This distribution was originally proposed by Poondi Kumaraswamy[1] for variables that are lower and upper bounded with a zero-inflation.

In this first article of the distribution, the natural lower bound of zero for rainfall was modelled using a discrete probability, as rainfall in many places, especially in tropics, has significant nonzero probability.

This was extended to inflations at both extremes [0,1] in the work of Fletcher and Ponnambalam.[2].

The probability density function of the Kumaraswamy distribution without considering any inflation is and where a and b are non-negative shape parameters.

In a more general form, the normalized variable x is replaced with the unshifted and unscaled variable z where: The raw moments of the Kumaraswamy distribution are given by:[3][4] where B is the Beta function and Γ(.)

The variance, skewness, and excess kurtosis can be calculated from these raw moments.

[6] Assume that Xa,b is a Kumaraswamy distributed random variable with parameters a and b.

Then Xa,b is the a-th root of a suitably defined Beta distributed random variable.

More formally, Let Y1,b denote a Beta distributed random variable with parameters

One may introduce generalised Kumaraswamy distributions by considering random variables of the form

denotes a Beta distributed random variable with parameters

However, in general, the cumulative distribution function does not have a closed form solution.