scipy.special.nbdtri#

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

Returns the inverse with respect to the parameter p of y = nbdtr(k, n, p), the negative binomial cumulative distribution function.

Parameters:

karray_like: The maximum number of allowed failures (nonnegative int).
narray_like: The target number of successes (positive int).
yarray_like: The probability of k or fewer failures before n successes (float).
outndarray, optional: Optional output array for the function results

Returns:

pscalar or ndarray: Probability of success in a single event (float) such that nbdtr(k, n, p) = y.

See also

nbdtr: Cumulative distribution function of the negative binomial.
nbdtrc: Negative binomial survival function.
scipy.stats.nbinom: negative binomial distribution.
nbdtrik: Inverse with respect to k of nbdtr(k, n, p).
nbdtrin: Inverse with respect to n of nbdtr(k, n, p).
scipy.stats.nbinom: Negative binomial distribution

Notes

Wrapper for the Cephes [1] routine nbdtri.

The negative binomial distribution is also available as scipy.stats.nbinom. Using nbdtri directly can improve performance compared to the ppf method of scipy.stats.nbinom.

References

[1]

Cephes Mathematical Functions Library, http://www.netlib.org/cephes/

Examples

nbdtri is the inverse of nbdtr with respect to p. Up to floating point errors the following holds: nbdtri(k, n, nbdtr(k, n, p))=p.

>>> import numpy as np
>>> from scipy.special import nbdtri, nbdtr
>>> k, n, y = 5, 10, 0.2
>>> cdf_val = nbdtr(k, n, y)
>>> nbdtri(k, n, cdf_val)
0.20000000000000004

Compute the function for k=10 and n=5 at several points by providing a NumPy array or list for y.

>>> y = np.array([0.1, 0.4, 0.8])
>>> nbdtri(3, 5, y)
array([0.34462319, 0.51653095, 0.69677416])

Plot the function for three different parameter sets.

>>> import matplotlib.pyplot as plt
>>> n_parameters = [5, 20, 30, 30]
>>> k_parameters = [20, 20, 60, 80]
>>> linestyles = ['solid', 'dashed', 'dotted', 'dashdot']
>>> parameters_list = list(zip(n_parameters, k_parameters, linestyles))
>>> cdf_vals = np.linspace(0, 1, 1000)
>>> fig, ax = plt.subplots(figsize=(8, 8))
>>> for parameter_set in parameters_list:
...     n, k, style = parameter_set
...     nbdtri_vals = nbdtri(k, n, cdf_vals)
...     ax.plot(cdf_vals, nbdtri_vals, label=rf"$k={k},\ n={n}$",
...             ls=style)
>>> ax.legend()
>>> ax.set_ylabel("$p$")
>>> ax.set_xlabel("$CDF$")
>>> title = "nbdtri: inverse of negative binomial CDF with respect to $p$"
>>> ax.set_title(title)
>>> plt.show()

../../_images/scipy-special-nbdtri-1_00_00.png

nbdtri 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]])
>>> y = np.array([0.3, 0.5, 0.7, 0.9])
>>> k.shape, y.shape
((3, 1), (4,))

>>> nbdtri(k, 5, y)
array([[0.37258157, 0.45169416, 0.53249956, 0.64578407],
       [0.24588501, 0.30451981, 0.36778453, 0.46397088],
       [0.18362101, 0.22966758, 0.28054743, 0.36066188]])