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Module « scipy.stats »

Fonction nbinom - module scipy.stats

Signature de la fonction nbinom

def nbinom(*args, **kwds) 

Description

help(scipy.stats.nbinom)

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(confidence, n, p, loc=0)
    Confidence interval with equal areas around the median.

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 :math:`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}

This distribution uses routines from the Boost Math C++ library for
the computation of the ``pmf``, ``cdf``, ``sf``, ``ppf``, ``isf``
and ``stats`` methods. [1]_

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)``.

References
----------
.. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.

Examples
--------
>>> import numpy as np
>>> 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

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