Consider a box containing M balls:, :math:`n` red and M-n blue. We randomly sample balls from the box, one at a time and without replacement, until we have picked r blue balls. nhypergeom is the distribution of the number of red balls k we have picked.

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

The probability mass function is defined as,

for k \in [0, n], n \in [0, M], r \in [0, M-n], and the binomial coefficient is:

It is equivalent to observing k successes in k+r-1 samples with k+r'th sample being a failure. The former can be modelled as a hypergeometric distribution. The probability of the latter is simply the number of failures remaining M-n-(r-1) divided by the size of the remaining population M-(k+r-1). This relationship can be shown as:

where NHG is probability mass function (PMF) of the negative hypergeometric distribution and HG is the PMF of the hypergeometric distribution.

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A negative hypergeometric discrete random variable.

import numpy as np
from scipy.stats import nhypergeom
import matplotlib.pyplot as plt

M, n, r = [20, 7, 12]
rv = nhypergeom(M, n, r)
x = np.arange(0, n+2)
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 with given 12 failures')
ax.set_ylabel('nhypergeom PMF')
plt.show()

prb = nhypergeom.pmf(x, M, n, r)

R = nhypergeom.rvs(M, n, r, size=10)

from scipy.stats import hypergeom, nhypergeom
M, n, r = 45, 13, 8
k = 6
nhypergeom.pmf(k, M, n, r)

hypergeom.pmf(k, M, n, k+r-1) * (M - n - (r-1)) / (M - (k+r-1))