scipy.special.log_ndtr#

scipy.special.log_ndtr(x, out=None) = <ufunc 'log_ndtr'>#

Logarithm of Gaussian cumulative distribution function.

Returns the log of the area under the standard Gaussian probability density function, integrated from minus infinity to x:

log(1/sqrt(2*pi) * integral(exp(-t**2 / 2), t=-inf..x))

Parameters:

xarray_like, real or complex: Argument
outndarray, optional: Optional output array for the function results

Returns:

scalar or ndarray: The value of the log of the normal CDF evaluated at x

See also

erf
erfc
scipy.stats.norm
ndtr

Notes

Array API Standard Support

log_ndtr 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 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.

See Support for the array API standard for more information.

Examples

>>> import numpy as np
>>> from scipy.special import log_ndtr, ndtr

The benefit of log_ndtr(x) over the naive implementation np.log(ndtr(x)) is most evident with moderate to large positive values of x:

>>> x = np.array([6, 7, 9, 12, 15, 25])
>>> log_ndtr(x)
array([-9.86587646e-010, -1.27981254e-012, -1.12858841e-019,
       -1.77648211e-033, -3.67096620e-051, -3.05669671e-138])

The results of the naive calculation for the moderate x values have only 5 or 6 correct significant digits. For values of x greater than approximately 8.3, the naive expression returns 0:

>>> np.log(ndtr(x))
array([-9.86587701e-10, -1.27986510e-12,  0.00000000e+00,
        0.00000000e+00,  0.00000000e+00,  0.00000000e+00])