Math 2.0

Probability theory and stochastic processes Definition

Expected value

Builds on Random variableProbability spaceDiscrete random variableRandom variable with a density

Definition

Let XX be a real random variable on a probability space (Ω,ℱ,P)(\Omega, \mathcal{F}, P). If X≥0X \ge 0, the expected value of XX is the Lebesgue integral E[X]=∫ΩX dP∈[0,∞].E[X] = \int_\Omega X \, dP \in [0, \infty]. In general, write X=X+−X−X = X^+ - X^- with X+=max⁡(X,0)X^+ = \max(X, 0) and X−=max⁡(−X,0)X^- = \max(-X, 0). If E[∣X∣]<∞E[|X|] < \infty, then XX is integrable and E[X]=E[X+]−E[X−]E[X] = E[X^+] - E[X^-].

Remarks

For a discrete random variable with values x1,x2,…x_1, x_2, \dots, E[X]=∑ixi P(X=xi)E[X] = \sum_i x_i \, P(X = x_i) when the series converges absolutely. For a random variable with density ff, E[X]=∫−∞∞xf(x) dxE[X] = \int_{-\infty}^{\infty} x f(x) \, dx when ∫∣x∣f(x) dx<∞\int |x| f(x) \, dx < \infty. Other notations are 𝔼[X]\mathbb{E}[X], 𝔼X\mathbb{E}X and μX\mu_X. The expected value depends only on the law of XX. If E[X+]=∞E[X^+] = \infty and E[X−]<∞E[X^-] < \infty, one sets E[X]=+∞E[X] = +\infty. If both are infinite, E[X]E[X] is not defined.

Reading path

To understand this page from the ground up, read these 43 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 space definition
  40. Random variable definition
  41. Discrete random variable definition
  42. Random variable with a density definition
  43. Expected value definition

Where this leads

177 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Expected_value, Arbitrary real-valued random variables
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