Math 2.0

Probability theory and stochastic processes Proof

Proof of the formula for the variance of a sum of independent random variables

Builds on Independence of random variablesIndependence of two random variablesVariance of a sumIndependent random variables are uncorrelated

Proof

We use the general formula for the variance of a sum and show that all covariances vanish.

  1. Let X1,…,XnX_1, \dots, X_n be independent real random variables with E[Xi2]<∞E[X_i^2] < \infty for each ii.
  2. Fix i<ji < j. Apply the definition of independence to the set J={i,j}J = \set{i, j}. For all Borel sets AA and BB, it gives P(Xi∈A,Xj∈B)=P(Xi∈A) P(Xj∈B)P(X_i \in A, X_j \in B) = P(X_i \in A) \, P(X_j \in B).
  3. So XiX_i and XjX_j are independent as a pair.
  4. Both have finite second moments. Since independent random variables are uncorrelated, Cov⁡(Xi,Xj)=0\operatorname{Cov}(X_i, X_j) = 0.
  5. By the formula for the variance of a sum, Var⁡(X1+⋯+Xn)=∑i=1nVar⁡(Xi)+2∑1≤i<j≤nCov⁡(Xi,Xj).\operatorname{Var}(X_1 + \dots + X_n) = \sum_{i=1}^n \operatorname{Var}(X_i) + 2 \sum_{1 \le i < j \le n} \operatorname{Cov}(X_i, X_j).
  6. By step 4, every covariance term is 00. So Var⁡(X1+⋯+Xn)=Var⁡(X1)+⋯+Var⁡(Xn)\operatorname{Var}(X_1 + \dots + X_n) = \operatorname{Var}(X_1) + \dots + \operatorname{Var}(X_n).

Idea

The variance of a sum is the sum of the variances plus twice the sum of all covariances. Independence of each pair kills every covariance, so only the variances remain.

Reading path

To understand this page from the ground up, read these 115 pages in order. The foundations come first. The path takes the proofs Proof of the formula for the variance of a sum, 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 independent random variables are uncorrelated, 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. Mutual independence of events definition
  88. Independence of two random variables definition
  89. Joint probability distribution definition
  90. Triangle inequality statement
  91. Absolute value is multiplicative statement
  92. Expected value definition
  93. Epsilon-N definition of the limit of a sequence statement
  94. Proof that the reals are Archimedean from the completeness axiom proof
  95. Independent sigma-algebras definition
  96. Marginal distribution definition
  97. Variance definition
  98. Linearity of expectation statement
  99. Proof that convergent sequences are bounded proof
  100. The reals are Archimedean statement
  101. Independence of random variables definition
  102. Covariance definition
  103. Proof that independence means the joint law is the product of the marginals proof
  104. Floor function definition
  105. Convergent sequences are bounded statement
  106. Independence means the joint law is the product of the marginals statement
  107. Proof of the formula for the variance of a sum proof
  108. Proof of the limit of a product of sequences proof
  109. Variance of a sum statement
  110. Limit of a product of sequences statement
  111. Proof of the expectation of a product of independent random variables by simple functions proof
  112. Expectation of a product of independent random variables statement
  113. Proof that independent random variables are uncorrelated proof
  114. Independent random variables are uncorrelated statement
  115. Proof of the formula for the variance of a sum of independent random variables proof

Where this leads

63 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Variance, Sum of uncorrelated variables
About this page
TagTCV030YM49
URIm2://TCV030YM49
Kindproof
WritingW0 stub
MSC60A05
ProvesVariance of a sum of independent random variables
UsesIndependence of random variables, Independence of two random variables, Variance of a sum, Independent random variables are uncorrelated
Changed4 November 2026

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