Math 2.0

Probability theory and stochastic processes Inequalities Theorem

Markov's inequality

Builds on Expected valueMonotonicity of probability

Statement

Let XX be a random variable with values in [0,∞)[0, \infty) and let a>0a > 0. Then P(X≥a)≤E[X]a.P(X \ge a) \le \frac{E[X]}{a}.

Remarks

Proof: a 1{X≥a}≤Xa \, 1_{\{X \ge a\}} \le X pointwise. Take expectations: a P(X≥a)≤E[X]a \, P(X \ge a) \le E[X]. More generally, for each nondecreasing g ⁣:[0,∞)→[0,∞)g \colon [0, \infty) \to [0, \infty) with g(a)>0g(a) > 0, P(X≥a)≤E[g(X)]/g(a)P(X \ge a) \le E[g(X)]/g(a). Applied to (X−E[X])2(X - E[X])^2, the inequality gives Chebyshev's inequality. Applied to etXe^{tX}, it gives the Chernoff bound.

History

The inequality is named after Andrey Markov. It appears earlier in the work of Pafnuty Chebyshev, the teacher of Markov.

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. Monotonicity of probability statement
  41. Random variable definition
  42. Discrete random variable definition
  43. Random variable with a density definition
  44. Expected value definition
  45. Markov's inequality statement

Where this leads

18 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Markov's_inequality, Statement
About this page
Tag05PWWHYBX5
URIm2://05PWWHYBX5
Kindstatement, theorem
WritingW1 stated
MSC60E15
StatusThis copy of the site has no build data. Run m2 status 05PWWHYBX5.
UsesExpected value, Monotonicity of probability
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