Math 2.0

Probability theory and stochastic processes Definition

Covariance

Builds on Expected valueVariance

Definition

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. The covariance of XX and YY is Cov⁡(X,Y)=E[(X−E[X]) (Y−E[Y])].\operatorname{Cov}(X, Y) = E\left[(X - E[X]) \, (Y - E[Y])\right]. XX and YY are uncorrelated if Cov⁡(X,Y)=0\operatorname{Cov}(X, Y) = 0.

Remarks

An equivalent formula is Cov⁡(X,Y)=E[XY]−E[X] E[Y]\operatorname{Cov}(X, Y) = E[XY] - E[X] \, E[Y]. The covariance of XX with itself is the variance: Cov⁡(X,X)=Var⁡(X)\operatorname{Cov}(X, X) = \operatorname{Var}(X). The covariance is symmetric and bilinear: Cov⁡(aX+bY,Z)=aCov⁡(X,Z)+bCov⁡(Y,Z)\operatorname{Cov}(aX + bY, Z) = a \operatorname{Cov}(X, Z) + b \operatorname{Cov}(Y, Z). It does not change when one adds constants to XX or YY. The Cauchy–Schwarz inequality gives ∣Cov⁡(X,Y)∣≤σXσY|\operatorname{Cov}(X, Y)| \le \sigma_X \sigma_Y.

Reading path

To understand this page from the ground up, read these 45 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

Where this leads

65 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Covariance, Definition
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