Hypergeometric Distribution

The hypergeometric distribution models the number of successes in a fixed number of draws from a finite population without replacement. It is also known as the discrete hypergeometric distribution. This distribution is used when sampling without replacement from a population consisting of two types of objects.

The hypergeometric distribution has three parameters: the number of tagged objects $K$, the number of untagged objects $N-K$, and the number of selections $n$. The probability mass function (PMF) is given by:

$$P(X=k)=\frac{(\genfrac{}{}{0ex}{}{K}{k})(\genfrac{}{}{0ex}{}{N-K}{n-k})}{(\genfrac{}{}{0ex}{}{N}{n})}$$

The cumulative distribution function (CDF) is:

$$F(k;N,K,n)=\sum _{i=0}^k\frac{(\genfrac{}{}{0ex}{}{K}{i})(\genfrac{}{}{0ex}{}{N-K}{n-i})}{(\genfrac{}{}{0ex}{}{N}{n})}$$

The domain of the hypergeometric distribution is $k\in \{0,1,2,\dots ,min(K,n)\}$. The parameters must satisfy $0\le K\le N$ and $0\le n\le N$.

The hypergeometric distribution is widely used in various fields due to its applicability to sampling without replacement. Common applications include:

• In quality control, the number of defective products out of a sample of fixed size follows a hypergeometric distribution.

• In biology, the size of an animal population can be estimated by first tagging a known number of animals, and a short time later recapturing a fixed number of animals. The number of tagged animals in the recaptured samples follows a hypergeometric distribution.

The hypergeometric distribution has several important statistical properties:

The hypergeometric distribution is closely related to several other distributions:

• The Binomial Distribution models the number of successes in a fixed number of Bernoulli trials with replacement.

• The Negative Binomial Distribution models the number of failures before the $n$th success in a series of Bernoulli trials.

The hypergeometric distribution is implemented by the HypergeometricDistribution class. It has one constructor that takes three arguments. The first argument is an integer that specifies the number of items in the untagged population. The second argument specifies the number of items in the tagged population. The third parameter specifies the number of samples drawn from the total population. The following constructs a hypergeometric distribution for 20 tagged items, 10 untagged items, and 15 samples:

var hyper = new HypergeometricDistribution(20, 10, 15);

Dim hyper = New HypergeometricDistribution(20, 10, 15)

No code example is currently available or this language may not be supported.

let hyper = HypergeometricDistribution(20, 10, 15)

The HypergeometricDistribution class has three specific properties: TaggedPopulation returns the number of items in the tagged population. UntaggedPopulation returns the number of items in the untagged population. NumberOfSamples returns the number of samples drawn from the entire population.

HypergeometricDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. The first argument is the random number generator. The remaining 3 parameters correspond to the parameters of the constructor.

var random = new Pcg32();
int sample = HypergeometricDistribution.Sample(random, 20, 10, 15);

Dim random = New Pcg32()
Dim sample = HypergeometricDistribution.Sample(random, 20, 10, 15)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample = HypergeometricDistribution.Sample(random, 20, 10, 15)

The above example uses the Pcg32 class to generate uniform random numbers.

For details of the properties and methods common to all discrete probability distribution classes, see the topic on Discrete Probability Distributions.

• "Introduction to Probability Models" by Sheldon M. Ross.

• "Probability and Statistics" by Morris H. DeGroot and Mark J. Schervish.

Other Resources