Math 2.0

Probability theory and stochastic processes Definition

Independence of random variables

Builds on Random variableMutual independence of eventsIndependent sigma-algebras

Definition

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space and let II be an index set. For each i∈Ii \in I, let XiX_i be a random variable with values in (Ei,ℰi)(E_i, \mathcal{E}_i). The family (Xi)i∈I(X_i)_{i \in I} is independent if for each finite nonempty set J⊆IJ \subseteq I and each choice of Bj∈ℰjB_j \in \mathcal{E}_j for j∈Jj \in J, P(⋂j∈J{Xj∈Bj})=∏j∈JP(Xj∈Bj).P\left(\bigcap_{j \in J} \{X_j \in B_j\}\right) = \prod_{j \in J} P(X_j \in B_j).

Remarks

Equivalently, the σ-algebras σ(Xi)={Xi−1(B):B∈ℰi}\sigma(X_i) = \{X_i^{-1}(B) : B \in \mathcal{E}_i\} are independent. For two random variables this is the independence of two random variables. Mutual independence implies pairwise independence, but the converse is false, as for events. For real random variables X1,…,XnX_1, \dots, X_n, independence holds if and only if the joint cumulative distribution function is the product of the marginal ones.

Reading path

To understand this page from the ground up, read these 43 pages in order. The foundations come first.

  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. Ordered pair definition
  15. Union definition
  16. Sample space definition
  17. The rationals form a field statement
  18. Existential quantifier definition
  19. Universal quantifier definition
  20. Cartesian product definition
  21. Subset definition
  22. The rationals form an ordered field statement
  23. Uniqueness quantifier definition
  24. Binary relation definition
  25. Universal set definition
  26. Order on the reals definition
  27. Power set definition
  28. Complement definition
  29. Function definition
  30. Interval definition
  31. Preimage of a set definition
  32. Sigma-algebra definition
  33. Event definition
  34. Borel sigma-algebra definition
  35. Probability axioms concept
  36. Probability measure definition
  37. Probability space definition
  38. Conditional probability definition
  39. Random variable definition
  40. Independence of two events definition
  41. Mutual independence of events definition
  42. Independent sigma-algebras definition
  43. Independence of random variables definition

Where this leads

148 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Independence_(probability_theory), More than two random variables
About this page
Tag0XCE070V8B
URIm2://0XCE070V8B
Kinddefinition
WritingW1 stated
MSC60A05
DefinesIndependence of random variables
UsesRandom variable, Mutual independence of events, Independent sigma-algebras

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • 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