Math 2.0

Probability theory and stochastic processes Proposition

Limit of the cumulative distribution function at infinity

Builds on Cumulative distribution functionContinuity of probability measures

Statement

Let XX be a real random variable with cumulative distribution function FXF_X. Then lim⁡x→+∞FX(x)=1.\lim_{x \to +\infty} F_X(x) = 1.

Remarks

The proof: the events {X≤n}\{X \le n\} increase with nn and their union is Ω\Omega. So continuity from below gives FX(n)→1F_X(n) \to 1, and FXF_X is nondecreasing. In the same way, lim⁡x→−∞FX(x)=0\lim_{x \to -\infty} F_X(x) = 0, because the events {X≤−n}\{X \le -n\} decrease to ∅\varnothing. The proof uses that XX takes only real values. For a random variable with values in [−∞,+∞][-\infty, +\infty], the limit is 1−P(X=+∞)1 - P(X = +\infty).

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. Continuity of probability measures statement
  40. Random variable definition
  41. Cumulative distribution function definition
  42. Limit of the cumulative distribution function at infinity statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cumulative_distribution_function, Properties
About this page
Tag0RJ5TZH6G0
URIm2://0RJ5TZH6G0
Kindstatement, proposition
WritingW0 stub
MSC60E05
StatusThis copy of the site has no build data. Run m2 status 0RJ5TZH6G0.
UsesCumulative distribution function, Continuity of probability measures

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