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.


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.



Beta prime distribution, Wikipedia,


>>> 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))

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)


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.


Parameter estimates for generic data. See 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.