# scipy.stats.betaprime#

scipy.stats.betaprime = <scipy.stats._continuous_distns.betaprime_gen object>[source]#

A beta prime continuous random variable.

As an instance of the rv_continuous class, betaprime 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.

Methods

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

Notes

The probability density function for betaprime is:

$f(x, a, b) = \frac{x^{a-1} (1+x)^{-a-b}}{\beta(a, b)}$

for $$x >= 0$$, $$a > 0$$, $$b > 0$$, where $$\beta(a, b)$$ is the beta function (see scipy.special.beta).

betaprime takes a and b as shape parameters.

The distribution is related to the beta distribution as follows: If $$X$$ follows a beta distribution with parameters $$a, b$$, then $$Y = X/(1-X)$$ has a beta prime distribution with parameters $$a, b$$ ([1]).

The beta prime distribution is a reparametrized version of the F distribution. The beta prime distribution with shape parameters a and b and scale = s is equivalent to the F distribution with parameters d1 = 2*a, d2 = 2*b and scale = (a/b)*s. For example,

>>> from scipy.stats import betaprime, f
>>> x = [1, 2, 5, 10]
>>> a = 12
>>> b = 5
>>> betaprime.pdf(x, a, b, scale=2)
array([0.00541179, 0.08331299, 0.14669185, 0.03150079])
>>> f.pdf(x, 2*a, 2*b, scale=(a/b)*2)
array([0.00541179, 0.08331299, 0.14669185, 0.03150079])


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

Beta prime distribution, Wikipedia, https://en.wikipedia.org/wiki/Beta_prime_distribution

Examples

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


Calculate the first four moments:

>>> a, b = 5, 6
>>> mean, var, skew, kurt = betaprime.stats(a, b, moments='mvsk')


Display the probability density function (pdf):

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


Check accuracy of cdf and ppf:

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


Generate random numbers:

>>> r = betaprime.rvs(a, b, 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()