The Probability Handbook by McShane-Vaughn Mary
Author:McShane-Vaughn, Mary
Language: eng
Format: epub
Publisher: ASQ Quality Press
Published: 2016-03-09T00:00:00+00:00
4.5 The Hypergeometric Distribution
The hypergeometric distribution can be thought of as the binomial distribution’s more complicated cousin. Like the binomial, the hypergeometric distribution is used when counting the number of defective units in a sample, and is the basis of some ANSI Z1.4 sampling plans. Unlike the trials of the binomial, however, the trials of a hypergeometric are not independent. The probability of each trial is dependent on the outcomes of the trials before it. The difference between the binomial and hypergeometric distributions can be likened to the “socks in a drawer” problem described in Section 3.8.
In the binomial case, sampling from a small population is performed with replacement so that the probability of choosing a black sock remains constant, or in the case of sampling for quality, the population is considered to be so large relative to the sample size that the probability of success remains essentially constant, even without replacing samples.
In the hypergeometric case, we choose samples without replacement from a relatively small population, in which case the probability of a black sock changes from trial to trial. As we will see in Section 4.5.3, the hypergeometric distribution converges to a binomial when the ratio of the sample size n to the lot size N approaches zero. This ratio is called the sampling fraction.
The hypergeometric random variable X denotes the number of defectives (or any outcome of interest) found in a sample of size n from a lot size of N. For the hypergeometric distribution,
There are only two possible outcomes for each trial
The population size is denoted as N, and the population is finite
The sample size n is less than or equal to the population size N
There are D defectives (or any outcome of interest) in the population
There are (N – D) non-defectives in the population
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