Math 2.0

Probability theory and stochastic processes Counterexample

Uncorrelated random variables that are not independent

Builds on CovarianceIndependent random variables are uncorrelatedIndependence of two random variablesNormal distribution

Counterexample

The claim "uncorrelated random variables are independent" is false. Here are two counterexamples. First: let X∼𝒩(0,1)X \sim \mathcal{N}(0, 1) and Y=X2Y = X^2. Then Cov⁡(X,Y)=E[X3]−E[X] E[X2]=0−0=0\operatorname{Cov}(X, Y) = E[X^3] - E[X] \, E[X^2] = 0 - 0 = 0, since the odd moments of XX are 00. But XX and YY are not independent: P(∣X∣≤1, Y>1)=0P(|X| \le 1, \, Y > 1) = 0, while P(∣X∣≤1) P(Y>1)>0P(|X| \le 1) \, P(Y > 1) > 0. Second: let X∼𝒩(0,1)X \sim \mathcal{N}(0, 1), and let WW be independent of XX with P(W=1)=P(W=−1)=1/2P(W = 1) = P(W = -1) = 1/2. Put Y=WXY = WX. Then Y∼𝒩(0,1)Y \sim \mathcal{N}(0, 1) and Cov⁡(X,Y)=E[W] E[X2]=0\operatorname{Cov}(X, Y) = E[W] \, E[X^2] = 0. But ∣Y∣=∣X∣|Y| = |X|, so XX and YY are not independent.

Remarks

In the second example, both XX and YY are normal, but the pair (X,Y)(X, Y) is not multivariate normal: X+YX + Y is 00 with probability 1/21/2. So the rule "uncorrelated implies independent" needs a joint normal law, not only normal marginals. The direction that is true is that independent variables are uncorrelated.

Reading path

To understand this page from the ground up, read these 62 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. Indicator function definition
  32. Injective function definition
  33. Preimage of a set definition
  34. Sigma-algebra definition
  35. Countable set definition
  36. Event definition
  37. Borel sigma-algebra definition
  38. Probability axioms concept
  39. Probability measure definition
  40. Dirac measure definition
  41. Probability distribution definition
  42. Probability space definition
  43. Conditional probability definition
  44. Random variable definition
  45. Discrete random variable definition
  46. Cumulative distribution function definition
  47. Independence of two events definition
  48. Law of a random variable definition
  49. Random variable with a density definition
  50. Joint probability distribution definition
  51. Independence of two random variables definition
  52. Expected value definition
  53. Probability density function definition
  54. Marginal distribution definition
  55. Variance definition
  56. Density of the normal distribution definition
  57. Independence means the joint law is the product of the marginals statement
  58. Covariance definition
  59. Normal distribution definition
  60. Expectation of a product of independent random variables statement
  61. Independent random variables are uncorrelated statement
  62. Uncorrelated random variables that are not independent counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Misconceptions_about_the_normal_distribution, A symmetric example
  2. Web https://en.wikipedia.org/wiki/Uncorrelatedness, Examples of dependence without correlation
About this page
Tag0C6J8BQXVR
URIm2://0C6J8BQXVR
Aliasesuncorrelated-does-not-imply-independent
Kindcounterexample
WritingW0 stub
MSC60A05
UsesCovariance, Independent random variables are uncorrelated, Independence of two random variables, Normal distribution

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