scipy.stats.nct#

scipy.stats.nct = <scipy.stats._continuous_distns.nct_gen object>[source]#

A non-central Student’s t continuous random variable.

As an instance of the rv_continuous class, nct 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

If $$Y$$ is a standard normal random variable and $$V$$ is an independent chi-square random variable (chi2) with $$k$$ degrees of freedom, then

$X = \frac{Y + c}{\sqrt{V/k}}$

has a non-central Student’s t distribution on the real line. The degrees of freedom parameter $$k$$ (denoted df in the implementation) satisfies $$k > 0$$ and the noncentrality parameter $$c$$ (denoted nc in the implementation) is a real number.

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, nct.pdf(x, df, nc, loc, scale) is identically equivalent to nct.pdf(y, df, nc) / scale with 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.

Examples

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


Calculate the first four moments:

>>> df, nc = 14, 0.24
>>> mean, var, skew, kurt = nct.stats(df, nc, moments='mvsk')


Display the probability density function (pdf):

>>> x = np.linspace(nct.ppf(0.01, df, nc),
...                 nct.ppf(0.99, df, nc), 100)
>>> ax.plot(x, nct.pdf(x, df, nc),
...        'r-', lw=5, alpha=0.6, label='nct 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 = nct(df, nc)
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')


Check accuracy of cdf and ppf:

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


Generate random numbers:

>>> r = nct.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.