scipy.stats.invgauss#
- 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.Notes
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
takesmu
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
andscale
parameters. Specifically,invgauss.pdf(x, mu, loc, scale)
is identically equivalent toinvgauss.pdf(y, mu) / scale
withy = (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 \(\nu\) andlam
for \(\lambda\), this parameterization is equivalent to the one above withmu = nu/lam
,loc = 0
, andscale = lam
.Examples
>>> 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
andppf
:>>> vals = invgauss.ppf([0.001, 0.5, 0.999], mu) >>> np.allclose([0.001, 0.5, 0.999], invgauss.cdf(vals, mu)) True
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) >>> plt.show()
Methods
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.
fit(data)
Parameter estimates for generic data. See scipy.stats.rv_continuous.fit 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.