scipy.stats.invgauss = <scipy.stats._continuous_distns.invgauss_gen object>[source]#

An inverse Gaussian continuous random variable.

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

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

for \(x \ge 0\) and \(\mu > 0\).

invgauss takes mu as a shape parameter for \(\mu\).

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

A common shape-scale parameterization of the inverse Gaussian distribution has density

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

Using nu for \(\nu\) and lam for \(\lambda\), this parameterization is equivalent to the one above with mu = nu/lam, loc = 0, and scale = lam.


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

Calculate the first four moments:

>>> mu = 0.145
>>> mean, var, skew, kurt = invgauss.stats(mu, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

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

Generate random numbers:

>>> r = invgauss.rvs(mu, 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(mu, loc=0, scale=1, size=1, random_state=None)

Random variates.

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

Probability density function.

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

Log of the probability density function.

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

Cumulative distribution function.

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

Log of the cumulative distribution function.

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

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

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

Log of the survival function.

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

Percent point function (inverse of cdf — percentiles).

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

Inverse survival function (inverse of sf).

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

Non-central moment of the specified order.

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

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

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

Median of the distribution.

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

Mean of the distribution.

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

Variance of the distribution.

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

Standard deviation of the distribution.

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

Confidence interval with equal areas around the median.