Math 2.0

Probability theory and stochastic processes Concept

Law of large numbers

Builds on Independent and identically distributed random variables

Summary

The law of large numbers says that the average of many independent repetitions of a random experiment tends to the expected value. For an i.i.d. sequence X1,X2,…X_1, X_2, \dots with finite mean μ\mu, the sample mean Xˉn=(X1+⋯+Xn)/n\bar{X}_n = (X_1 + \dots + X_n)/n tends to μ\mu. The weak law gives convergence in probability, and the strong law gives almost sure convergence.

Remarks

The law fails without a finite mean. For i.i.d. standard Cauchy variables, Xˉn\bar{X}_n is standard Cauchy for each nn and does not converge. For indicator variables of an event AA, the law says that the relative frequency of AA tends to P(A)P(A). This links probability with long-run frequency.

Reading path

To understand this page from the ground up, read these 48 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. Preimage of a set definition
  32. Sigma-algebra definition
  33. Event definition
  34. Borel sigma-algebra definition
  35. Probability axioms concept
  36. Probability measure definition
  37. Probability distribution definition
  38. Probability space definition
  39. Conditional probability definition
  40. Random variable definition
  41. Independence of two events definition
  42. Law of a random variable definition
  43. Mutual independence of events definition
  44. Identically distributed random variables definition
  45. Independent sigma-algebras definition
  46. Independence of random variables definition
  47. Independent and identically distributed random variables definition
  48. Law of large numbers concept

Where this leads

11 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Law_of_large_numbers, lead section
  2. Web https://en.wikipedia.org/wiki/Law_of_large_numbers, Forms
About this page
Tag0ZVEAYVRC3
URIm2://0ZVEAYVRC3
Aliaseslln
Kindconcept
WritingW1 stated
MSC60F05, 60F15
UsesIndependent and identically distributed random variables
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