scipy.special.poisson_binom_cdf#

scipy.special.poisson_binom_cdf = <wrapped_ufunc 'poisson_binom_cdf'>#

Poisson binomial cumulative distribution function.

The Poisson binomial distribution is the discrete probability distribution of a sum of independent Bernoulli trials that are not necessarily identically distributed [1].

Parameters:
karray_like of int

Number of successes at which to evaluate the CDF.

parray_like of float

Success probabilities of independent Bernoulli trials.

outndarray, optional

Optional output array for the function results.

**kwargs

For other keyword-only arguments, see the NumPy ufunc docs.

Returns:
scalar or ndarray

Values of the Poisson binomial cumulative distribution function.

See also

scipy.stats.poisson_binom

Notes

For \(\mathbf{p} = \left(p_1, p_2, \ldots, p_n\right)\) giving the success probabilities for a sequence of \(n\) independent Bernoulli trials

\[\mathrm{CDF}(k, \mathbf{p}) = \sum_{l=0}^{k}\sum_{A \in F_l}\prod_{i \in A} p_i \prod_{j \in A^c} (1 - p_j)\]

where \(F_l\) is the set of all subsets of size \(l\) of \(\{1, 2, \ldots, n\}\) and \(A^c\) is the complement of \(A\) in \(\{1, 2, \ldots, n\}\).

Added in version 2.0.0.

Array API Standard Support

poisson_binom_cdf 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. poisson_binom_cdf does not currently support out for the PyTorch backend. It supports the axis kwarg for all supported backends.

See Support for the array API standard for more information.

References

Examples

>>> import numpy as np
>>> from scipy.special import poisson_binom_cdf

A batch of two Poisson binomial distributions involving three trials each.

>>> p = np.asarray([[0.2, 0.4, 0.6], [0.3, 0.3, 0.1]])

Evaluate the cdf across the entire support. An extra dimension is added to k to broadcast against the batch dimension of p.

>>> k = np.asarray([0, 1, 2, 3])[:, None]
>>> poisson_binom_cdf(k, p)
array([[0.192, 0.441],
       [0.656, 0.868],
       [0.952, 0.991],
       [1.   , 1.   ]])
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