scipy.stats.exponnorm = <scipy.stats._continuous_distns.exponnorm_gen object>[source]#

An exponentially modified Normal continuous random variable.

Also known as the exponentially modified Gaussian distribution [1].

As an instance of the rv_continuous class, exponnorm 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 exponnorm is:

\[f(x, K) = \frac{1}{2K} \exp\left(\frac{1}{2 K^2} - x / K \right) \text{erfc}\left(-\frac{x - 1/K}{\sqrt{2}}\right)\]

where \(x\) is a real number and \(K > 0\).

It can be thought of as the sum of a standard normal random variable and an independent exponentially distributed random variable with rate 1/K.

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

An alternative parameterization of this distribution (for example, in the Wikipedia article [1]) involves three parameters, \(\mu\), \(\lambda\) and \(\sigma\).

In the present parameterization this corresponds to having loc and scale equal to \(\mu\) and \(\sigma\), respectively, and shape parameter \(K = 1/(\sigma\lambda)\).

Added in version 0.16.0.


[1] (1,2)

Exponentially modified Gaussian distribution, Wikipedia,


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

Calculate the first four moments:

>>> K = 1.5
>>> mean, var, skew, kurt = exponnorm.stats(K, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = exponnorm.rvs(K, 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(K, loc=0, scale=1, size=1, random_state=None)

Random variates.

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

Probability density function.

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

Log of the probability density function.

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

Cumulative distribution function.

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

Log of the cumulative distribution function.

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

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

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

Log of the survival function.

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

Percent point function (inverse of cdf — percentiles).

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

Inverse survival function (inverse of sf).

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

Non-central moment of the specified order.

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

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

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

Median of the distribution.

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

Mean of the distribution.

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

Variance of the distribution.

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

Standard deviation of the distribution.

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

Confidence interval with equal areas around the median.