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Module « scipy.stats »

Fonction nhypergeom - module scipy.stats

Signature de la fonction nhypergeom

def nhypergeom(*args, **kwds) 

Description

help(scipy.stats.nhypergeom)

A negative hypergeometric discrete random variable.

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

As an instance of the `rv_discrete` class, `nhypergeom` 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, r, loc=0, size=1, random_state=None)
    Random variates.
pmf(k, M, n, r, loc=0)
    Probability mass function.
logpmf(k, M, n, r, loc=0)
    Log of the probability mass function.
cdf(k, M, n, r, loc=0)
    Cumulative distribution function.
logcdf(k, M, n, r, loc=0)
    Log of the cumulative distribution function.
sf(k, M, n, r, loc=0)
    Survival function  (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
logsf(k, M, n, r, loc=0)
    Log of the survival function.
ppf(q, M, n, r, loc=0)
    Percent point function (inverse of ``cdf`` --- percentiles).
isf(q, M, n, r, loc=0)
    Inverse survival function (inverse of ``sf``).
stats(M, n, r, loc=0, moments='mv')
    Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
entropy(M, n, r, loc=0)
    (Differential) entropy of the RV.
expect(func, args=(M, n, r), loc=0, lb=None, ub=None, conditional=False)
    Expected value of a function (of one argument) with respect to the distribution.
median(M, n, r, loc=0)
    Median of the distribution.
mean(M, n, r, loc=0)
    Mean of the distribution.
var(M, n, r, loc=0)
    Variance of the distribution.
std(M, n, r, loc=0)
    Standard deviation of the distribution.
interval(confidence, M, n, r, loc=0)
    Confidence interval with equal areas around the median.

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,

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

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

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

It is equivalent to observing :math:`k` successes in :math:`k+r-1`
samples with :math:`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
:math:`M-n-(r-1)` divided by the size of the remaining population
:math:`M-(k+r-1)`. This relationship can be shown as:

.. math:: NHG(k;M,n,r) = HG(k;M,n,k+r-1)\frac{(M-n-(r-1))}{(M-(k+r-1))}

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

The probability mass function above is defined in the "standardized" form.
To shift distribution use the ``loc`` parameter.
Specifically, ``nhypergeom.pmf(k, M, n, r, loc)`` is identically
equivalent to ``nhypergeom.pmf(k - loc, M, n, r)``.

Examples
--------
>>> import numpy as np
>>> from scipy.stats import nhypergeom
>>> 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 (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)

>>> 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()

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)

And to generate random numbers:

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

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

>>> from scipy.stats import hypergeom, nhypergeom
>>> M, n, r = 45, 13, 8
>>> k = 6
>>> nhypergeom.pmf(k, M, n, r)
0.06180776620271643
>>> hypergeom.pmf(k, M, n, k+r-1) * (M - n - (r-1)) / (M - (k+r-1))
0.06180776620271644

See Also
--------
hypergeom, binom, nbinom

References
----------
.. [1] Negative Hypergeometric Distribution on Wikipedia
       https://en.wikipedia.org/wiki/Negative_hypergeometric_distribution

.. [2] Negative Hypergeometric Distribution from
       http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Negativehypergeometric.pdf

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