Math 2.0

Probability theory and stochastic processes Proof

Proof of Chebyshev's inequality

Builds on Markov's inequalityVarianceStandard deviation

Proof

We apply Markov's inequality to the squared deviation.

  1. Let XX be a real random variable with finite mean μ\mu and finite variance σ2\sigma^2. Let a>0a > 0.
  2. Let Y=(X−μ)2Y = (X - \mu)^2. Then YY takes values in [0,∞)[0, \infty), and E[Y]=σ2E[Y] = \sigma^2 by the definition of the variance.
  3. For u≥0u \ge 0 and a>0a > 0, u≥au \ge a holds if and only if u2≥a2u^2 \ge a^2, since squaring is increasing on [0,∞)[0, \infty).
  4. Apply step 3 to u=∣X−μ∣u = \abs{X - \mu}. This gives the equality of events {∣X−μ∣≥a}={Y≥a2}\set{\abs{X - \mu} \ge a} = \set{Y \ge a^2}.
  5. By Markov's inequality for YY and the level a2>0a^2 > 0, P(Y≥a2)≤E[Y]/a2=σ2/a2P(Y \ge a^2) \le E[Y]/a^2 = \sigma^2/a^2.
  6. By steps 4 and 5, P(∣X−μ∣≥a)≤σ2/a2P(\abs{X - \mu} \ge a) \le \sigma^2/a^2.
  7. Now let σ>0\sigma > 0 be the standard deviation and let k>0k > 0. Then a=kσ>0a = k\sigma > 0.
  8. Step 6 with this aa gives P(∣X−μ∣≥kσ)≤σ2/(k2σ2)=1/k2P(\abs{X - \mu} \ge k\sigma) \le \sigma^2/(k^2\sigma^2) = 1/k^2.

Idea

A large deviation ∣X−μ∣≥a\abs{X - \mu} \ge a is the same event as a large squared deviation (X−μ)2≥a2(X - \mu)^2 \ge a^2. Markov's inequality bounds the second event by the mean of the squared deviation, which is the variance.

Reading path

To understand this page from the ground up, read these 74 pages in order. The foundations come first. The path takes the proofs Proof of the approximation property of the supremum, Proof of the probability of the complement, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the existence of square roots of nonnegative reals, Proof of the monotonicity of probability, Proof that the reals are Archimedean from the completeness axiom, Proof of Markov's inequality.

  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. Archimedean ordered field definition
  13. Disjunction definition
  14. Implication definition
  15. Predicate definition
  16. The reals form a field statement
  17. Supremum definition
  18. Ordered pair definition
  19. Union definition
  20. Sample space definition
  21. The rationals form a field statement
  22. Existential quantifier definition
  23. Universal quantifier definition
  24. Completeness axiom statement
  25. Cartesian product definition
  26. Subset definition
  27. The rationals form an ordered field statement
  28. Uniqueness quantifier definition
  29. Binary relation definition
  30. Universal set definition
  31. Order on the reals definition
  32. Power set definition
  33. Antisymmetric relation definition
  34. Reflexive relation definition
  35. Transitive relation definition
  36. Complement definition
  37. Function definition
  38. Interval definition
  39. Preorder definition
  40. Sigma-algebra definition
  41. Indicator function definition
  42. Preimage of a set definition
  43. Injective function definition
  44. Partial order definition
  45. Event definition
  46. Countable set definition
  47. Borel sigma-algebra definition
  48. Total order definition
  49. Probability axioms concept
  50. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  51. Probability measure definition
  52. Probability space definition
  53. The reals form an ordered field statement
  54. Random variable definition
  55. Proof of the approximation property of the supremum proof
  56. Proof of the existence of square roots of nonnegative reals proof
  57. Discrete random variable definition
  58. Random variable with a density definition
  59. Existence of square roots of nonnegative reals statement
  60. Approximation property of the supremum statement
  61. Expected value definition
  62. Square root definition
  63. Proof that the reals are Archimedean from the completeness axiom proof
  64. Linearity of expectation statement
  65. Variance definition
  66. The reals are Archimedean statement
  67. Proof of Markov's inequality proof
  68. Standard deviation definition
  69. Proof of the probability of the complement proof
  70. Probability of the complement statement
  71. Proof of the monotonicity of probability proof
  72. Monotonicity of probability statement
  73. Markov's inequality statement
  74. Proof of Chebyshev's inequality proof

Where this leads

19 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Chebyshev's_inequality, Proof
About this page
TagD5T6KJSR02
URIm2://D5T6KJSR02
Kindproof
WritingW0 stub
MSC60E15
ProvesChebyshev's inequality
UsesMarkov's inequality, Variance, Standard deviation
Changed4 November 2026

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