scipy.stats.nbinom(*args, **kwds) = <scipy.stats._discrete_distns.nbinom_gen object>[source]

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.


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:

\[f(k) = \binom{k+n-1}{n-1} p^n (1-p)^k\]

for \(k \ge 0\).

nbinom takes \(n\) and \(p\) as shape parameters where n is the number of successes, whereas p is the probability of a single success.

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).


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

Calculate a few first moments:

>>> n, p = 0.4, 0.4
>>> 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)

Check accuracy of cdf and ppf:

>>> prob = nbinom.cdf(x, n, p)
>>> np.allclose(x, nbinom.ppf(prob, n, p))

Generate random numbers:

>>> r = nbinom.rvs(n, p, size=1000)


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(alpha, n, p, loc=0)

Endpoints of the range that contains alpha percent of the distribution

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