`scipy.stats._discrete_distns:nhypergeom_gen`

source: /scipy/stats/_discrete_distns.py :738

Signature

   class     nhypergeom_gen (    a    =  0   ,    b    =  inf   ,    name    =  None   ,    badvalue    =  None   ,    moment_tol    =  1e-08   ,    values    =  None   ,    inc    =  1   ,    longname    =  None   ,    shapes    =  None   ,    seed    =  None     )

Members

Summary

A negative hypergeometric discrete random variable.

Extended Summary

Consider a box containing $M$ balls:, $n$ 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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Notes

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,

f (k; M, n, r) = \frac{( k k + r - 1 ) ( n - k M - r - k )}{( n M )}

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

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

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:

N H G (k; M, n, r) = H G (k; M, n, k + r - 1) \frac{( M - n - ( r - 1 ))}{( M - ( k + r - 1 ))}

where $N H G$ is probability mass function (PMF) of the negative hypergeometric distribution and $H G$ is the PMF of the hypergeometric distribution.

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Examples

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

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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 (successes) in a sample with exactly 12 animals that aren't dogs (failures), we can initialize a frozen distribution and plot the probability mass function:

M, n, r = [20, 7, 12]
rv = nhypergeom(M, n, r)
x = np.arange(0, n+2)
pmf_dogs = rv.pmf(x)

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fig = plt.figure()
ax = fig.add_subplot(111)

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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')

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plt.show()

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Instead of using a frozen distribution we can also use `nhypergeom` methods directly. To for example obtain the probability mass function, use:

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

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And to generate random numbers:

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

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To verify the relationship between `hypergeom` and `nhypergeom`, use:

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

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

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Aliases

scipy.stats._discrete_distns.nhypergeom_gen