Math 2.0

Probability theory and stochastic processes Conditioning Theorem

Law of total variance

Builds on Conditional varianceConditional expectationLaw of total expectationVariance as the mean of the square minus the square of the mean

Statement

Let XX be a real random variable with E[X2]<∞E[X^2] < \infty, and let YY be a random variable on the same probability space. Then Var⁡(X)=E[Var⁡(X∣Y)]+Var⁡(E[X∣Y]).\operatorname{Var}(X) = E\left[\operatorname{Var}(X \mid Y)\right] + \operatorname{Var}\left(E[X \mid Y]\right).

Remarks

Proof: by the variance formula in its conditional form, E[X2∣Y]=Var⁡(X∣Y)+(E[X∣Y])2E[X^2 \mid Y] = \operatorname{Var}(X \mid Y) + (E[X \mid Y])^2. Take expectations with the law of total expectation, and subtract (E[X])2=(E[E[X∣Y]])2(E[X])^2 = (E[E[X \mid Y]])^2. The same result holds with a σ-algebra 𝒢\mathcal{G} in place of YY. The first term is the mean variance inside the groups. The second term is the variance between the group means. A corollary: Var⁡(E[X∣Y])≤Var⁡(X)\operatorname{Var}(E[X \mid Y]) \le \operatorname{Var}(X).

Reading path

To understand this page from the ground up, read these 59 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. Conjunction definition
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Empty set object
  16. Family of sets definition
  17. Ordered pair definition
  18. Intersection definition
  19. Union definition
  20. Sample space definition
  21. The rationals form a field statement
  22. Existential quantifier definition
  23. Universal quantifier definition
  24. Disjoint sets definition
  25. Cartesian product definition
  26. Subset definition
  27. Union of a family of sets definition
  28. The rationals form an ordered field statement
  29. Uniqueness quantifier definition
  30. Universal set definition
  31. Binary relation definition
  32. Partition of a set definition
  33. Order on the reals definition
  34. Power set definition
  35. Complement definition
  36. Function definition
  37. Interval definition
  38. Injective function definition
  39. Preimage of a set definition
  40. Sigma-algebra definition
  41. Countable set definition
  42. Event definition
  43. Borel sigma-algebra definition
  44. Probability axioms concept
  45. Probability measure definition
  46. Probability space definition
  47. Conditional probability definition
  48. Random variable definition
  49. Discrete random variable definition
  50. Law of total probability statement
  51. Random variable with a density definition
  52. Expected value definition
  53. Linearity of expectation statement
  54. Variance definition
  55. Conditional expectation definition
  56. Variance as the mean of the square minus the square of the mean statement
  57. Law of total expectation statement
  58. Conditional variance definition
  59. Law of total variance statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Law_of_total_variance, General case
About this page
Tag03BY5XTNA4
URIm2://03BY5XTNA4
Aliasesvariance-decomposition-formula, eve-law
Kindstatement, theorem
WritingW0 stub
MSC60A10
StatusThis copy of the site has no build data. Run m2 status 03BY5XTNA4.
UsesConditional variance, Conditional expectation, Law of total expectation, Variance as the mean of the square minus the square of the mean
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