Math 2.0

Probability theory and stochastic processes Theorem

Cauchy–Schwarz inequality for expectations

Builds on Expected valueCovariance

Statement

Let XX and YY be real random variables on one probability space with E[X2]<∞E[X^2] < \infty and E[Y2]<∞E[Y^2] < \infty. Then XYXY is integrable and (E[XY])2≤E[X2] E[Y2].\left(E[XY]\right)^2 \le E[X^2] \, E[Y^2]. Equality holds if and only if aX+bY=0aX + bY = 0 almost surely for some reals aa and bb, not both 00.

Remarks

Proof: for each real ss, 0≤E[(X+sY)2]=E[X2]+2sE[XY]+s2E[Y2]0 \le E[(X + sY)^2] = E[X^2] + 2s E[XY] + s^2 E[Y^2]. A quadratic polynomial in ss that is never negative has a discriminant that is not positive. This is the Cauchy–Schwarz inequality for the inner product ⟨X,Y⟩=E[XY]\langle X, Y \rangle = E[XY] on the space L2L^2 of square-integrable random variables. Applied to X−E[X]X - E[X] and Y−E[Y]Y - E[Y], it gives Cov⁡(X,Y)2≤Var⁡(X)Var⁡(Y)\operatorname{Cov}(X, Y)^2 \le \operatorname{Var}(X) \operatorname{Var}(Y). So the correlation coefficient is in [−1,1][-1, 1].

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 space definition
  40. Random variable definition
  41. Discrete random variable definition
  42. Random variable with a density definition
  43. Expected value definition
  44. Variance definition
  45. Covariance definition
  46. Cauchy–Schwarz inequality for expectations statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Cauchy–Schwarz_inequality, Probability theory
About this page
Tag03PGHKAJG3
URIm2://03PGHKAJG3
Aliasescauchy-schwarz-inequality-for-random-variables
Kindstatement, theorem
WritingW0 stub
MSC60E15
StatusThis copy of the site has no build data. Run m2 status 03PGHKAJG3.
UsesExpected value, Covariance

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