scipy.stats.levy#

scipy.stats.levy = <scipy.stats._continuous_distns.levy_gen object>[source]#

A Levy continuous random variable.

As an instance of the rv_continuous class, levy 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.

See also

levy_stable, levy_l

Notes

The probability density function for levy is:

\[f(x) = \frac{1}{\sqrt{2\pi x^3}} \exp\left(-\frac{1}{2x}\right)\]

for \(x > 0\).

This is the same as the Levy-stable distribution with \(a=1/2\) and \(b=1\).

The probability density above is defined in the “standardized” form. To shift and/or scale the distribution use the loc and scale parameters. Specifically, levy.pdf(x, loc, scale) is identically equivalent to levy.pdf(y) / 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 levy
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)

Calculate the first four moments:

>>> mean, var, skew, kurt = levy.stats(moments='mvsk')

Display the probability density function (pdf):

>>> # `levy` is very heavy-tailed.
>>> # To show a nice plot, let's cut off the upper 40 percent.
>>> a, b = levy.ppf(0), levy.ppf(0.6)
>>> x = np.linspace(a, b, 100)
>>> ax.plot(x, levy.pdf(x),
...        'r-', lw=5, alpha=0.6, label='levy 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 = levy()
>>> ax.plot(x, rv.pdf(x), 'k-', lw=2, label='frozen pdf')

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = levy.rvs(size=1000)

And compare the histogram:

>>> # manual binning to ignore the tail
>>> bins = np.concatenate((np.linspace(a, b, 20), [np.max(r)]))
>>> ax.hist(r, bins=bins, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.set_xlim([x[0], x[-1]])
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

Methods

rvs(loc=0, scale=1, size=1, random_state=None)	Random variates.
pdf(x, loc=0, scale=1)	Probability density function.
logpdf(x, loc=0, scale=1)	Log of the probability density function.
cdf(x, loc=0, scale=1)	Cumulative distribution function.
logcdf(x, loc=0, scale=1)	Log of the cumulative distribution function.
sf(x, loc=0, scale=1)	Survival function (also defined as `1 - cdf`, but sf is sometimes more accurate).
logsf(x, loc=0, scale=1)	Log of the survival function.
ppf(q, loc=0, scale=1)	Percent point function (inverse of `cdf` — percentiles).
isf(q, loc=0, scale=1)	Inverse survival function (inverse of `sf`).
moment(order, loc=0, scale=1)	Non-central moment of the specified order.
stats(loc=0, scale=1, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(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=(), 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(loc=0, scale=1)	Median of the distribution.
mean(loc=0, scale=1)	Mean of the distribution.
var(loc=0, scale=1)	Variance of the distribution.
std(loc=0, scale=1)	Standard deviation of the distribution.
interval(confidence, loc=0, scale=1)	Confidence interval with equal areas around the median.