Math 2.0

Probability theory and stochastic processes Theorem

Linearity of expectation

Builds on Expected value

Statement

Let XX and YY be integrable real random variables on one probability space, and let a,b∈ℝa, b \in \mathbb{R}. Then aX+bYaX + bY is integrable and E[aX+bY]=a E[X]+b E[Y].E[aX + bY] = a \, E[X] + b \, E[Y].

Remarks

The statement is the linearity of the Lebesgue integral. It holds for all integrable XX and YY, also when they are not independent. By induction, E[X1+⋯+Xn]=E[X1]+⋯+E[Xn]E[X_1 + \dots + X_n] = E[X_1] + \dots + E[X_n]. A standard use is the method of indicators. If NN counts how many of the events A1,…,AnA_1, \dots, A_n occur, then N=∑i1AiN = \sum_i 1_{A_i} and E[N]=∑iP(Ai)E[N] = \sum_i P(A_i). For nonnegative random variables, additivity E[X+Y]=E[X]+E[Y]E[X + Y] = E[X] + E[Y] holds without integrability, in [0,∞][0, \infty].

Reading path

To understand this page from the ground up, read these 44 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
  44. Linearity of expectation statement

Where this leads

63 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Expected_value, Properties
About this page
Tag0P7PTC8DDE
URIm2://0P7PTC8DDE
Aliasesexpectation-is-linear
Kindstatement, theorem
WritingW0 stub
MSC60A05
StatusThis copy of the site has no build data. Run m2 status 0P7PTC8DDE.
UsesExpected value

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