scipy.stats.poisson_binom#

scipy.stats.poisson_binom = <scipy.stats._discrete_distns.poisson_binom_gen object>[source]#

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:

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

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

poisson_binom accepts a single array argument p for shape parameters \(0 ≤ p_i ≤ 1\), where the last axis corresponds with the index \(i\) 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()

../../_images/scipy-stats-poisson_binom-1_00_00.png

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)