scipy.stats.gausshyper#
- scipy.stats.gausshyper = <scipy.stats._continuous_distns.gausshyper_gen object>[source]#
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
andscale
parameters. Specifically,gausshyper.pdf(x, a, b, c, z, loc, scale)
is identically equivalent togausshyper.pdf(y, a, b, c, z) / 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.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
>>> import numpy as np >>> 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
andppf
:>>> 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, bins='auto', histtype='stepfilled', alpha=0.2) >>> ax.set_xlim([x[0], x[-1]]) >>> 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.