Math 2.0

Probability theory and stochastic processes Proposition

Tail sum formula for expectation

Builds on Expected valueCumulative distribution function

Statement

Let XX be a random variable with values in [0,∞)[0, \infty). Then E[X]=∫0∞P(X>t) dt,E[X] = \int_0^\infty P(X > t) \, dt, with both sides in [0,∞][0, \infty]. If XX takes values in {0,1,2,… }\{0, 1, 2, \dots\}, then E[X]=∑n=1∞P(X≥n).E[X] = \sum_{n=1}^{\infty} P(X \ge n).

Remarks

The proof writes X=∫0∞1{X>t} dtX = \int_0^\infty 1_{\{X > t\}} \, dt and exchanges expectation and integral by Tonelli's theorem. In terms of the cumulative distribution function FF, the first formula reads E[X]=∫0∞(1−F(t)) dtE[X] = \int_0^\infty (1 - F(t)) \, dt. For X∼Exp⁡(λ)X \sim \operatorname{Exp}(\lambda), the formula gives E[X]=∫0∞e−λt dt=1/λE[X] = \int_0^\infty e^{-\lambda t} \, dt = 1/\lambda.

Reading path

To understand this page from the ground up, read these 46 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. Random variable definition
  42. Cumulative distribution function definition
  43. Discrete random variable definition
  44. Random variable with a density definition
  45. Expected value definition
  46. Tail sum formula for expectation statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Expected_value, Tail-sum formula
About this page
Tag0Y6QNF1J49
URIm2://0Y6QNF1J49
Aliasestail-sum-formula, layer-cake-formula-for-expectation
Kindstatement, proposition
WritingW0 stub
MSC60A05
StatusThis copy of the site has no build data. Run m2 status 0Y6QNF1J49.
UsesExpected value, Cumulative distribution function

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