scipy.stats.gausshyper

scipy.stats.gausshyper = <scipy.stats._continuous_distns.gausshyper_gen object>

A Gauss hypergeometric continuous random variable.

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

\[f(x, a, b, c, z) = C x^{a-1} (1-x)^{b-1} (1+zx)^{-c}\]

for \(0 \le x \le 1\), \(a,b > 0\), \(c\) a real number, \(z > -1\), and \(C = \frac{1}{B(a, b) F[2, 1](c, a; a+b; -z)}\). \(F[2, 1]\) is the Gauss hypergeometric function scipy.special.hyp2f1.

gausshyper takes \(a\), \(b\), \(c\) and \(z\) as shape parameters.

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

References

1

Armero, C., and M. J. Bayarri. “Prior Assessments for Prediction in Queues.” Journal of the Royal Statistical Society. Series D (The Statistician) 43, no. 1 (1994): 139-53. doi:10.2307/2348939

Examples

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

Calculate the first four moments:

>>> a, b, c, z = 13.8, 3.12, 2.51, 5.18
>>> mean, var, skew, kurt = gausshyper.stats(a, b, c, z, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

>>> vals = gausshyper.ppf([0.001, 0.5, 0.999], a, b, c, z)
>>> np.allclose([0.001, 0.5, 0.999], gausshyper.cdf(vals, a, b, c, z))
True

Generate random numbers:

>>> r = gausshyper.rvs(a, b, c, z, size=1000)

And compare the histogram:

>>> ax.hist(r, density=True, histtype='stepfilled', alpha=0.2)
>>> ax.legend(loc='best', frameon=False)
>>> plt.show()

Methods

rvs(a, b, c, z, loc=0, scale=1, size=1, random_state=None)	Random variates.
pdf(x, a, b, c, z, loc=0, scale=1)	Probability density function.
logpdf(x, a, b, c, z, loc=0, scale=1)	Log of the probability density function.
cdf(x, a, b, c, z, loc=0, scale=1)	Cumulative distribution function.
logcdf(x, a, b, c, z, loc=0, scale=1)	Log of the cumulative distribution function.
sf(x, a, b, c, z, loc=0, scale=1)	Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).
logsf(x, a, b, c, z, loc=0, scale=1)	Log of the survival function.
ppf(q, a, b, c, z, loc=0, scale=1)	Percent point function (inverse of cdf — percentiles).
isf(q, a, b, c, z, loc=0, scale=1)	Inverse survival function (inverse of sf).
moment(order, a, b, c, z, loc=0, scale=1)	Non-central moment of the specified order.
stats(a, b, c, z, loc=0, scale=1, moments=’mv’)	Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).
entropy(a, b, c, z, 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=(a, b, c, z), 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(a, b, c, z, loc=0, scale=1)	Median of the distribution.
mean(a, b, c, z, loc=0, scale=1)	Mean of the distribution.
var(a, b, c, z, loc=0, scale=1)	Variance of the distribution.
std(a, b, c, z, loc=0, scale=1)	Standard deviation of the distribution.
interval(confidence, a, b, c, z, loc=0, scale=1)	Confidence interval with equal areas around the median.