scipy.stats.chi2=<scipy.stats._continuous_distns.chi2_gen object>[source]#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 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(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.
|