Math 2.0

Probability theory and stochastic processes Conditioning Theorem

Law of total expectation

Builds on Conditional expectationExpected valueLaw of total probability

Statement

Let XX be an integrable real random variable on (Ω,ℱ,P)(\Omega, \mathcal{F}, P) and let 𝒢⊆ℱ\mathcal{G} \subseteq \mathcal{F} be a σ-algebra. Then E[E[X∣𝒢]]=E[X].E\left[E[X \mid \mathcal{G}]\right] = E[X]. In particular, E[E[X∣Y]]=E[X]E[E[X \mid Y]] = E[X] for each random variable YY on the same space.

Remarks

Proof: put G=ΩG = \Omega in the defining property of the conditional expectation. For a partition of Ω\Omega into events A1,A2,…A_1, A_2, \dots with P(Ai)>0P(A_i) > 0, it gives E[X]=∑iE[X∣Ai] P(Ai)E[X] = \sum_i E[X \mid A_i] \, P(A_i). This extends the law of total probability. More generally, for σ-algebras ℋ⊆𝒢\mathcal{H} \subseteq \mathcal{G}, E[E[X∣𝒢]∣ℋ]=E[X∣ℋ]E[E[X \mid \mathcal{G}] \mid \mathcal{H}] = E[X \mid \mathcal{H}]. This is the tower property.

Reading path

To understand this page from the ground up, read these 54 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. Conditional expectation definition
  54. Law of total expectation statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Law_of_total_expectation, Proof in the general case
About this page
Tag04RRB6HM00
URIm2://04RRB6HM00
Aliasestower-property, law-of-iterated-expectations, adam-law
Kindstatement, theorem
WritingW0 stub
MSC60A10
StatusThis copy of the site has no build data. Run m2 status 04RRB6HM00.
UsesConditional expectation, Expected value, Law of total probability
Same as

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by