Math 2.0

Probability theory and stochastic processes Proof

Proof that independent random variables are uncorrelated

Builds on Independence of two random variablesExpectation of a product of independent random variablesCovarianceLinearity of expectationExpected value

Proof

We expand the covariance and use the product rule for independent variables.

  1. Let XX and YY be independent real random variables with E[X2]<∞E[X^2] < \infty and E[Y2]<∞E[Y^2] < \infty.
  2. For each real xx, ∣x∣≤(1+x2)/2\abs{x} \le (1 + x^2)/2, because (∣x∣−1)2≥0(\abs{x} - 1)^2 \ge 0. The integral of nonnegative functions is monotone.
  3. So E[∣X∣]≤(1+E[X2])/2<∞E[\abs{X}] \le (1 + E[X^2])/2 < \infty. Thus XX is integrable, and in the same way YY is integrable. Let m=E[X]m = E[X] and n=E[Y]n = E[Y] be their expected values.
  4. By the product rule for independent variables, XYXY is integrable and E[XY]=mnE[XY] = mn.
  5. Pointwise, (X−m)(Y−n)=XY−nX−mY+mn(X - m)(Y - n) = XY - nX - mY + mn.
  6. Each term on the right is integrable. By linearity of expectation, E[(X−m)(Y−n)]=E[XY]−nE[X]−mE[Y]+mn=E[XY]−mn.E[(X - m)(Y - n)] = E[XY] - n E[X] - m E[Y] + mn = E[XY] - mn.
  7. By the definition of the covariance and steps 4 and 6, Cov⁡(X,Y)=mn−mn=0\operatorname{Cov}(X, Y) = mn - mn = 0.
  8. So XX and YY are uncorrelated.

Idea

The covariance equals E[XY]−E[X]E[Y]E[XY] - E[X] E[Y]. Independence makes the mean of the product equal to the product of the means, so this difference is 00.

Reading path

To understand this page from the ground up, read these 108 pages in order. The foundations come first. The path takes the proofs Proof of the approximation property of the supremum, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the expectation of a product of independent random variables by simple functions, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof that independence means the joint law is the product of the marginals.

  1. Cartesian product concept
  2. Construction of the reals from Cauchy sequences concept
  3. Element definition
  4. Extended real number line object
  5. Integers object
  6. Integral of a non-negative function concept
  7. Measurable function concept
  8. Measurable space concept
  9. Measure concept
  10. Measure space concept
  11. Ordered field definition
  12. Product sigma-algebra concept
  13. Proposition concept
  14. Random experiment concept
  15. Rational numbers object
  16. Real numbers object
  17. Sequence definition
  18. Set bounded above definition
  19. Sigma-algebra concept
  20. Sigma-finite measure concept
  21. Simple function concept
  22. Set definition
  23. Ring of integers statement
  24. Monotone convergence theorem statement
  25. Product sigma-algebra definition
  26. Archimedean ordered field definition
  27. Disjunction definition
  28. Implication definition
  29. Predicate definition
  30. Product measure definition
  31. The reals form a field statement
  32. Real sequence definition
  33. Supremum definition
  34. Ordered pair definition
  35. Union definition
  36. Sample space definition
  37. Existential quantifier definition
  38. Universal quantifier definition
  39. The rationals form a field statement
  40. Integral of a non-negative function definition
  41. Bounded sequence definition
  42. Limit of a sequence definition
  43. Completeness axiom statement
  44. Cartesian product definition
  45. Subset definition
  46. Uniqueness quantifier definition
  47. The rationals form an ordered field statement
  48. Convergent sequence definition
  49. Binary relation definition
  50. Universal set definition
  51. Power set definition
  52. Order on the reals definition
  53. Antisymmetric relation definition
  54. Reflexive relation definition
  55. Transitive relation definition
  56. Function definition
  57. Complement definition
  58. Interval definition
  59. Preorder definition
  60. Sigma-algebra definition
  61. Indicator function definition
  62. Preimage of a set definition
  63. Injective function definition
  64. Partial order definition
  65. Event definition
  66. Countable set definition
  67. Borel sigma-algebra definition
  68. Total order definition
  69. Probability axioms concept
  70. Probability measure definition
  71. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  72. Probability space definition
  73. Probability distribution definition
  74. The reals form an ordered field statement
  75. Conditional probability definition
  76. Absolute value definition
  77. Random variable definition
  78. Proof of the approximation property of the supremum proof
  79. Independence of two events definition
  80. Law of a random variable definition
  81. Proof of the triangle inequality for real numbers proof
  82. Proof that the absolute value is multiplicative proof
  83. Discrete random variable definition
  84. Random variable with a density definition
  85. Proof of the epsilon-N definition of the limit of a sequence proof
  86. Approximation property of the supremum statement
  87. Independence of two random variables definition
  88. Joint probability distribution definition
  89. Triangle inequality statement
  90. Absolute value is multiplicative statement
  91. Expected value definition
  92. Epsilon-N definition of the limit of a sequence statement
  93. Proof that the reals are Archimedean from the completeness axiom proof
  94. Marginal distribution definition
  95. Variance definition
  96. Linearity of expectation statement
  97. Proof that convergent sequences are bounded proof
  98. The reals are Archimedean statement
  99. Covariance definition
  100. Proof that independence means the joint law is the product of the marginals proof
  101. Floor function definition
  102. Convergent sequences are bounded statement
  103. Independence means the joint law is the product of the marginals statement
  104. Proof of the limit of a product of sequences proof
  105. Limit of a product of sequences statement
  106. Proof of the expectation of a product of independent random variables by simple functions proof
  107. Expectation of a product of independent random variables statement
  108. Proof that independent random variables are uncorrelated proof

Where this leads

66 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Covariance, Uncorrelatedness and independence
About this page
TagNABYVABHTT
URIm2://NABYVABHTT
Kindproof
WritingW0 stub
MSC60A05
ProvesIndependent random variables are uncorrelated
UsesIndependence of two random variables, Expectation of a product of independent random variables, Covariance, Linearity of expectation, Expected value
Changed4 November 2026

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