scipy.stats.moyal = <scipy.stats._continuous_distns.moyal_gen object>[source]#

A Moyal continuous random variable.

As an instance of the rv_continuous class, moyal 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.


The probability density function for moyal is:

\[f(x) = \exp(-(x + \exp(-x))/2) / \sqrt{2\pi}\]

for a real number \(x\).

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

This distribution has utility in high-energy physics and radiation detection. It describes the energy loss of a charged relativistic particle due to ionization of the medium [1]. It also provides an approximation for the Landau distribution. For an in depth description see [2]. For additional description, see [3].



J.E. Moyal, “XXX. Theory of ionization fluctuations”, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol 46, 263-280, (1955). DOI:10.1080/14786440308521076 (gated)


G. Cordeiro et al., “The beta Moyal: a useful skew distribution”, International Journal of Research and Reviews in Applied Sciences, vol 10, 171-192, (2012).


C. Walck, “Handbook on Statistical Distributions for Experimentalists; International Report SUF-PFY/96-01”, Chapter 26, University of Stockholm: Stockholm, Sweden, (2007).

Added in version 1.1.0.


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

Calculate the first four moments:

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

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = moyal.rvs(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)


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.


Parameter estimates for generic data. See 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.