scipy.stats.exponnorm(*args, **kwds) = <scipy.stats._continuous_distns.exponnorm_gen object>[source]

An exponentially modified Normal continuous random variable.

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.

An alternative parameterization of this distribution (for example, in Wikipedia) 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 \(K = 1/(\sigma\lambda)\).

New in version 0.16.0.


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

Calculate a few first 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, histtype='stepfilled', alpha=0.2)
>>> 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(n, K, loc=0, scale=1)

Non-central moment of order n

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

Endpoints of the range that contains alpha percent of the distribution

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