Statistics
Textbooks
Boundless Statistics
Probability and Variability
Other Random Variables
Statistics Textbooks Boundless Statistics Probability and Variability Other Random Variables
Statistics Textbooks Boundless Statistics Probability and Variability
Statistics Textbooks Boundless Statistics
Statistics Textbooks
Statistics
Concept Version 9
Created by Boundless

The Hypergeometric Random Variable

A hypergeometric random variable is a discrete random variable characterized by a fixed number of trials with differing probabilities of success.

Learning Objective

  • Contrast hypergeometric distribution and binomial distribution


Key Points

    • The hypergeometric distribution applies to sampling without replacement from a finite population whose elements can be classified into two mutually exclusive categories like pass/fail, male/female or employed/unemployed.
    • As random selections are made from the population, each subsequent draw decreases the population causing the probability of success to change with each draw.
    • It is in contrast to the binomial distribution, which describes the probability of kkk successes in nnn draws with replacement.

Terms

  • binomial distribution

    the discrete probability distribution of the number of successes in a sequence of nnn independent yes/no experiments, each of which yields success with probability ppp

  • hypergeometric distribution

    a discrete probability distribution that describes the number of successes in a sequence of nnn draws from a finite population without replacement

  • Bernoulli Trial

    an experiment whose outcome is random and can be either of two possible outcomes, "success" or "failure"


Full Text

The hypergeometric distribution is a discrete probability distribution that describes the probability of kkk successes in nnn draws without replacement from a finite population of size NNN containing a maximum of KKK successes. This is in contrast to the binomial distribution, which describes the probability of kkk successes in nnn draws with replacement.

The hypergeometric distribution applies to sampling without replacement from a finite population whose elements can be classified into two mutually exclusive categories like pass/fail, male/female or employed/unemployed. As random selections are made from the population, each subsequent draw decreases the population causing the probability of success to change with each draw. The following conditions characterize the hypergeometric distribution:

  • The result of each draw can be classified into one or two categories.
  • The probability of a success changes on each draw.

A random variable follows the hypergeometric distribution if its probability mass function is given by:

P(X=k)=(Kk)(N−Kn−k)(Nn)\displaystyle P(X=k) = \frac{{{K}\choose{k}}{{N-K}\choose{n-k}}}{{{N}\choose{n}}}P(X=k)=​(​n​N​​)​​(​k​K​​)(​n−k​N−K​​)​​

Where:

  • NNN is the population size,
  • KKK is the number of success states in the population,
  • nnn is the number of draws,
  • kkk is the number of successes, and
  • (ab)\displaystyle {{a}\choose{b}}(​b​a​​) is a binomial coefficient.

A hypergeometric probability distribution is the outcome resulting from a hypergeometric experiment. The characteristics of a hypergeometric experiment are:

  1. You take samples from 2 groups.
  2. You are concerned with a group of interest, called the first group.
  3. You sample without replacement from the combined groups. For example, you want to choose a softball team from a combined group of 11 men and 13 women. The team consists of 10 players.
  4. Each pick is not independent, since sampling is without replacement. In the softball example, the probability of picking a women first is 1324\frac{13}{24}​24​​13​​. The probability of picking a man second is 1123\frac{11}{23}​23​​11​​, if a woman was picked first. It is 1023\frac{10}{23}​23​​10​​ if a man was picked first. The probability of the second pick depends on what happened in the first pick.
  5. You are not dealing with Bernoulli Trials.
[ edit ]
Edit this content
Prev Concept
The Poisson Random Variable
Continuous Probability Distributions
Next Concept
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.