Math 2.0

Probability theory and stochastic processes Definition

Probability distribution

Builds on Probability measureBorel sigma-algebra

Definition

Let EE be a set with a σ-algebra ℰ\mathcal{E}. A probability distribution on (E,ℰ)(E, \mathcal{E}) is a probability measure μ\mu on (E,ℰ)(E, \mathcal{E}). A probability distribution on ℝ\mathbb{R} is a probability measure on ℝ\mathbb{R} with its Borel σ-algebra ℬ(ℝ)\mathcal{B}(\mathbb{R}).

Remarks

Each random variable XX with values in EE gives a distribution, its law B↦P(X∈B)B \mapsto P(X \in B). Conversely, each distribution μ\mu is the law of some random variable, for example the identity map on (E,ℰ,μ)(E, \mathcal{E}, \mu). A distribution on ℝ\mathbb{R} is determined by its cumulative distribution function. Important types are discrete distributions, given by a probability mass function, and absolutely continuous distributions, given by a probability density function.

Reading path

To understand this page from the ground up, read these 30 pages in order. The foundations come first.

Show the path as a graph
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 numbersobjectReal numbers, object, no evidence levelReal 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 levelEventdefinitionBorel sigma-algebra, definition, no evidence levelBorel sigma-algebradefinitionProbability axioms, concept, no evidence levelProbability axiomsconceptProbability measure, definition, no evidence levelProbability measuredefinitionProbability distribution, definition, no evidence levelProbability distribut…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. Real numbers object
  9. Set definition
  10. Ring of integers statement
  11. Disjunction definition
  12. Implication definition
  13. Predicate definition
  14. Union definition
  15. Sample space definition
  16. The rationals form a field statement
  17. Universal quantifier definition
  18. Subset definition
  19. The rationals form an ordered field statement
  20. Power set definition
  21. Universal set definition
  22. Order on the reals definition
  23. Complement definition
  24. Interval definition
  25. Sigma-algebra definition
  26. Event definition
  27. Borel sigma-algebra definition
  28. Probability axioms concept
  29. Probability measure definition
  30. Probability distribution definition

Where this leads

207 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Probability_distribution, General probability definition
About this page
Tag0QB913WM5E
URIm2://0QB913WM5E
Kinddefinition
WritingW1 stated
MSC60E05
DefinesProbability distribution
UsesProbability measure, Borel sigma-algebra

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by