Math 2.0

Probability theory and stochastic processes Theorem

The cumulative distribution function determines the distribution

Builds on Cumulative distribution functionProbability distributionBorel sigma-algebra

Statement

Let μ\mu and ν\nu be probability distributions on ℝ\mathbb{R} with its Borel σ-algebra. If μ((−∞,x])=ν((−∞,x])\mu((-\infty, x]) = \nu((-\infty, x]) for each x∈ℝx \in \mathbb{R}, then μ(B)=ν(B)\mu(B) = \nu(B) for each Borel set B⊆ℝB \subseteq \mathbb{R}.

Remarks

In other words, two real random variables with the same cumulative distribution function are identically distributed. The half-lines (−∞,x](-\infty, x] are closed under finite intersections and generate the Borel σ-algebra. The proof uses Dynkin's π-λ theorem to pass from these half-lines to all Borel sets. So to describe a real distribution, it is enough to give its cumulative distribution function.

Reading path

To understand this page from the ground up, read these 41 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. Random variable definition
  40. Cumulative distribution function definition
  41. The cumulative distribution function determines the distribution statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cumulative_distribution_function, Properties
About this page
Tag0JEVZAFDY6
URIm2://0JEVZAFDY6
Kindstatement, theorem
WritingW0 stub
MSC60E05
StatusThis copy of the site has no build data. Run m2 status 0JEVZAFDY6.
UsesCumulative distribution function, Probability distribution, Borel sigma-algebra

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.