Math 2.0

Probability theory and stochastic processes Inequalities Theorem

Chebyshev's inequality

Builds on Markov's inequalityVarianceStandard deviation

Statement

Let XX be a real random variable with finite mean μ\mu and finite variance σ2\sigma^2. Then for each a>0a > 0, P(∣X−μ∣≥a)≤σ2a2.P(|X - \mu| \ge a) \le \frac{\sigma^2}{a^2}. Equivalently, if σ>0\sigma > 0, then P(∣X−μ∣≥kσ)≤1/k2P(|X - \mu| \ge k\sigma) \le 1/k^2 for each k>0k > 0.

Remarks

Proof: apply Markov's inequality to the nonnegative variable (X−μ)2(X - \mu)^2 and the level a2a^2. For k=2k = 2, at least 3/43/4 of the mass is within two standard deviations of the mean, for every distribution with finite variance. The bound is sharp: some distributions reach equality for a given k≥1k \ge 1. Chebyshev's inequality gives a short proof of the weak law of large numbers for variables with finite variance.

History

The inequality is named after Pafnuty Chebyshev. Irénée-Jules Bienaymé formulated it first.

Reading path

To understand this page from the ground up, read these 53 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 bounded above definition
  10. Set definition
  11. Ring of integers statement
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Supremum definition
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. The rationals form a field statement
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Completeness axiom statement
  23. Cartesian product definition
  24. Subset definition
  25. The rationals form an ordered field statement
  26. Uniqueness quantifier definition
  27. Existence of square roots of nonnegative reals statement
  28. Binary relation definition
  29. Universal set definition
  30. Order on the reals definition
  31. Power set definition
  32. Square root definition
  33. Complement definition
  34. Function definition
  35. Interval definition
  36. Injective function definition
  37. Preimage of a set definition
  38. Sigma-algebra definition
  39. Countable set definition
  40. Event definition
  41. Borel sigma-algebra definition
  42. Probability axioms concept
  43. Probability measure definition
  44. Probability space definition
  45. Monotonicity of probability statement
  46. Random variable definition
  47. Discrete random variable definition
  48. Random variable with a density definition
  49. Expected value definition
  50. Markov's inequality statement
  51. Variance definition
  52. Standard deviation definition
  53. Chebyshev's inequality statement

Where this leads

11 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Chebyshev's_inequality, Probabilistic statement
About this page
Tag02HZ1BK3MW
URIm2://02HZ1BK3MW
Aliasesbienayme-chebyshev-inequality
Kindstatement, theorem
WritingW1 stated
MSC60E15
StatusThis copy of the site has no build data. Run m2 status 02HZ1BK3MW.
UsesMarkov's inequality, Variance, Standard deviation
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