scipy.stats.gennorm = <scipy.stats._continuous_distns.gennorm_gen object>[source]#

A generalized normal continuous random variable.

As an instance of the rv_continuous class, gennorm 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


Laplace distribution


normal distribution


The probability density function for gennorm is [1]:

\[f(x, \beta) = \frac{\beta}{2 \Gamma(1/\beta)} \exp(-|x|^\beta),\]

where \(x\) is a real number, \(\beta > 0\) and \(\Gamma\) is the gamma function (scipy.special.gamma).

gennorm takes beta as a shape parameter for \(\beta\). For \(\beta = 1\), it is identical to a Laplace distribution. For \(\beta = 2\), it is identical to a normal distribution (with scale=1/sqrt(2)).



“Generalized normal distribution, Version 1”,


Nardon, Martina, and Paolo Pianca. “Simulation techniques for generalized Gaussian densities.” Journal of Statistical Computation and Simulation 79.11 (2009): 1317-1329


Wicklin, Rick. “Simulate data from a generalized Gaussian distribution” in The DO Loop blog, September 21, 2016,


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

Calculate the first four moments:

>>> beta = 1.3
>>> mean, var, skew, kurt = gennorm.stats(beta, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = gennorm.rvs(beta, 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(beta, loc=0, scale=1, size=1, random_state=None)

Random variates.

pdf(x, beta, loc=0, scale=1)

Probability density function.

logpdf(x, beta, loc=0, scale=1)

Log of the probability density function.

cdf(x, beta, loc=0, scale=1)

Cumulative distribution function.

logcdf(x, beta, loc=0, scale=1)

Log of the cumulative distribution function.

sf(x, beta, loc=0, scale=1)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(x, beta, loc=0, scale=1)

Log of the survival function.

ppf(q, beta, loc=0, scale=1)

Percent point function (inverse of cdf — percentiles).

isf(q, beta, loc=0, scale=1)

Inverse survival function (inverse of sf).

moment(order, beta, loc=0, scale=1)

Non-central moment of the specified order.

stats(beta, loc=0, scale=1, moments=’mv’)

Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).

entropy(beta, 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=(beta,), 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(beta, loc=0, scale=1)

Median of the distribution.

mean(beta, loc=0, scale=1)

Mean of the distribution.

var(beta, loc=0, scale=1)

Variance of the distribution.

std(beta, loc=0, scale=1)

Standard deviation of the distribution.

interval(confidence, beta, loc=0, scale=1)

Confidence interval with equal areas around the median.