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Fonction nhypergeom - module scipy.stats

Signature de la fonction nhypergeom

def nhypergeom(*args, **kwds) 

Description

nhypergeom.__doc__

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(alpha, M, n, r, loc=0)
        Endpoints of the range that contains fraction alpha [0, 1] of the
        distribution

    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
    --------
    >>> 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