scipy.special.ndtr#
- scipy.special.ndtr(x, out=None) = <ufunc 'ndtr'>#
Cumulative distribution of the standard normal distribution.
Returns the area under the standard Gaussian probability density function, integrated from minus infinity to x
\[\frac{1}{\sqrt{2\pi}} \int_{-\infty}^x \exp(-t^2/2) dt\]- Parameters:
- xarray_like, real or complex
Argument
- outndarray, optional
Optional output array for the function results
- Returns:
- scalar or ndarray
The value of the normal CDF evaluated at x
See also
log_ndtr
Logarithm of ndtr
ndtri
Inverse of ndtr, standard normal percentile function
erf
Error function
erfc
1 - erf
scipy.stats.norm
Normal distribution
Notes
ndtr
has experimental support for Python Array API Standard compatible backends in addition to NumPy. Please consider testing these features by setting an environment variableSCIPY_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
✅
✅
Dask
✅
n/a
See Support for the array API standard for more information.
Examples
Evaluate
ndtr
at one point.>>> import numpy as np >>> from scipy.special import ndtr >>> ndtr(0.5) 0.6914624612740131
Evaluate the function at several points by providing a NumPy array or list for x.
>>> ndtr([0, 0.5, 2]) array([0.5 , 0.69146246, 0.97724987])
Plot the function.
>>> import matplotlib.pyplot as plt >>> x = np.linspace(-5, 5, 100) >>> fig, ax = plt.subplots() >>> ax.plot(x, ndtr(x)) >>> ax.set_title(r"Standard normal cumulative distribution function $\Phi$") >>> plt.show()