= Power transform of a uniform random variable
{title2=$U\sim\operatorname{Uniform}(0,1),\ a>0\ \Longrightarrow\ U^a\sim\operatorname{Beta}(1/a,1)$}
The <cumulative distribution function> of $U^a$ on $[0,1]$ is $z^{1/a}$. Differentiating gives density $z^{1/a-1}/a$ on $(0,1)$, which is the <beta distribution> <probability density function> with parameters $(1/a,1)$. This also gives a simple instance of <inverse transform sampling>: to sample a density $\alpha z^{\alpha-1}$ on $(0,1)$, raise a uniform draw to the power $1/\alpha$.
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