Math 2.0

Probability theory and stochastic processes Definition

Independence of two random variables

Builds on Random variableIndependence of two events

Definition

Let (Ω,ℱ,P)(\Omega, \mathcal{F}, P) be a probability space. Let XX be a random variable with values in (E1,ℰ1)(E_1, \mathcal{E}_1) and YY a random variable with values in (E2,ℰ2)(E_2, \mathcal{E}_2). XX and YY are independent if for all A∈ℰ1A \in \mathcal{E}_1 and B∈ℰ2B \in \mathcal{E}_2, P(X∈A, Y∈B)=P(X∈A) P(Y∈B).P(X \in A, \, Y \in B) = P(X \in A) \, P(Y \in B).

Remarks

In other words, the events {X∈A}\{X \in A\} and {Y∈B}\{Y \in B\} are independent for all AA and BB. For real random variables, it is enough to check P(X≤x,Y≤y)=P(X≤x) P(Y≤y)P(X \le x, Y \le y) = P(X \le x) \, P(Y \le y) for all real xx and yy. For discrete random variables, independence is equivalent to P(X=x,Y=y)=P(X=x) P(Y=y)P(X = x, Y = y) = P(X = x) \, P(Y = y) for all xx and yy. If XX and YY are independent and gg, hh are measurable functions, then g(X)g(X) and h(Y)h(Y) are independent.

Reading path

To understand this page from the ground up, read these 41 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. Independence of two random variables definition

Where this leads

69 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Independence_(probability_theory), Two random variables
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