Kumaraswamy Distribution
X ~ Kumaraswamy(a, b)
Unit-Interval Family • Published July 22, 2026
The already-published Beta distribution's closed-form cousin: same two-shape-parameter flexibility on [0,1], but with an elementary CDF F(x)=1-(1-x^a)^b that inverts in closed form with no incomplete Beta function needed, making it the standard substitute for Beta in simulation studies and hydrological models, verified via exact agreement with numerical differentiation and a 5-million-draw simulation, no closed-form MLE despite the closed-form CDF, and simulation via a genuinely closed-form inverse-transform construction (no rejection sampling required, unlike Beta).
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About This Reference Sheet
The already-published Beta distribution's closed-form cousin: same two-shape-parameter flexibility on [0,1], but with an elementary CDF F(x)=1-(1-x^a)^b that inverts in closed form with no incomplete Beta function needed, making it the standard substitute for Beta in simulation studies and hydrological models, verified via exact agreement with numerical differentiation and a 5-million-draw simulation, no closed-form MLE despite the closed-form CDF, and simulation via a genuinely closed-form inverse-transform construction (no rejection sampling required, unlike Beta).
Support
x ∈ [0, 1]
Parameters
a > 0, b > 0 (shape parameters)