Kumaraswamy Distribution - Relation To The Beta Distribution

Relation To The Beta Distribution

The Kuramaswamy distribution is closely related to Beta distribution. Assume that X_a,b is a Kumaraswamy distributed random variable with parameters a and b. Then X_a,b is the a-th root of a suitably defined Beta distributed random variable. More formally, Let Y_1,b denote a Beta distributed random variable with parameters and . One has the following relation between X_a,b and Y_1,b.

with equality in distribution.

$\operatorname{P}\{X_{a,b}\le x\}=\int_0^x ab t^{a-1}(1-t^a)^{b-1}dt= \int_0^{x^a} b(1-t)^{b-1}dt=\operatorname{P}\{Y_{1,b}\le x^a\} =\operatorname{P}\{Y^{1/a}_{1,b}\le x\} .$

One may introduce generalised Kuramaswamy distributions by considering random variables of the form, with and where denotes a Beta distributed random variable with parameters and . The raw moments of this generalized Kumaraswamy distribution are given by:

Note that we can reobtain the original moments setting, and . However, in general the cumulative distribution function does not have a closed form solution.

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