Kumaraswamy Distribution Variance Calculator

For the Kumaraswamy distribution with parameters \(a>0\) and \(b>0\), the \(k\)th moment is $$ E[X^k] = b\,\frac{\Gamma\Bigl(1+\frac{k}{a}\Bigr)\,\Gamma(b)}{\Gamma\Bigl(1+\frac{k}{a}+b\Bigr)}. $$ In particular, the mean is $$ \mu = b\,\frac{\Gamma\Bigl(1+\frac{1}{a}\Bigr)\,\Gamma(b)}{\Gamma\Bigl(1+\frac{1}{a}+b\Bigr)}, $$ and the variance is $$ \operatorname{Var}(X) = E[X^2] – \mu^2, $$ where $$ E[X^2] = b\,\frac{\Gamma\Bigl(1+\frac{2}{a}\Bigr)\,\Gamma(b)}{\Gamma\Bigl(1+\frac{2}{a}+b\Bigr)}. $$

* Enter parameters \(a > 0\) and \(b > 0\). The PDF is $$ f(x;a,b)= a\,b\,x^{a-1}\Bigl(1-x^a\Bigr)^{b-1},\quad 0\le x\le 1. $$