Module « scipy.stats »

Fonction nbinom - module scipy.stats

Signature de la fonction nbinom

def nbinom(*args, **kwds) 

Description

nbinom.__doc__

A negative binomial discrete random variable.

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

    Notes
    -----
    Negative binomial distribution describes a sequence of i.i.d. Bernoulli
    trials, repeated until a predefined, non-random number of successes occurs.

    The probability mass function of the number of failures for `nbinom` is:

    .. math::

       f(k) = \binom{k+n-1}{n-1} p^n (1-p)^k

    for :math:`k \ge 0`, :math:`0 < p \leq 1`

    `nbinom` takes :math:`n` and :math:`p` as shape parameters where n is the
    number of successes, :math:`p` is the probability of a single success,
    and :math:`1-p` is the probability of a single failure.

    Another common parameterization of the negative binomial distribution is
    in terms of the mean number of failures :math:`\mu` to achieve :math:`n`
    successes. The mean :math:`\mu` is related to the probability of success
    as

    .. math::

       p = \frac{n}{n + \mu}

    The number of successes :math:`n` may also be specified in terms of a
    "dispersion", "heterogeneity", or "aggregation" parameter :math:`\alpha`,
    which relates the mean :math:`\mu` to the variance :math:`\sigma^2`,
    e.g. :math:`\sigma^2 = \mu + \alpha \mu^2`. Regardless of the convention
    used for :math:`\alpha`,

    .. math::

       p &= \frac{\mu}{\sigma^2} \\
       n &= \frac{\mu^2}{\sigma^2 - \mu}

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

    Examples
    --------
    >>> from scipy.stats import nbinom
    >>> import matplotlib.pyplot as plt
    >>> fig, ax = plt.subplots(1, 1)
    
    Calculate the first four moments:
    
    >>> n, p = 5, 0.5
    >>> mean, var, skew, kurt = nbinom.stats(n, p, moments='mvsk')
    
    Display the probability mass function (``pmf``):
    
    >>> x = np.arange(nbinom.ppf(0.01, n, p),
    ...               nbinom.ppf(0.99, n, p))
    >>> ax.plot(x, nbinom.pmf(x, n, p), 'bo', ms=8, label='nbinom pmf')
    >>> ax.vlines(x, 0, nbinom.pmf(x, n, p), 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 = nbinom(n, p)
    >>> 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 = nbinom.cdf(x, n, p)
    >>> np.allclose(x, nbinom.ppf(prob, n, p))
    True
    
    Generate random numbers:
    
    >>> r = nbinom.rvs(n, p, size=1000)

    See Also
    --------
    hypergeom, binom, nhypergeom