scipy.stats.hypergeom = <scipy.stats._discrete_distns.hypergeom_gen object at 0x2b2318ef6390>[source]

A hypergeometric discrete random variable.

The hypergeometric distribution models drawing objects from a bin. M is the total number of objects, n is total number of Type I objects. The random variate represents the number of Type I objects in N drawn without replacement from the total population.

As an instance of the rv_discrete class, hypergeom object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.


The symbols used to denote the shape parameters (M, n, and N) are not universally accepted. See the Examples for a clarification of the definitions used here.

The probability mass function is defined as,

\[p(k, M, n, N) = \frac{\binom{n}{k} \binom{M - n}{N - k}}{\binom{M}{N}}\]

for \(k \in [\max(0, N - M + n), \min(n, N)]\), where the binomial coefficients are defined as,

\[\binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.\]

The probability mass function above is defined in the “standardized” form. To shift distribution use the loc parameter. Specifically, hypergeom.pmf(k, M, n, N, loc) is identically equivalent to hypergeom.pmf(k - loc, M, n, N).


>>> from scipy.stats import hypergeom
>>> import matplotlib.pyplot as plt

Suppose we have a collection of 20 animals, of which 7 are dogs. Then if we want to know the probability of finding a given number of dogs if we choose at random 12 of the 20 animals, we can initialize a frozen distribution and plot the probability mass function:

>>> [M, n, N] = [20, 7, 12]
>>> rv = hypergeom(M, n, N)
>>> x = np.arange(0, n+1)
>>> pmf_dogs = rv.pmf(x)
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(x, pmf_dogs, 'bo')
>>> ax.vlines(x, 0, pmf_dogs, lw=2)
>>> ax.set_xlabel('# of dogs in our group of chosen animals')
>>> ax.set_ylabel('hypergeom PMF')

(Source code)


Instead of using a frozen distribution we can also use hypergeom methods directly. To for example obtain the cumulative distribution function, use:

>>> prb = hypergeom.cdf(x, M, n, N)

And to generate random numbers:

>>> R = hypergeom.rvs(M, n, N, size=10)


rvs(M, n, N, loc=0, size=1, random_state=None) Random variates.
pmf(k, M, n, N, loc=0) Probability mass function.
logpmf(k, M, n, N, loc=0) Log of the probability mass function.
cdf(k, M, n, N, loc=0) Cumulative distribution function.
logcdf(k, M, n, N, loc=0) Log of the cumulative distribution function.
sf(k, M, n, N, loc=0) Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).
logsf(k, M, n, N, loc=0) Log of the survival function.
ppf(q, M, n, N, loc=0) Percent point function (inverse of cdf — percentiles).
isf(q, M, n, N, loc=0) Inverse survival function (inverse of sf).
stats(M, n, N, loc=0, moments='mv') Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(M, n, N, loc=0) (Differential) entropy of the RV.
expect(func, args=(M, n, N), loc=0, lb=None, ub=None, conditional=False) Expected value of a function (of one argument) with respect to the distribution.
median(M, n, N, loc=0) Median of the distribution.
mean(M, n, N, loc=0) Mean of the distribution.
var(M, n, N, loc=0) Variance of the distribution.
std(M, n, N, loc=0) Standard deviation of the distribution.
interval(alpha, M, n, N, loc=0) Endpoints of the range that contains alpha percent of the distribution

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