Kumaraswamy Distribution#

Univariate, Continuous, Bounded

The Kumaraswamy distribution is a continuous probability distribution bounded between 0 and 1. It is characterized by two positive shape parameters: \(a\) and \(b\).

The Kumaraswamy distribution is a flexible distribution that can adopt a wide range of shapes, including uniform, U-shape, exponential-like, and many others, always restricted to the unit interval [0, 1].

Key properties and parameters:#

Support

\(x \in (0, 1)\)

Mean

\(b B(1 + \tfrac{1}{a}, b)\)

Variance

\(b B(1 + \tfrac{2}{a}, b) - (b B(1 + \tfrac{1}{a}, b))^2\)

Parameters:

  • \(a\) : (float) First shape parameter, \(a > 0\).

  • \(b\) : (float) Second shape parameter, \(b > 0\).

Probability Density Function (PDF)#

\[ f(x|a, b) = a b x^{a-1} (1 - x^{a})^{b-1} \]
/home/docs/checkouts/readthedocs.org/user_builds/preliz/envs/stable/lib/python3.11/site-packages/pytensor/link/numba/dispatch/basic.py:211: UserWarning: Numba will use object mode to run XlogY0's perform method. Set `pytensor.config.compiler_verbose = True` to see more details.
  warnings.warn(
../../_images/af8c8286a6b65ffe08087dac3753a8d18d0af49bc80c62133ebdadfee4882ad9.png

Cumulative Distribution Function (CDF)#

\[ F(x|a, b) = 1 - (1 - x^{a})^{b} \]
../../_images/2af827401e0b45e649df178fcae696fc81eb79b3559387003b41dc558344f8af.png

See also

Common Alternatives:

  • Beta - The Kumaraswamy distribution is similar to the Beta distribution, but with closed-form expressions for its probability density function, cumulative distribution function and quantile function.

Related Distributions:

  • Uniform - The Uniform distribution on the interval [0, 1] is a special case of the Kumaraswamy distribution with \(a = b = 1\).

References#