the length of intervals as PP is a probability space. It models a point chosen uniformly at random in [0,1][0, 1].

Reading path

To understand this page from the ground up, read these 28 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 spacedefinition
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

Where this leads

359 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Probability_space, Definition
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Kinddefinition
WritingW1 stated
MSC60A05
DefinesProbability space
UsesSample space, Sigma-algebra, Probability measure

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