scipy.stats.betanbinom#

scipy.stats.betanbinom = <scipy.stats._discrete_distns.betanbinom_gen object>[source]#

A beta-negative-binomial discrete random variable.

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

See also

betabinom: Beta binomial distribution

Notes

The beta-negative-binomial distribution is a negative binomial distribution with a probability of success p that follows a beta distribution.

The probability mass function for betanbinom is:

\[f(k) = \binom{n + k - 1}{k} \frac{B(a + n, b + k)}{B(a, b)}\]

for \(k \ge 0\), \(n \geq 0\), \(a > 0\), \(b > 0\), where \(B(a, b)\) is the beta function.

betanbinom takes \(n\), \(a\), and \(b\) as shape parameters.

References

[1]

https://en.wikipedia.org/wiki/Beta_negative_binomial_distribution

The probability mass function above is defined in the “standardized” form. To shift distribution use the loc parameter. Specifically, betanbinom.pmf(k, n, a, b, loc) is identically equivalent to betanbinom.pmf(k - loc, n, a, b).

Added in version 1.12.0.

Examples

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

Calculate the first four moments:

>>> n, a, b = 5, 9.3, 1
>>> mean, var, skew, kurt = betanbinom.stats(n, a, b, moments='mvsk')

Display the probability mass function (pmf):

>>> x = np.arange(betanbinom.ppf(0.01, n, a, b),
...               betanbinom.ppf(0.99, n, a, b))
>>> ax.plot(x, betanbinom.pmf(x, n, a, b), 'bo', ms=8, label='betanbinom pmf')
>>> ax.vlines(x, 0, betanbinom.pmf(x, n, a, b), 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 = betanbinom(n, a, b)
>>> 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-betanbinom-1_00_00.png

Check accuracy of cdf and ppf:

>>> prob = betanbinom.cdf(x, n, a, b)
>>> np.allclose(x, betanbinom.ppf(prob, n, a, b))
True

Generate random numbers:

>>> r = betanbinom.rvs(n, a, b, size=1000)