Math 2.0

Probability theory and stochastic processes Definition

Quantile function

Builds on Cumulative distribution functionInfimum

Definition

Let F ⁣:ℝ→[0,1]F \colon \mathbb{R} \to [0, 1] be the cumulative distribution function of a real random variable XX. The quantile function of XX is the function Q ⁣:(0,1)→ℝQ \colon (0, 1) \to \mathbb{R} with Q(p)=inf⁡{x∈ℝ:F(x)≥p}.Q(p) = \inf \{x \in \mathbb{R} : F(x) \ge p\}. Here inf⁡\inf is the infimum. The number Q(p)Q(p) is the pp-quantile of XX.

Remarks

The set in the definition is nonempty and bounded below for 0<p<10 < p < 1, and the infimum is a minimum because FF is right-continuous. So F(Q(p))≥pF(Q(p)) \ge p, and Q(p)≤xQ(p) \le x if and only if p≤F(x)p \le F(x). If FF is continuous and strictly increasing, then Q=F−1Q = F^{-1}. If UU is uniform on (0,1)(0, 1), then Q(U)Q(U) has the cumulative distribution function FF. This is inverse transform sampling. Q(1/2)Q(1/2) is a median of XX, and Q(1/4)Q(1/4) and Q(3/4)Q(3/4) are the lower and upper quartiles.

Reading path

To understand this page from the ground up, read these 43 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 bounded above definition
  10. Set definition
  11. Ring of integers statement
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Infimum definition
  16. Ordered pair definition
  17. Union definition
  18. Sample space definition
  19. The rationals form a field statement
  20. Existential quantifier definition
  21. Universal quantifier definition
  22. Cartesian product definition
  23. Subset definition
  24. The rationals form an ordered field statement
  25. Uniqueness quantifier definition
  26. Binary relation definition
  27. Universal set definition
  28. Order on the reals definition
  29. Power set definition
  30. Complement definition
  31. Function definition
  32. Interval definition
  33. Preimage of a set definition
  34. Sigma-algebra definition
  35. Event definition
  36. Borel sigma-algebra definition
  37. Probability axioms concept
  38. Probability measure definition
  39. Probability distribution definition
  40. Probability space definition
  41. Random variable definition
  42. Cumulative distribution function definition
  43. Quantile function definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quantile_function, General distribution function
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WritingW1 stated
MSC60E05
DefinesQuantile function
UsesCumulative distribution function, Infimum

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