Math 2.0

Probability theory and stochastic processes Definition

Hypergeometric distribution

Builds on Probability mass functionBinomial coefficientClassical definition of probability

Definition

Let NN, KK and nn be integers with N≥1N \ge 1, 0≤K≤N0 \le K \le N and 0≤n≤N0 \le n \le N. The hypergeometric distribution with parameters NN, KK and nn has the probability mass function p(k)=(Kk)(N−Kn−k)(Nn)for max⁡(0,n+K−N)≤k≤min⁡(n,K).p(k) = \frac{\binom{K}{k} \binom{N - K}{n - k}}{\binom{N}{n}} \quad \text{for } \max(0, n + K - N) \le k \le \min(n, K).

Remarks

A population has NN objects, and KK of them are successes. Draw nn objects without replacement, each subset of size nn equally likely. Then the number of successes in the sample has this distribution. The mean is nK/NnK/N. The variance is nKNN−KNN−nN−1n \frac{K}{N} \frac{N - K}{N} \frac{N - n}{N - 1} for N≥2N \ge 2. When NN is large compared with nn, the distribution is close to the binomial distribution with parameters nn and K/NK/N. The probability of exactly 22 aces in a hand of 55 cards is (42)(483)/(525)\binom{4}{2} \binom{48}{3} / \binom{52}{5}.

Reading path

To understand this page from the ground up, read these 50 pages in order. The foundations come first.

  1. Binomial coefficient definition
  2. Construction of the reals from Cauchy sequences concept
  3. Element 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. Power set definition
  28. Order on the reals definition
  29. Complement definition
  30. Function definition
  31. Interval definition
  32. Sigma-algebra definition
  33. Codomain definition
  34. Injective function definition
  35. Preimage of a set definition
  36. Event definition
  37. Surjective function definition
  38. Countable set definition
  39. Borel sigma-algebra definition
  40. Probability axioms concept
  41. Bijective function definition
  42. Probability measure definition
  43. Equinumerous sets definition
  44. Probability space definition
  45. Finite set definition
  46. Classical definition of probability definition
  47. Random variable definition
  48. Discrete random variable definition
  49. Probability mass function definition
  50. Hypergeometric distribution definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Hypergeometric_distribution, Probability mass function
About this page
Tag0SPBPW3K61
URIm2://0SPBPW3K61
Kinddefinition
WritingW1 stated
MSC60E05
DefinesHypergeometric distribution
UsesProbability mass function, Binomial coefficient, Classical definition of probability

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