Math 2.0

Probability theory and stochastic processes Proposition

Independent random variables are uncorrelated

Builds on CovarianceExpectation of a product of independent random variablesIndependence of two random variables

Statement

Let XX and YY be independent real random variables with E[X2]<∞E[X^2] < \infty and E[Y2]<∞E[Y^2] < \infty. Then Cov⁡(X,Y)=0\operatorname{Cov}(X, Y) = 0, so XX and YY are uncorrelated.

Remarks

Proof: by the product rule for independent variables, E[XY]=E[X] E[Y]E[XY] = E[X] \, E[Y]. So Cov⁡(X,Y)=E[XY]−E[X] E[Y]=0\operatorname{Cov}(X, Y) = E[XY] - E[X] \, E[Y] = 0. The converse is false. The node uncorrelated random variables that are not independent gives a counterexample. For a pair (X,Y)(X, Y) with a joint multivariate normal law, uncorrelated does imply independent.

Reading path

To understand this page from the ground up, read these 55 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. Injective function definition
  32. Preimage of a set definition
  33. Sigma-algebra definition
  34. Countable set definition
  35. Event definition
  36. Borel sigma-algebra definition
  37. Probability axioms concept
  38. Probability measure definition
  39. Probability distribution definition
  40. Probability space definition
  41. Conditional probability definition
  42. Random variable definition
  43. Discrete random variable definition
  44. Law of a random variable definition
  45. Independence of two events definition
  46. Random variable with a density definition
  47. Joint probability distribution definition
  48. Expected value definition
  49. Independence of two random variables definition
  50. Marginal distribution definition
  51. Variance definition
  52. Independence means the joint law is the product of the marginals statement
  53. Covariance definition
  54. Expectation of a product of independent random variables statement
  55. Independent random variables are uncorrelated statement

Where this leads

57 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Uncorrelatedness, Definition for two real random variables
  2. Web https://en.wikipedia.org/wiki/Covariance, Uncorrelatedness and independence
About this page
Tag077GN9HDR6
URIm2://077GN9HDR6
Kindstatement, proposition
WritingW0 stub
MSC60A05
StatusThis copy of the site has no build data. Run m2 status 077GN9HDR6.
UsesCovariance, Expectation of a product of independent random variables, Independence of two random variables

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