A chi-squared continuous random variable.
For the noncentral chi-square distribution, see
ncx2.
As an instance of the rv_continuous class,
chi2 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
chi2 is:
\[f[x, k] = \frac{1}{2^{k/2} \Gamma \left[ k/2 \right]} x^{k/2-1} \exp \left[ -x/2 \right]\]
for \[x > 0\] and \[k > 0\] [degrees of freedom, denoted df in the implementation].
chi2 takes df as a shape parameter.
The chi-squared distribution is a special case of the gamma distribution, with gamma parameters a = df/2, loc = 0 and scale = 2.
The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically,
chi2.pdf[x, df, loc, scale] is identically equivalent to chi2.pdf[y, df] / 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
>>> from scipy.stats import chi2 >>> import matplotlib.pyplot as plt >>> fig, ax = plt.subplots[1, 1]
Calculate the first four moments:
>>> df = 55 >>> mean, var, skew, kurt = chi2.stats[df, moments='mvsk']
Display the probability density function [pdf]:
>>> x = np.linspace[chi2.ppf[0.01, df], ... chi2.ppf[0.99, df], 100] >>> ax.plot[x, chi2.pdf[x, df], ... 'r-', lw=5, alpha=0.6, label='chi2 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 = chi2[df] >>> ax.plot[x, rv.pdf[x], 'k-', lw=2, label='frozen pdf']
Check accuracy of cdf and ppf:
>>> vals = chi2.ppf[[0.001, 0.5, 0.999], df] >>> np.allclose[[0.001, 0.5, 0.999], chi2.cdf[vals, df]] True
Generate random numbers:
>>> r = chi2.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 |
logsf[x, df, loc=0, scale=1] | Log of the survival function. |
ppf[q, df, loc=0, scale=1] | Percent point function [inverse of |
isf[q, df, loc=0, scale=1] | Inverse survival function [inverse of |
moment[order, df, loc=0, scale=1] | Non-central moment of the specified order. |
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] | Parameter estimates for generic data. See scipy.stats.rv_continuous.fit for detailed documentation of the keyword arguments. |
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[confidence, df, loc=0, scale=1] | Confidence interval with equal areas around the median. |