Math 2.0

Probability theory and stochastic processes Definition

Poisson distribution

Builds on Probability mass functionFactorial

Definition

Let λ>0\lambda > 0. The Poisson distribution with parameter λ\lambda is the distribution on {0,1,2,… }\{0, 1, 2, \dots\} with the probability mass function p(k)=λke−λk!for k=0,1,2,…p(k) = \frac{\lambda^k e^{-\lambda}}{k!} \quad \text{for } k = 0, 1, 2, \dots Here k!k! is the factorial of kk. One writes X∼Poisson⁡(λ)X \sim \operatorname{Poisson}(\lambda).

Remarks

The values add up to 11 because ∑k≥0λk/k!=eλ\sum_{k \ge 0} \lambda^k / k! = e^{\lambda}. The mean and the variance are both λ\lambda. It is the limit of the binomial distribution with parameters nn and λ/n\lambda / n as n→∞n \to \infty. This is the Poisson limit theorem. Siméon Denis Poisson introduced the distribution in 1837.

Reading path

To understand this page from the ground up, read these 44 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 space definition
  41. Random variable definition
  42. Discrete random variable definition
  43. Probability mass function definition
  44. Poisson distribution definition

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Poisson_distribution, Probability mass function
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DefinesPoisson distribution
UsesProbability mass function, Factorial

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