scipy.stats.burr = <scipy.stats._continuous_distns.burr_gen object>[source]#

A Burr (Type III) continuous random variable.

As an instance of the rv_continuous class, burr 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.

See also


a special case of either burr or burr12 with d=1


Burr Type XII distribution


Mielke Beta-Kappa / Dagum distribution


The probability density function for burr is:

\[f(x; c, d) = c d \frac{x^{-c - 1}} {{(1 + x^{-c})}^{d + 1}}\]

for \(x >= 0\) and \(c, d > 0\).

burr takes c and d as shape parameters for \(c\) and \(d\).

This is the PDF corresponding to the third CDF given in Burr’s list; specifically, it is equation (11) in Burr’s paper [1]. The distribution is also commonly referred to as the Dagum distribution [2]. If the parameter \(c < 1\) then the mean of the distribution does not exist and if \(c < 2\) the variance does not exist [2]. The PDF is finite at the left endpoint \(x = 0\) if \(c * d >= 1\).

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



Burr, I. W. “Cumulative frequency functions”, Annals of Mathematical Statistics, 13(2), pp 215-232 (1942).


Kleiber, Christian. “A guide to the Dagum distributions.” Modeling Income Distributions and Lorenz Curves pp 97-117 (2008).


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

Calculate the first four moments:

>>> c, d = 10.5, 4.3
>>> mean, var, skew, kurt = burr.stats(c, d, moments='mvsk')

Display the probability density function (pdf):

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

Check accuracy of cdf and ppf:

>>> vals = burr.ppf([0.001, 0.5, 0.999], c, d)
>>> np.allclose([0.001, 0.5, 0.999], burr.cdf(vals, c, d))

Generate random numbers:

>>> r = burr.rvs(c, d, 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(c, d, loc=0, scale=1, size=1, random_state=None)

Random variates.

pdf(x, c, d, loc=0, scale=1)

Probability density function.

logpdf(x, c, d, loc=0, scale=1)

Log of the probability density function.

cdf(x, c, d, loc=0, scale=1)

Cumulative distribution function.

logcdf(x, c, d, loc=0, scale=1)

Log of the cumulative distribution function.

sf(x, c, d, loc=0, scale=1)

Survival function (also defined as 1 - cdf, but sf is sometimes more accurate).

logsf(x, c, d, loc=0, scale=1)

Log of the survival function.

ppf(q, c, d, loc=0, scale=1)

Percent point function (inverse of cdf — percentiles).

isf(q, c, d, loc=0, scale=1)

Inverse survival function (inverse of sf).

moment(order, c, d, loc=0, scale=1)

Non-central moment of the specified order.

stats(c, d, loc=0, scale=1, moments=’mv’)

Mean(‘m’), variance(‘v’), skew(‘s’), and/or kurtosis(‘k’).

entropy(c, d, 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=(c, d), 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(c, d, loc=0, scale=1)

Median of the distribution.

mean(c, d, loc=0, scale=1)

Mean of the distribution.

var(c, d, loc=0, scale=1)

Variance of the distribution.

std(c, d, loc=0, scale=1)

Standard deviation of the distribution.

interval(confidence, c, d, loc=0, scale=1)

Confidence interval with equal areas around the median.