# scipy.stats.chi¶

scipy.stats.chi(*args, **kwds) = <scipy.stats._continuous_distns.chi_gen object>[source]

A chi continuous random variable.

As an instance of the rv_continuous class, chi 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 chi is:

$f(x, k) = \frac{1}{2^{k/2-1} \Gamma \left( k/2 \right)} x^{k-1} \exp \left( -x^2/2 \right)$

for $$x > 0$$ and $$k > 0$$ (degrees of freedom, denoted df in the implementation). $$\Gamma$$ is the gamma function (scipy.special.gamma).

Special cases of chi are:

chi takes df as a shape parameter.

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

Examples

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


Calculate a few first moments:

>>> df = 78
>>> mean, var, skew, kurt = chi.stats(df, moments='mvsk')


Display the probability density function (pdf):

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


Check accuracy of cdf and ppf:

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


Generate random numbers:

>>> r = chi.rvs(df, size=1000)


And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()


Methods

 rvs(df, loc=0, scale=1, size=1, random_state=None) Random variates. pdf(x, df, loc=0, scale=1) Probability density function. logpdf(x, df, loc=0, scale=1) Log of the probability density function. cdf(x, df, loc=0, scale=1) Cumulative distribution function. logcdf(x, df, loc=0, scale=1) Log of the cumulative distribution function. sf(x, df, loc=0, scale=1) Survival function (also defined as 1 - cdf, but sf is sometimes more accurate). logsf(x, df, loc=0, scale=1) Log of the survival function. ppf(q, df, loc=0, scale=1) Percent point function (inverse of cdf — percentiles). isf(q, df, loc=0, scale=1) Inverse survival function (inverse of sf). moment(n, df, loc=0, scale=1) Non-central moment of order n stats(df, loc=0, scale=1, moments=’mv’) Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’). entropy(df, loc=0, scale=1) (Differential) entropy of the RV. fit(data, df, loc=0, scale=1) Parameter estimates for generic data. expect(func, args=(df,), 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, loc=0, scale=1) Median of the distribution. mean(df, loc=0, scale=1) Mean of the distribution. var(df, loc=0, scale=1) Variance of the distribution. std(df, loc=0, scale=1) Standard deviation of the distribution. interval(alpha, df, loc=0, scale=1) Endpoints of the range that contains alpha percent of the distribution

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