scipy.special.nbdtr(k, n, p, out=None) = <ufunc 'nbdtr'>#

Negative binomial cumulative distribution function.

Returns the sum of the terms 0 through k of the negative binomial distribution probability mass function,

\[F = \sum_{j=0}^k {{n + j - 1}\choose{j}} p^n (1 - p)^j.\]

In a sequence of Bernoulli trials with individual success probabilities p, this is the probability that k or fewer failures precede the nth success.


The maximum number of allowed failures (nonnegative int).


The target number of successes (positive int).


Probability of success in a single event (float).

outndarray, optional

Optional output array for the function results

Fscalar or ndarray

The probability of k or fewer failures before n successes in a sequence of events with individual success probability p.

See also


Negative binomial survival function


Negative binomial quantile function


Negative binomial distribution


If floating point values are passed for k or n, they will be truncated to integers.

The terms are not summed directly; instead the regularized incomplete beta function is employed, according to the formula,

\[\mathrm{nbdtr}(k, n, p) = I_{p}(n, k + 1).\]

Wrapper for the Cephes [1] routine nbdtr.

The negative binomial distribution is also available as scipy.stats.nbinom. Using nbdtr directly can improve performance compared to the cdf method of scipy.stats.nbinom (see last example).



Cephes Mathematical Functions Library,


Compute the function for k=10 and n=5 at p=0.5.

>>> import numpy as np
>>> from scipy.special import nbdtr
>>> nbdtr(10, 5, 0.5)

Compute the function for n=10 and p=0.5 at several points by providing a NumPy array or list for k.

>>> nbdtr([5, 10, 15], 10, 0.5)
array([0.15087891, 0.58809853, 0.88523853])

Plot the function for four different parameter sets.

>>> import matplotlib.pyplot as plt
>>> k = np.arange(130)
>>> n_parameters = [20, 20, 20, 80]
>>> p_parameters = [0.2, 0.5, 0.8, 0.5]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(p_parameters, n_parameters,
...                            linestyles))
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> for parameter_set in parameters_list:
...     p, n, style = parameter_set
...     nbdtr_vals = nbdtr(k, n, p)
...     ax.plot(k, nbdtr_vals, label=rf"$n={n},\, p={p}$",
...             ls=style)
>>> ax.legend()
>>> ax.set_xlabel("$k$")
>>> ax.set_title("Negative binomial cumulative distribution function")

The negative binomial distribution is also available as scipy.stats.nbinom. Using nbdtr directly can be much faster than calling the cdf method of scipy.stats.nbinom, especially for small arrays or individual values. To get the same results one must use the following parametrization: nbinom(n, p).cdf(k)=nbdtr(k, n, p).

>>> from scipy.stats import nbinom
>>> k, n, p = 5, 3, 0.5
>>> nbdtr_res = nbdtr(k, n, p)  # this will often be faster than below
>>> stats_res = nbinom(n, p).cdf(k)
>>> stats_res, nbdtr_res  # test that results are equal
(0.85546875, 0.85546875)

nbdtr can evaluate different parameter sets by providing arrays with shapes compatible for broadcasting for k, n and p. Here we compute the function for three different k at four locations p, resulting in a 3x4 array.

>>> k = np.array([[5], [10], [15]])
>>> p = np.array([0.3, 0.5, 0.7, 0.9])
>>> k.shape, p.shape
((3, 1), (4,))
>>> nbdtr(k, 5, p)
array([[0.15026833, 0.62304687, 0.95265101, 0.9998531 ],
       [0.48450894, 0.94076538, 0.99932777, 0.99999999],
       [0.76249222, 0.99409103, 0.99999445, 1.        ]])