scipy.special.bdtrik#

scipy.special.bdtrik(y, n, p, out=None) = <wrapped_ufunc 'bdtrik'>#

Binomial distribution quantile.

Finds the number of successes k such that the sum of the terms 0 through k of the Binomial probability density for n events with probability p is equal to the given cumulative probability y.

Parameters:

yarray_like: Cumulative probability (probability of k or fewer successes in n events).
narray_like: Number of events (float).
parray_like: Success probability (float).
outndarray, optional: Optional output array for the function values

Returns:

kscalar or ndarray: The number of successes k such that bdtr(k, n, p) = y.

See also

bdtr: Binomial distribution cumulative distribution function

Notes

Formula 26.5.24 of [1] is used to reduce the binomial distribution to the cumulative incomplete beta distribution.

This function uses routines from the Boost.Math C++ library [3] which rely on numerical inversion of the binomial distribution CDF [4].

Array API Standard Support

bdtrik has support for Python Array API Standard compatible backends in addition to NumPy. The following combinations of backend and device (or other capability) are supported.

Library	CPU	GPU
NumPy	✅	n/a
CuPy	n/a	⛔
PyTorch	✅	⛔
JAX	✅	⛔
Dask	✅	n/a

For the NumPy backend, this function supports all NumPy ufunc keyword arguments. Other backends may support out, but typically none of the other ufunc kwargs. out is typically supported for CuPy and PyTorch, but not currently in cases where SciPy relies on a generic Array API implementation or, for PyTorch on CPU, falls back to the NumPy backend. out is never supported for JAX because JAX arrays are immutable. bdtrik does not currently support out for the PyTorch backend.

See Support for the array API standard for more information.

References

[1]

Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972.

[2]

NIST Digital Library of Mathematical Functions https://dlmf.nist.gov/8.17.5#E5

[3]

The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/.

[4]

https://www.boost.org/doc/libs/latest/libs/math/doc/html/math_toolkit/dist_ref/dists/binomial_dist.html

Examples

We have a coin for which the probability of showing heads when flipped is 0.525. The coin is flipped 8 times. Find the largest value of k such that the probability that X <= k is not greater than 0.2, where X is the number of heads.

In fact, there is no integer value of k that will give a probability of exactly 0.2, as this plot of the cumulative distribution function shows.

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from scipy.special import bdtr, bdtrik

>>> n = 8
>>> p = 0.525
>>> k = np.arange(0, n + 1)
>>> plt.plot(k, bdtr(k, n, p), 'o')
>>> plt.grid(True, alpha=0.5)
>>> plt.xlabel('k')
>>> plt.axhline(0.2, linestyle='--', color='k', alpha=0.5)
>>> plt.title(f"bdtr(k, {n}, {p})")
>>> plt.show()

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

From the graph we can see that we would choose k = 2. The function bdtrik lets us find this value directly.

bdtrik returns a floating point value that is like a continuous extension of k. This computes k as a noninteger floating point value:

>>> bdtrik(0.2, n, p)
np.float64(2.508332751475262)

For our final answer we need an integer k, and since we want to ensure that the probability at k does not exceed 0.2, we truncate the fractional part of this value with np.floor:

>>> np.floor(bdtrik(0.2, n, p))
np.float64(2.0)