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Fonction nchypergeom_fisher - module scipy.stats

Signature de la fonction nchypergeom_fisher

def nchypergeom_fisher(*args, **kwds) 

Description

nchypergeom_fisher.__doc__

A Fisher's noncentral hypergeometric discrete random variable.

    Fisher's noncentral hypergeometric distribution models drawing objects of
    two types from a bin. `M` is the total number of objects, `n` is the
    number of Type I objects, and `odds` is the odds ratio: the odds of
    selecting a Type I object rather than a Type II object when there is only
    one object of each type.
    The random variate represents the number of Type I objects drawn if we
    take a handful of objects from the bin at once and find out afterwards
    that we took `N` objects.

    As an instance of the `rv_discrete` class, `nchypergeom_fisher` 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, odds, loc=0, size=1, random_state=None)
        Random variates.
    pmf(k, M, n, N, odds, loc=0)
        Probability mass function.
    logpmf(k, M, n, N, odds, loc=0)
        Log of the probability mass function.
    cdf(k, M, n, N, odds, loc=0)
        Cumulative distribution function.
    logcdf(k, M, n, N, odds, loc=0)
        Log of the cumulative distribution function.
    sf(k, M, n, N, odds, loc=0)
        Survival function  (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
    logsf(k, M, n, N, odds, loc=0)
        Log of the survival function.
    ppf(q, M, n, N, odds, loc=0)
        Percent point function (inverse of ``cdf`` --- percentiles).
    isf(q, M, n, N, odds, loc=0)
        Inverse survival function (inverse of ``sf``).
    stats(M, n, N, odds, loc=0, moments='mv')
        Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
    entropy(M, n, N, odds, loc=0)
        (Differential) entropy of the RV.
    expect(func, args=(M, n, N, odds), 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, odds, loc=0)
        Median of the distribution.
    mean(M, n, N, odds, loc=0)
        Mean of the distribution.
    var(M, n, N, odds, loc=0)
        Variance of the distribution.
    std(M, n, N, odds, loc=0)
        Standard deviation of the distribution.
    interval(alpha, M, n, N, odds, loc=0)
        Endpoints of the range that contains fraction alpha [0, 1] of the
        distribution

    See Also
    --------
    nchypergeom_wallenius, hypergeom, nhypergeom

    Notes
    -----
    Let mathematical symbols :math:`N`, :math:`n`, and :math:`M` correspond
    with parameters `N`, `n`, and `M` (respectively) as defined above.

    The probability mass function is defined as

    .. math::

        p(x; M, n, N, \omega) =
        \frac{\binom{n}{x}\binom{M - n}{N-x}\omega^x}{P_0},

    for
    :math:`x \in [x_l, x_u]`,
    :math:`M \in {\mathbb N}`,
    :math:`n \in [0, M]`,
    :math:`N \in [0, M]`,
    :math:`\omega > 0`,
    where
    :math:`x_l = \max(0, N - (M - n))`,
    :math:`x_u = \min(N, n)`,

    .. math::

        P_0 = \sum_{y=x_l}^{x_u} \binom{n}{y}\binom{M - n}{N-y}\omega^y,

    and the binomial coefficients are defined as

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

    `nchypergeom_fisher` uses the BiasedUrn package by Agner Fog with
    permission for it to be distributed under SciPy's license.

    The symbols used to denote the shape parameters (`N`, `n`, and `M`) are not
    universally accepted; they are chosen for consistency with `hypergeom`.

    Note that Fisher's noncentral hypergeometric distribution is distinct
    from Wallenius' noncentral hypergeometric distribution, which models
    drawing a pre-determined `N` objects from a bin one by one.
    When the odds ratio is unity, however, both distributions reduce to the
    ordinary hypergeometric distribution.

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

    References
    ----------
    .. [1] Agner Fog, "Biased Urn Theory".
           https://cran.r-project.org/web/packages/BiasedUrn/vignettes/UrnTheory.pdf

    .. [2] "Fisher's noncentral hypergeometric distribution", Wikipedia,
           https://en.wikipedia.org/wiki/Fisher's_noncentral_hypergeometric_distribution

    Examples
    --------
    >>> from scipy.stats import nchypergeom_fisher
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)
    
    Calculate the first four moments:
    
    >>> M, n, N, odds = 140, 80, 60, 0.5
    >>> mean, var, skew, kurt = nchypergeom_fisher.stats(M, n, N, odds, moments='mvsk')
    
    Display the probability mass function (``pmf``):
    
    >>> x = np.arange(nchypergeom_fisher.ppf(0.01, M, n, N, odds),
    ...               nchypergeom_fisher.ppf(0.99, M, n, N, odds))
    >>> ax.plot(x, nchypergeom_fisher.pmf(x, M, n, N, odds), 'bo', ms=8, label='nchypergeom_fisher pmf')
    >>> ax.vlines(x, 0, nchypergeom_fisher.pmf(x, M, n, N, odds), colors='b', lw=5, alpha=0.5)
    
    Alternatively, the distribution object can be called (as a function)
    to fix the shape and location. This returns a "frozen" RV object holding
    the given parameters fixed.
    
    Freeze the distribution and display the frozen ``pmf``:
    
    >>> rv = nchypergeom_fisher(M, n, N, odds)
    >>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-', lw=1,
    ...         label='frozen pmf')
    >>> ax.legend(loc='best', frameon=False)
    >>> plt.show()
    
    Check accuracy of ``cdf`` and ``ppf``:
    
    >>> prob = nchypergeom_fisher.cdf(x, M, n, N, odds)
    >>> np.allclose(x, nchypergeom_fisher.ppf(prob, M, n, N, odds))
    True
    
    Generate random numbers:
    
    >>> r = nchypergeom_fisher.rvs(M, n, N, odds, size=1000)