scipy.special.pdtr#

scipy.special.pdtr(k, m, out=None) = <ufunc 'pdtr'>#

Poisson cumulative distribution function.

Defined as the probability that a Poisson-distributed random variable with event rate \(m\) is less than or equal to \(k\). More concretely, this works out to be [1]

\[\exp(-m) \sum_{j = 0}^{\lfloor{k}\rfloor} \frac{m^j}{j!}.\]
Parameters:
karray_like

Number of occurrences (nonnegative, real)

marray_like

Shape parameter (nonnegative, real)

outndarray, optional

Optional output array for the function results

Returns:
scalar or ndarray

Values of the Poisson cumulative distribution function

See also

pdtrc

Poisson survival function

pdtrik

inverse of pdtr with respect to k

pdtri

inverse of pdtr with respect to m

Notes

pdtr has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variable SCIPY_ARRAY_API=1 and providing CuPy, PyTorch, JAX, or Dask arrays as array arguments. The following combinations of backend and device (or other capability) are supported.

Library

CPU

GPU

NumPy

✅

n/a

CuPy

n/a

✅

PyTorch

✅

⛔

JAX

⚠️ no JIT

⛔

Dask

✅

n/a

See Support for the array API standard for more information.

References

Examples

>>> import numpy as np
>>> import scipy.special as sc

It is a cumulative distribution function, so it converges to 1 monotonically as k goes to infinity.

>>> sc.pdtr([1, 10, 100, np.inf], 1)
array([0.73575888, 0.99999999, 1.        , 1.        ])

It is discontinuous at integers and constant between integers.

>>> sc.pdtr([1, 1.5, 1.9, 2], 1)
array([0.73575888, 0.73575888, 0.73575888, 0.9196986 ])
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