Math 2.0

Probability theory and stochastic processes Expectation, moments and transforms Theorem

Variance of a sum of independent random variables

Builds on Variance of a sumIndependent random variables are uncorrelatedIndependence of random variables

Statement

Let X1,…,XnX_1, \dots, X_n be independent real random variables with E[Xi2]<∞E[X_i^2] < \infty for each ii. Then Var⁡(X1+⋯+Xn)=Var⁡(X1)+⋯+Var⁡(Xn).\operatorname{Var}(X_1 + \dots + X_n) = \operatorname{Var}(X_1) + \dots + \operatorname{Var}(X_n).

Remarks

The proof uses the variance of a sum: independent variables are uncorrelated, so each covariance term is 00. So the formula holds more generally for pairwise uncorrelated XiX_i. For i.i.d. XiX_i with variance σ2\sigma^2, the sample mean Xˉn=(X1+⋯+Xn)/n\bar{X}_n = (X_1 + \dots + X_n)/n has variance σ2/n\sigma^2 / n. For X∼Bin⁡(n,p)X \sim \operatorname{Bin}(n, p), the formula gives Var⁡(X)=np(1−p)\operatorname{Var}(X) = np(1 - p), as XX is a sum of nn independent Bernoulli variables.

History

The formula is also called the Bienaymé formula, after Irénée-Jules Bienaymé.

Reading path

To understand this page from the ground up, read these 61 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. Mutual independence of events definition
  50. Independence of two random variables definition
  51. Marginal distribution definition
  52. Linearity of expectation statement
  53. Variance definition
  54. Independent sigma-algebras definition
  55. Independence means the joint law is the product of the marginals statement
  56. Covariance definition
  57. Independence of random variables definition
  58. Expectation of a product of independent random variables statement
  59. Variance of a sum statement
  60. Independent random variables are uncorrelated statement
  61. Variance of a sum of independent random variables statement

Where this leads

55 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Variance, Sum of uncorrelated variables
About this page
Tag0V0778DCW9
URIm2://0V0778DCW9
Aliasesbienayme-formula
Kindstatement, theorem
WritingW0 stub
MSC60A05
StatusThis copy of the site has no build data. Run m2 status 0V0778DCW9.
UsesVariance of a sum, Independent random variables are uncorrelated, Independence of random variables

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