numpy.random.Generator.hypergeometric#

method

random.Generator.hypergeometric(ngood, nbad, nsample, size=None)#

Draw samples from a Hypergeometric distribution.

Samples are drawn from a hypergeometric distribution with specified parameters, ngood (ways to make a good selection), nbad (ways to make a bad selection), and nsample (number of items sampled, which is less than or equal to the sum ngood + nbad).

Parameters:

ngoodint or array_like of ints: Number of ways to make a good selection. Must be nonnegative and less than 10**9.
nbadint or array_like of ints: Number of ways to make a bad selection. Must be nonnegative and less than 10**9.
nsampleint or array_like of ints: Number of items sampled. Must be nonnegative and less than ngood + nbad.
sizeint or tuple of ints, optional: Output shape. If the given shape is, e.g., (m, n, k), then m * n * k samples are drawn. If size is None (default), a single value is returned if ngood, nbad, and nsample are all scalars. Otherwise, np.broadcast(ngood, nbad, nsample).size samples are drawn.

Returns:

outndarray or scalar: Drawn samples from the parameterized hypergeometric distribution. Each sample is the number of good items within a randomly selected subset of size nsample taken from a set of ngood good items and nbad bad items.

See also

multivariate_hypergeometric: Draw samples from the multivariate hypergeometric distribution.
scipy.stats.hypergeom: probability density function, distribution or cumulative density function, etc.

Notes

The probability mass function (PMF) for the Hypergeometric distribution is

\[P(x) = \frac{\binom{g}{x}\binom{b}{n-x}}{\binom{g+b}{n}},\]

where \(0 \le x \le n\) and \(n-b \le x \le g\)

for P(x) the probability of x good results in the drawn sample, g = ngood, b = nbad, and n = nsample.

Consider an urn with black and white marbles in it, ngood of them are black and nbad are white. If you draw nsample balls without replacement, then the hypergeometric distribution describes the distribution of black balls in the drawn sample.

Note that this distribution is very similar to the binomial distribution, except that in this case, samples are drawn without replacement, whereas in the Binomial case samples are drawn with replacement (or the sample space is infinite). As the sample space becomes large, this distribution approaches the binomial.

The arguments ngood and nbad each must be less than 10**9. For extremely large arguments, the algorithm that is used to compute the samples [4] breaks down because of loss of precision in floating point calculations. For such large values, if nsample is not also large, the distribution can be approximated with the binomial distribution, binomial(n=nsample, p=ngood/(ngood + nbad)).

References

[1]

Lentner, Marvin, “Elementary Applied Statistics”, Bogden and Quigley, 1972.

[2]

Weisstein, Eric W. “Hypergeometric Distribution.” From MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/HypergeometricDistribution.html

[3]

Wikipedia, “Hypergeometric distribution”, https://en.wikipedia.org/wiki/Hypergeometric_distribution

[4]

Stadlober, Ernst, “The ratio of uniforms approach for generating discrete random variates”, Journal of Computational and Applied Mathematics, 31, pp. 181-189 (1990).

Examples

Draw samples from the distribution:

>>> rng = np.random.default_rng()
>>> ngood, nbad, nsamp = 100, 2, 10
# number of good, number of bad, and number of samples
>>> s = rng.hypergeometric(ngood, nbad, nsamp, 1000)
>>> from matplotlib.pyplot import hist
>>> hist(s)
#   note that it is very unlikely to grab both bad items

Suppose you have an urn with 15 white and 15 black marbles. If you pull 15 marbles at random, how likely is it that 12 or more of them are one color?

>>> s = rng.hypergeometric(15, 15, 15, 100000)
>>> sum(s>=12)/100000. + sum(s<=3)/100000.
#   answer = 0.003 ... pretty unlikely!