Math 2.0

Probability theory and stochastic processes Definition

Bernoulli process

Builds on Independent and identically distributed random variablesBernoulli distributionStochastic process

Definition

Let p∈[0,1]p \in [0, 1]. A Bernoulli process with parameter pp is a stochastic process (Xn)n≥1(X_n)_{n \ge 1} of i.i.d. random variables with the Bernoulli distribution Bernoulli⁡(p)\operatorname{Bernoulli}(p): P(Xn=1)=pandP(Xn=0)=1−pfor each n≥1.P(X_n = 1) = p \quad \text{and} \quad P(X_n = 0) = 1 - p \quad \text{for each } n \ge 1.

Remarks

One calls Xn=1X_n = 1 a success and Xn=0X_n = 0 a failure in trial nn. The number of successes in the first nn trials has the binomial distribution Bin⁡(n,p)\operatorname{Bin}(n, p). For 0<p≤10 < p \le 1, the number of failures before the first success has the geometric distribution with parameter pp. The times between successes are i.i.d. The Bernoulli process is a discrete-time analogue of the Poisson process.

Reading path

To understand this page from the ground up, read these 55 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. Bernoulli trial concept
  15. Ordered pair definition
  16. Union definition
  17. Sample space definition
  18. The rationals form a field statement
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. Cartesian product definition
  22. Subset definition
  23. The rationals form an ordered field statement
  24. Uniqueness quantifier definition
  25. Binary relation definition
  26. Universal set definition
  27. Order on the reals definition
  28. Power set definition
  29. Complement definition
  30. Function definition
  31. Interval definition
  32. Injective function definition
  33. Preimage of a set definition
  34. Sigma-algebra definition
  35. Countable set definition
  36. Event definition
  37. Borel sigma-algebra definition
  38. Probability axioms concept
  39. Probability measure definition
  40. Probability distribution definition
  41. Probability space definition
  42. Conditional probability definition
  43. Random variable definition
  44. Discrete random variable definition
  45. Law of a random variable definition
  46. Independence of two events definition
  47. Stochastic process concept
  48. Probability mass function definition
  49. Identically distributed random variables definition
  50. Mutual independence of events definition
  51. Independent sigma-algebras definition
  52. Bernoulli distribution definition
  53. Independence of random variables definition
  54. Independent and identically distributed random variables definition
  55. Bernoulli process definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Bernoulli_process, Definition
About this page
Tag02XPXR1FBK
URIm2://02XPXR1FBK
Kinddefinition
WritingW0 stub
MSC60G50
DefinesBernoulli process
UsesIndependent and identically distributed random variables, Bernoulli distribution, Stochastic process

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