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# scipy.stats.hypergeom

scipy.stats.hypergeom = <scipy.stats._discrete_distns.hypergeom_gen object>

  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.

  Methods

  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(confidence, M, n, N, loc=0) | Confidence interval with equal areas around the median.

  See also

  `nhypergeom`, `binom`, `nbinom`

  Notes

  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,

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

  This distribution uses routines from the Boost Math C++ library for the computation of the `pmf`, `cdf`, `sf` and `stats` methods. [1]

  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)`.

  References

  [1]

  The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/.

  Examples

      >>> import numpy as np
      >>> 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')
      >>> plt.show()

  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)