Math 2.0

Probability theory and stochastic processes Definition

Probability density function

Builds on Random variable with a densityCumulative distribution function

Definition

Let XX be a real random variable on a probability space (Ω,ℱ,P)(\Omega, \mathcal{F}, P). A probability density function of XX is a Borel measurable function f ⁣:ℝ→[0,∞)f \colon \mathbb{R} \to [0, \infty) such that P(X∈B)=∫Bf(x) dxP(X \in B) = \int_B f(x) \, dx for each Borel set B⊆ℝB \subseteq \mathbb{R}. So XX has a probability density function if and only if XX is a random variable with a density.

Remarks

Each density satisfies ∫−∞∞f(x) dx=1\int_{-\infty}^{\infty} f(x) \, dx = 1. Two densities of XX are equal except on a set of Lebesgue measure zero. P(a≤X≤b)=∫abf(x) dxP(a \le X \le b) = \int_a^b f(x) \, dx for a≤ba \le b. If the cumulative distribution function FXF_X is differentiable at xx and ff is continuous at xx, then FX′(x)=f(x)F_X'(x) = f(x). A density can be larger than 11. The uniform distribution on [0,1/2][0, 1/2] has the density 22 on this interval.

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. Random variable definition
  40. Cumulative distribution function definition
  41. Random variable with a density definition
  42. Probability density function definition

Where this leads

110 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Probability_density_function, Formal definition
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MSC60E05
DefinesProbability density function
UsesRandom variable with a density, Cumulative distribution function

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