Find the distribution of $X$.

Question

$U \sim \mathrm {Unif} (0,1)$. 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.

Answer 1

The correct answer is this: $F(x)=0$ for $x<1$ and $F(x)=1-x^{-\alpha}$ for $x \geq 1$. The density is given by $f(x)=0$ for $x<1$ and $f(x)=\alpha x^{-\alpha -1}$ for $x >1$