Math 2.0

Probability theory and stochastic processes Definition

Geometric distribution

Builds on Probability mass functionBernoulli trial

Definition

Let p∈(0,1]p \in (0, 1]. The geometric distribution with parameter pp is the distribution on {0,1,2,… }\{0, 1, 2, \dots\} with the probability mass function p(k)=(1−p)k pfor k=0,1,2,…p(k) = (1 - p)^k \, p \quad \text{for } k = 0, 1, 2, \dots

Remarks

It is the distribution of the number of failures before the first success in independent Bernoulli trials with success probability pp. The other common convention counts the trials up to and including the first success. That number Y=X+1Y = X + 1 has P(Y=k)=(1−p)k−1pP(Y = k) = (1 - p)^{k-1} p for k≥1k \ge 1. For XX as in the definition, P(X≥k)=(1−p)kP(X \ge k) = (1 - p)^k, the mean is (1−p)/p(1 - p)/p and the variance is (1−p)/p2(1 - p)/p^2. The geometric distribution is the only distribution on {0,1,2,… }\{0, 1, 2, \dots\} with the memoryless property.

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. 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 space definition
  41. Random variable definition
  42. Discrete random variable definition
  43. Probability mass function definition
  44. Geometric distribution definition

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Geometric_distribution, Definition
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WritingW1 stated
MSC60E05
DefinesGeometric distribution
UsesProbability mass function, Bernoulli trial

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