$U \sim \mathrm {Unif} (0,1)$. Let $\alpha > 0$. Then find the density function of $X=U^{-\frac 1 {\alpha}}$. I have found that if $F$ is the cumulative distribution function of the random variable $X$ then $$ F(x) = \left\{ \begin{array}{ll} 0 & \quad \frac {1} {x^{\alpha}} < 0 \\ 1 - x^{-\alpha} & \quad 0 < \frac 1 {x^{\alpha}} < 1 \\ 1 & \quad \frac {1} {x^{\alpha}} > 1 \end{array} \right. $$
Can it be simplified more? Please help me in this regard.
Thank you very much.