Power of Random Variable with Continuous Uniform Distribution has Beta Distribution

Theorem
Let $X \sim \ContinuousUniform 0 1$ where $\ContinuousUniform 0 1$ is the continuous uniform distribution on $\closedint 0 1$.

Let $n$ be a positive real number.

Then:


 * $X^n \sim \BetaDist {\dfrac 1 n} 1$

where $\operatorname {Beta}$ is the beta distribution.

Proof
Let:


 * $Y \sim \BetaDist {\dfrac 1 n} 1$

We aim to show that:


 * $\map \Pr {Y < x^n} = \map \Pr {X < x}$

for all $x \in \closedint 0 1$.

We have: