Math 2.0

Probability theory and stochastic processes Definition

Convolution of probability distributions

Builds on Probability distributionIndependence of two random variablesProbability mass functionProbability density function

Definition

Let μ\mu and ν\nu be probability distributions on ℝ\mathbb{R}. Their convolution is the probability distribution μ∗ν\mu * \nu on ℝ\mathbb{R} with (μ∗ν)(B)=∫ℝν(B−x) μ(dx)(\mu * \nu)(B) = \int_{\mathbb{R}} \nu(B - x) \, \mu(dx) for each Borel set B⊆ℝB \subseteq \mathbb{R}, where B−x={b−x:b∈B}B - x = \{b - x : b \in B\}.

Remarks

If X∼μX \sim \mu and Y∼νY \sim \nu are independent, then X+Y∼μ∗νX + Y \sim \mu * \nu. For distributions on the integers with mass functions pp and qq, μ∗ν\mu * \nu has the mass function (p∗q)(n)=∑kp(k) q(n−k)(p * q)(n) = \sum_{k} p(k) \, q(n - k). If μ\mu and ν\nu have densities ff and gg, then μ∗ν\mu * \nu has the density (f∗g)(x)=∫f(y) g(x−y) dy(f * g)(x) = \int f(y) \, g(x - y) \, dy. The convolution is commutative and associative. Its characteristic function is the product: φμ∗ν=φμφν\varphi_{\mu * \nu} = \varphi_\mu \varphi_\nu.

Reading path

To understand this page from the ground up, read these 50 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. Injective function definition
  32. Preimage of a set definition
  33. Sigma-algebra definition
  34. Countable set 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. Conditional probability definition
  42. Random variable definition
  43. Cumulative distribution function definition
  44. Discrete random variable definition
  45. Independence of two events definition
  46. Random variable with a density definition
  47. Probability mass function definition
  48. Independence of two random variables definition
  49. Probability density function definition
  50. Convolution of probability distributions definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Convolution_of_probability_distributions, Introduction
About this page
Tag07CHJ5YX28
URIm2://07CHJ5YX28
Kinddefinition
WritingW0 stub
MSC60E05
DefinesConvolution of probability distributions
UsesProbability distribution, Independence of two random variables, Probability mass function, Probability density function

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