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Module « scipy.stats »

Fonction skellam - module scipy.stats

Signature de la fonction skellam

def skellam(*args, **kwds) 

Description

help(scipy.stats.skellam)

A  Skellam discrete random variable.

As an instance of the `rv_discrete` class, `skellam` 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(mu1, mu2, loc=0, size=1, random_state=None)
    Random variates.
pmf(k, mu1, mu2, loc=0)
    Probability mass function.
logpmf(k, mu1, mu2, loc=0)
    Log of the probability mass function.
cdf(k, mu1, mu2, loc=0)
    Cumulative distribution function.
logcdf(k, mu1, mu2, loc=0)
    Log of the cumulative distribution function.
sf(k, mu1, mu2, loc=0)
    Survival function  (also defined as ``1 - cdf``, but `sf` is sometimes more accurate).
logsf(k, mu1, mu2, loc=0)
    Log of the survival function.
ppf(q, mu1, mu2, loc=0)
    Percent point function (inverse of ``cdf`` --- percentiles).
isf(q, mu1, mu2, loc=0)
    Inverse survival function (inverse of ``sf``).
stats(mu1, mu2, loc=0, moments='mv')
    Mean('m'), variance('v'), skew('s'), and/or kurtosis('k').
entropy(mu1, mu2, loc=0)
    (Differential) entropy of the RV.
expect(func, args=(mu1, mu2), loc=0, lb=None, ub=None, conditional=False)
    Expected value of a function (of one argument) with respect to the distribution.
median(mu1, mu2, loc=0)
    Median of the distribution.
mean(mu1, mu2, loc=0)
    Mean of the distribution.
var(mu1, mu2, loc=0)
    Variance of the distribution.
std(mu1, mu2, loc=0)
    Standard deviation of the distribution.
interval(confidence, mu1, mu2, loc=0)
    Confidence interval with equal areas around the median.

Notes
-----
Probability distribution of the difference of two correlated or
uncorrelated Poisson random variables.

Let :math:`k_1` and :math:`k_2` be two Poisson-distributed r.v. with
expected values :math:`\lambda_1` and :math:`\lambda_2`. Then,
:math:`k_1 - k_2` follows a Skellam distribution with parameters
:math:`\mu_1 = \lambda_1 - \rho \sqrt{\lambda_1 \lambda_2}` and
:math:`\mu_2 = \lambda_2 - \rho \sqrt{\lambda_1 \lambda_2}`, where
:math:`\rho` is the correlation coefficient between :math:`k_1` and
:math:`k_2`. If the two Poisson-distributed r.v. are independent then
:math:`\rho = 0`.

Parameters :math:`\mu_1` and :math:`\mu_2` must be strictly positive.

For details see: https://en.wikipedia.org/wiki/Skellam_distribution

`skellam` takes :math:`\mu_1` and :math:`\mu_2` as shape parameters.

The probability mass function above is defined in the "standardized" form.
To shift distribution use the ``loc`` parameter.
Specifically, ``skellam.pmf(k, mu1, mu2, loc)`` is identically
equivalent to ``skellam.pmf(k - loc, mu1, mu2)``.

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

Calculate the first four moments:

>>> mu1, mu2 = 15, 8
>>> mean, var, skew, kurt = skellam.stats(mu1, mu2, moments='mvsk')

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

>>> x = np.arange(skellam.ppf(0.01, mu1, mu2),
...               skellam.ppf(0.99, mu1, mu2))
>>> ax.plot(x, skellam.pmf(x, mu1, mu2), 'bo', ms=8, label='skellam pmf')
>>> ax.vlines(x, 0, skellam.pmf(x, mu1, mu2), 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 = skellam(mu1, mu2)
>>> 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 = skellam.cdf(x, mu1, mu2)
>>> np.allclose(x, skellam.ppf(prob, mu1, mu2))
True

Generate random numbers:

>>> r = skellam.rvs(mu1, mu2, size=1000)

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