scipy.stats.hypergeom = <scipy.stats._discrete_distns.hypergeom_gen object at 0x2b45d2fe25d0>[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.

Discrete random variables are defined from a standard form and may require some shape parameters to complete its specification. Any optional keyword parameters can be passed to the methods of the RV object as given below:


x : array_like


q : array_like

lower or upper tail probability

M, n, N : array_like

shape parameters

loc : array_like, optional

location parameter (default=0)

size : int or tuple of ints, optional

shape of random variates (default computed from input arguments )

moments : str, optional

composed of letters [‘mvsk’] specifying which moments to compute where ‘m’ = mean, ‘v’ = variance, ‘s’ = (Fisher’s) skew and ‘k’ = (Fisher’s) kurtosis. (default=’mv’)

Alternatively, the object may be called (as a function) to fix the shape and

location parameters returning a “frozen” discrete RV object:

rv = hypergeom(M, n, N, loc=0)

  • Frozen RV object with the same methods but holding the given shape and location fixed.


The probability mass function is defined as:

pmf(k, M, n, N) = choose(n, k) * choose(M - n, N - k) / choose(M, N),
                               for max(0, N - (M-n)) <= k <= min(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 variates.
pmf(x, M, n, N, loc=0) Probability mass function.
logpmf(x, M, n, N, loc=0) Log of the probability mass function.
cdf(x, M, n, N, loc=0) Cumulative density function.
logcdf(x, M, n, N, loc=0) Log of the cumulative density function.
sf(x, M, n, N, loc=0) Survival function (1-cdf — sometimes more accurate).
logsf(x, 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, 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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