Power of Random Variable with Continuous Uniform Distribution has Beta Distribution
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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:
\(\ds \map \Pr {Y < x^n}\) | \(=\) | \(\ds \int_0^{x^n} \frac 1 {\map \Beta {\frac 1 n, 1} } u^{\frac 1 n - 1} \paren {1 - u}^{1 - 1} \rd u\) | Definition of Beta Distribution | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\map \Gamma {\frac 1 n + 1} } {\map \Gamma {\frac 1 n} \map \Gamma 1} \intlimits {n u^{\frac 1 n} } 0 {x^n}\) | Definition of Gamma Function, Primitive of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac n n x\) | Gamma Difference Equation | |||||||||||
\(\ds \) | \(=\) | \(\ds x\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 {1 - 0} \int_0^x \rd u\) | Primitive of Constant | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \Pr {X < x}\) | Definition of Continuous Uniform Distribution |
$\blacksquare$