scipy.stats.ncx2#
- 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_continuousclass,- ncx2object 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 - ncx2is:\[f(x, k, \lambda) = \frac{1}{2} \exp(-(\lambda+x)/2) (x/\lambda)^{(k-2)/4} I_{(k-2)/2}(\sqrt{\lambda x})\]- for \(x >= 0\), \(k > 0\) and \(\lambda \ge 0\). \(k\) specifies the degrees of freedom (denoted - dfin the implementation) and \(\lambda\) is the non-centrality parameter (denoted- ncin the implementation). \(I_\nu\) denotes the modified Bessel function of first order of degree \(\nu\) (- scipy.special.iv).- ncx2takes- dfand- ncas shape parameters.- This distribution uses routines from the Boost Math C++ library for the computation of the - pdf,- cdf,- ppf,- sfand- isfmethods. [1]- The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the - locand- scaleparameters. Specifically,- ncx2.pdf(x, df, nc, loc, scale)is identically equivalent to- ncx2.pdf(y, df, nc) / scalewith- y = (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 [1]- The Boost Developers. “Boost C++ Libraries”. https://www.boost.org/. - 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 - cdfand- 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 - rvs(df, nc, loc=0, scale=1, size=1, random_state=None) - Random variates. - pdf(x, df, nc, loc=0, scale=1) - Probability density function. - logpdf(x, df, nc, loc=0, scale=1) - Log of the probability density function. - cdf(x, df, nc, loc=0, scale=1) - Cumulative distribution function. - logcdf(x, df, nc, loc=0, scale=1) - Log of the cumulative distribution function. - sf(x, df, nc, loc=0, scale=1) - Survival function (also defined as - 1 - cdf, but sf is sometimes more accurate).- logsf(x, df, nc, loc=0, scale=1) - Log of the survival function. - ppf(q, df, nc, loc=0, scale=1) - Percent point function (inverse of - cdf— percentiles).- isf(q, df, nc, loc=0, scale=1) - Inverse survival function (inverse of - sf).- moment(order, df, nc, loc=0, scale=1) - Non-central moment of the specified order. - stats(df, nc, loc=0, scale=1, moments=’mv’) - Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). - entropy(df, nc, loc=0, scale=1) - (Differential) entropy of the RV. - fit(data) - Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments. - expect(func, args=(df, nc), loc=0, scale=1, lb=None, ub=None, conditional=False, **kwds) - Expected value of a function (of one argument) with respect to the distribution. - median(df, nc, loc=0, scale=1) - Median of the distribution. - mean(df, nc, loc=0, scale=1) - Mean of the distribution. - var(df, nc, loc=0, scale=1) - Variance of the distribution. - std(df, nc, loc=0, scale=1) - Standard deviation of the distribution. - interval(confidence, df, nc, loc=0, scale=1) - Confidence interval with equal areas around the median.