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Module « scipy.stats »

Fonction poisson_binom - module scipy.stats

Signature de la fonction poisson_binom

def poisson_binom(*args, **kwds) 

Description

help(scipy.stats.poisson_binom)

A Poisson Binomial discrete random variable.

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

See Also
--------
binom

Notes
-----
The probability mass function for `poisson_binom` is:

.. math::

 f(k; p_1, p_2, ..., p_n) = \sum_{A \in F_k} \prod_{i \in A} p_i \prod_{j \in A^C} 1 - p_j

where :math:`k \in \{0, 1, \dots, n-1, n\}`, :math:`F_k` is the set of all
subsets of :math:`k` integers that can be selected :math:`\{0, 1, \dots, n-1, n\}`,
and :math:`A^C` is the complement of a set :math:`A`.

`poisson_binom` accepts a single array argument ``p`` for shape parameters
:math:`0 ≤ p_i ≤ 1`, where the last axis corresponds with the index :math:`i` and
any others are for batch dimensions. Broadcasting behaves according to the usual
rules except that the last axis of ``p`` is ignored. Instances of this class do
not support serialization/unserialization.

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

References
----------
.. [1] "Poisson binomial distribution", Wikipedia,
       https://en.wikipedia.org/wiki/Poisson_binomial_distribution
.. [2] Biscarri, William, Sihai Dave Zhao, and Robert J. Brunner. "A simple and
       fast method for computing the Poisson binomial distribution function".
       Computational Statistics & Data Analysis 122 (2018) 92-100.
       :doi:`10.1016/j.csda.2018.01.007`

Examples
--------
>>> import numpy as np
>>> from scipy.stats import poisson_binom
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> p = [0.1, 0.6, 0.7, 0.8]
>>> mean, var, skew, kurt = poisson_binom.stats(p, moments='mvsk')

Display the probability mass function (``pmf``):

>>> x = np.arange(poisson_binom.ppf(0.01, p),
...               poisson_binom.ppf(0.99, p))
>>> ax.plot(x, poisson_binom.pmf(x, p), 'bo', ms=8, label='poisson_binom pmf')
>>> ax.vlines(x, 0, poisson_binom.pmf(x, 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 = poisson_binom(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 = poisson_binom.cdf(x, p)
>>> np.allclose(x, poisson_binom.ppf(prob, p))
True

Generate random numbers:

>>> r = poisson_binom.rvs(p, size=1000)

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