Math 2.0

Probability theory and stochastic processes Proposition

Monotonicity of the cumulative distribution function

Builds on Cumulative distribution functionMonotonicity of probability

Statement

Let XX be a real random variable with cumulative distribution function FXF_X. Then FXF_X is nondecreasing: if x≤yx \le y, then FX(x)≤FX(y)F_X(x) \le F_X(y).

Remarks

The proof: {X≤x}⊆{X≤y}\{X \le x\} \subseteq \{X \le y\}, so the claim follows from the monotonicity of probability. More precisely, FX(y)−FX(x)=P(x<X≤y)≥0F_X(y) - F_X(x) = P(x < X \le y) \ge 0. A nondecreasing function has at most countably many points of discontinuity. So P(X=x)>0P(X = x) > 0 for at most countably many xx.

Reading path

To understand this page from the ground up, read these 42 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. Preimage of a set definition
  32. Sigma-algebra definition
  33. Event definition
  34. Borel sigma-algebra definition
  35. Probability axioms concept
  36. Probability measure definition
  37. Probability distribution definition
  38. Probability space definition
  39. Monotonicity of probability statement
  40. Random variable definition
  41. Cumulative distribution function definition
  42. Monotonicity of the cumulative distribution function statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cumulative_distribution_function, Properties
About this page
Tag0XD22PBMWA
URIm2://0XD22PBMWA
Kindstatement, proposition
WritingW0 stub
MSC60E05
StatusThis copy of the site has no build data. Run m2 status 0XD22PBMWA.
UsesCumulative distribution function, Monotonicity of probability

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