Math 2.0

Probability theory and stochastic processes Proposition

Variance of a sum

Builds on VarianceCovarianceLinearity of expectation

Statement

Let X1,…,XnX_1, \dots, X_n be real random variables on one probability space with E[Xi2]<∞E[X_i^2] < \infty for each ii. Then Var⁡(∑i=1nXi)=∑i=1nVar⁡(Xi)+2∑1≤i<j≤nCov⁡(Xi,Xj).\operatorname{Var}\left(\sum_{i=1}^n X_i\right) = \sum_{i=1}^n \operatorname{Var}(X_i) + 2 \sum_{1 \le i < j \le n} \operatorname{Cov}(X_i, X_j). In particular, Var⁡(X+Y)=Var⁡(X)+Var⁡(Y)+2Cov⁡(X,Y)\operatorname{Var}(X + Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) + 2 \operatorname{Cov}(X, Y).

Remarks

Proof: Var⁡(S)=Cov⁡(S,S)\operatorname{Var}(S) = \operatorname{Cov}(S, S) for S=∑iXiS = \sum_i X_i. Expand by the bilinearity of the covariance: Cov⁡(S,S)=∑i,jCov⁡(Xi,Xj)\operatorname{Cov}(S, S) = \sum_{i, j} \operatorname{Cov}(X_i, X_j). Also Var⁡(X−Y)=Var⁡(X)+Var⁡(Y)−2Cov⁡(X,Y)\operatorname{Var}(X - Y) = \operatorname{Var}(X) + \operatorname{Var}(Y) - 2 \operatorname{Cov}(X, Y). More generally, Var⁡(∑iaiXi)=∑i,jaiajCov⁡(Xi,Xj)\operatorname{Var}\left(\sum_i a_i X_i\right) = \sum_{i, j} a_i a_j \operatorname{Cov}(X_i, X_j) for real aia_i.

Reading path

To understand this page from the ground up, read these 47 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 space definition
  40. Random variable definition
  41. Discrete random variable definition
  42. Random variable with a density definition
  43. Expected value definition
  44. Linearity of expectation statement
  45. Variance definition
  46. Covariance definition
  47. Variance of a sum statement

Where this leads

56 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Variance, Sum of correlated variables
About this page
Tag0D86GJEZ39
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Kindstatement, proposition
WritingW0 stub
MSC60A05
StatusThis copy of the site has no build data. Run m2 status 0D86GJEZ39.
UsesVariance, Covariance, Linearity of expectation

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