scipy.special.chndtr#

scipy.special.chndtr(x, df, nc, out=None) = <ufunc 'chndtr'>#

Non-central chi-squared cumulative distribution function.

The cumulative distribution function is given by

\[F_{\nu,\lambda}(x) = \sum_{j=0}^{\infty} e^{-\lambda / 2} \frac{(\lambda / 2)^j}{j!} F_{\chi^2_{\nu + 2j}}(x),\]

where \(\nu > 0\) is the degrees of freedom (df), \(\lambda \geq 0\) is the non-centrality parameter (nc), and \(F_{\chi^2_{\nu + 2j}}\) is the CDF of the central chi-squared distribution with \(\nu + 2j\) degrees of freedom.

Parameters:

xarray_like: Upper bound of the integral; must satisfy x >= 0.
dfarray_like: Degrees of freedom; must satisfy df > 0.
ncarray_like: Non-centrality parameter; must satisfy nc >= 0.
outndarray, optional: Optional output array for the function results.

Returns:

cdfscalar or ndarray: Value of the non-central chi-squared cumulative distribution function.

See also

chndtrix: Noncentral Chi Squared distribution quantile
chndtridf: Inverse of chndtr with respect to df
chndtrinc: Inverse of chndtr with respect to nc
scipy.stats.ncx2: Non-central chi-squared distribution

Notes

The noncentral chi squared distribution is also available in scipy.stats.ncx2. scipy.stats.ncx2.cdf is equivalent to chndtr.

This function wraps routines from the Boost Math C++ library [1].

References

[1]

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

Examples

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

Compute the noncentral chi squared distribution CDF at one point.

>>> x = 4.0
>>> df = 1.0
>>> nc = 5.0
>>> sc.chndtr(x, df, nc)
0.40667858759710945

Plot the noncentral chi squared distribution CDF for different parameters.

>>> import matplotlib.pyplot as plt
>>> x = np.linspace(0, 40, 1000)
>>> plt.plot(x, sc.chndtr(x, 1, 5), label=r"$df=1,\ nc=5$")
>>> plt.plot(x, sc.chndtr(x, 5, 10), label=r"$df=5,\ nc=10$")
>>> plt.legend()
>>> plt.show()

../../_images/scipy-special-chndtr-1.png