Math 2.0

Probability theory and stochastic processes Proof

Proof of the monotonicity of the cumulative distribution function

Builds on Cumulative distribution functionRandom variableBorel sigma-algebraClosed setOpen setMonotonicity of probability

Proof

  1. Let XX be a real random variable on a probability space (Ω,ℱ,P)(\Omega, \mathcal{F}, P), with cumulative distribution function FX(t)=P(X≤t)F_X(t) = P(X \le t).
  2. Let t∈ℝt \in \R. The complement of (−∞,t](-\infty, t] is (t,∞)(t, \infty), which is open. So (−∞,t](-\infty, t] is closed.
  3. The Borel σ-algebra contains the open set (t,∞)(t, \infty) and is closed under complements. So it contains (−∞,t](-\infty, t].
  4. Since XX is a random variable, {X≤t}=X−1((−∞,t])\set{X \le t} = X^{-1}((-\infty, t]) is in ℱ\mathcal{F}.
  5. Let x≤yx \le y be real numbers. If ω∈Ω\omega \in \Omega satisfies X(ω)≤xX(\omega) \le x, then X(ω)≤yX(\omega) \le y. So {X≤x}⊆{X≤y}\set{X \le x} \subseteq \set{X \le y}.
  6. By step 4 both sets are events. By monotonicity of probability, P(X≤x)≤P(X≤y)P(X \le x) \le P(X \le y), that is, FX(x)≤FX(y)F_X(x) \le F_X(y).

Idea

Raising the threshold from xx to yy can only enlarge the event {X≤x}\set{X \le x}. A larger event has at least the same probability.

Reading path

To understand this page from the ground up, read these 73 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 monotonicity of probability, Proof that the reals are Archimedean from the completeness axiom.

  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. Conjunction definition
  14. Disjunction definition
  15. Implication definition
  16. Predicate definition
  17. The reals form a field statement
  18. Supremum definition
  19. Empty set object
  20. Family of sets definition
  21. Ordered pair definition
  22. Intersection definition
  23. Union definition
  24. Sample space definition
  25. The rationals form a field statement
  26. Existential quantifier definition
  27. Universal quantifier definition
  28. Completeness axiom statement
  29. Cartesian product definition
  30. Subset definition
  31. Union of a family of sets definition
  32. The rationals form an ordered field statement
  33. Uniqueness quantifier definition
  34. Binary relation definition
  35. Universal set definition
  36. Power set definition
  37. Order on the reals definition
  38. Topological space definition
  39. Antisymmetric relation definition
  40. Reflexive relation definition
  41. Transitive relation definition
  42. Complement definition
  43. Function definition
  44. Interval definition
  45. Preorder definition
  46. Sigma-algebra definition
  47. Preimage of a set definition
  48. Metric space definition
  49. Partial order definition
  50. Event definition
  51. Open ball definition
  52. Borel sigma-algebra definition
  53. Total order definition
  54. Probability axioms concept
  55. Metric topology definition
  56. Probability measure definition
  57. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  58. Open set definition
  59. Probability distribution definition
  60. Probability space definition
  61. Closed set definition
  62. The reals form an ordered field statement
  63. Random variable definition
  64. Proof of the approximation property of the supremum proof
  65. Cumulative distribution function definition
  66. Approximation property of the supremum statement
  67. Proof that the reals are Archimedean from the completeness axiom proof
  68. The reals are Archimedean statement
  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. Proof of the monotonicity of the cumulative distribution function proof

Where this leads

3 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Cumulative_Distribution_Function_is_Increasing
About this page
Tag2TGSAHD1K1
URIm2://2TGSAHD1K1
Kindproof
WritingW0 stub
MSC60E05
ProvesMonotonicity of the cumulative distribution function
UsesCumulative distribution function, Random variable, Borel sigma-algebra, Closed set, Open set, Monotonicity of probability
Changed4 November 2026

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