Math 2.0

Probability theory and stochastic processes Definition

Poisson point process

Builds on Poisson distributionStochastic processExponential distributionIndependence of random variables

Definition

Let λ>0\lambda > 0. A homogeneous Poisson process with rate λ\lambda is a stochastic process (N(t))t≥0(N(t))_{t \ge 0} with values in {0,1,2,… }\{0, 1, 2, \dots\} such that:

  1. N(0)=0N(0) = 0.
  2. For all 0≤t0<t1<⋯<tn0 \le t_0 < t_1 < \dots < t_n, the increments N(t1)−N(t0),…,N(tn)−N(tn−1)N(t_1) - N(t_0), \dots, N(t_n) - N(t_{n-1}) are independent.
  3. For all 0≤s<t0 \le s < t, N(t)−N(s)N(t) - N(s) has the Poisson distribution with parameter λ(t−s)\lambda (t - s).

Remarks

N(t)N(t) counts the points, or events, in the interval (0,t](0, t]. One usually also asks that the paths t↦N(t)t \mapsto N(t) are right-continuous. The times between consecutive events are i.i.d. with the exponential distribution Exp⁡(λ)\operatorname{Exp}(\lambda). Conversely, i.i.d. Exp⁡(λ)\operatorname{Exp}(\lambda) gaps define a Poisson process with rate λ\lambda. Given N(t)=nN(t) = n, the nn event times in (0,t](0, t] have the law of nn independent uniform points in (0,t](0, t], put in increasing order. On a general space EE with a measure μ\mu, a Poisson point process has independent counts in disjoint sets. The count in a set BB has the law Poisson⁡(μ(B))\operatorname{Poisson}(\mu(B)) when μ(B)<∞\mu(B) < \infty.

Reading path

To understand this page from the ground up, read these 56 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Factorial definition
  4. Integers object
  5. Ordered field definition
  6. Proposition concept
  7. Random experiment concept
  8. Rational numbers object
  9. Real numbers object
  10. Set definition
  11. Ring of integers statement
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Ordered pair definition
  16. Union definition
  17. Sample space definition
  18. Existential quantifier definition
  19. The rationals form a field statement
  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. Cumulative distribution function definition
  45. Discrete random variable definition
  46. Independence of two events definition
  47. Stochastic process concept
  48. Random variable with a density definition
  49. Probability mass function definition
  50. Mutual independence of events definition
  51. Probability density function definition
  52. Poisson distribution definition
  53. Independent sigma-algebras definition
  54. Exponential distribution definition
  55. Independence of random variables definition
  56. Poisson point process definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Poisson_point_process, Interpreted as a counting process
About this page
Tag05TG0SE4HT
URIm2://05TG0SE4HT
Kinddefinition
WritingW1 stated
MSC60G55
DefinesPoisson point process
UsesPoisson distribution, Stochastic process, Exponential distribution, Independence of random variables

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

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

What links here

Used by