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Module « scipy.special »

Fonction nbdtri - module scipy.special

Signature de la fonction nbdtri

def nbdtri(*args, **kwargs) 

Description

help(scipy.special.nbdtri)

nbdtri(x1, x2, x3, /, out=None, *, where=True, casting='same_kind', order='K', dtype=None, subok=True[, signature])

nbdtri(k, n, y, out=None)

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

Parameters
----------
k : array_like
    The maximum number of allowed failures (nonnegative int).
n : array_like
    The target number of successes (positive int).
y : array_like
    The probability of `k` or fewer failures before `n` successes (float).
out : ndarray, optional
    Optional output array for the function results

Returns
-------
p : scalar 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()

`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]])

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