scipy.stats.ncx2 — SciPy v1.15.2 Manual (original) (raw)

scipy.stats.ncx2 = <scipy.stats._continuous_distns.ncx2_gen object>[source]#

A non-central chi-squared continuous random variable.

As an instance of the rv_continuous class, ncx2 object inherits from it a collection of generic methods (see below for the full list), and completes them with details specific for this particular distribution.

Notes

The probability density function for ncx2 is:

\[f(x, k, \lambda) = \frac{1}{2} \exp(-(\lambda+x)/2) (x/\lambda)^{(k-2)/4} I_{(k-2)/2}(\sqrt{\lambda x})\]

for \(x >= 0\), \(k > 0\) and \(\lambda \ge 0\).\(k\) specifies the degrees of freedom (denoted df in the implementation) and \(\lambda\) is the non-centrality parameter (denoted nc in the implementation). \(I_\nu\) denotes the modified Bessel function of first order of degree \(\nu\)(scipy.special.iv).

ncx2 takes df and nc as shape parameters.

This distribution uses routines from the Boost Math C++ library for the computation of the pdf, cdf, ppf, sf and isfmethods. [1]

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, ncx2.pdf(x, df, nc, loc, scale) is identically equivalent to ncx2.pdf(y, df, nc) / scale withy = (x - loc) / scale. Note that shifting the location of a distribution does not make it a “noncentral” distribution; noncentral generalizations of some distributions are available in separate classes.

References

Examples

import numpy as np from scipy.stats import ncx2 import matplotlib.pyplot as plt fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

df, nc = 21, 1.06 mean, var, skew, kurt = ncx2.stats(df, nc, moments='mvsk')

Display the probability density function (pdf):

x = np.linspace(ncx2.ppf(0.01, df, nc), ... ncx2.ppf(0.99, df, nc), 100) ax.plot(x, ncx2.pdf(x, df, nc), ... 'r-', lw=5, alpha=0.6, label='ncx2 pdf')

Alternatively, the distribution object can be called (as a function) to fix the shape, location and scale parameters. This returns a “frozen” RV object holding the given parameters fixed.

Freeze the distribution and display the frozen pdf:

rv = ncx2(df, nc) ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

vals = ncx2.ppf([0.001, 0.5, 0.999], df, nc) np.allclose([0.001, 0.5, 0.999], ncx2.cdf(vals, df, nc)) True

Generate random numbers:

r = ncx2.rvs(df, nc, size=1000)

And compare the histogram:

ax.hist(r, density=True, bins='auto', histtype='stepfilled', alpha=0.2) ax.set_xlim([x[0], x[-1]]) ax.legend(loc='best', frameon=False) plt.show()

Methods