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)#

\[ f(x|a, b) = a b x^{a-1} (1 - x^{a})^{b-1} \]
../../_images/d5648f0db29b23af900701e7b88b7254394a78136394347568c5271e2d0765f5.png

Cumulative Distribution Function (CDF)#

\[ F(x|a, b) = 1 - (1 - x^{a})^{b} \]
../../_images/af47563ee399fb041bf7782942320a1bdc83e45da6dd51d2b6e37f20ce871bee.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#