Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics and probability resit / 9 / c / i / Solution

Write $Z=U^a$. Since $a>0$, the map is increasing on $[0,1]$, and for $0⩽z⩽1$,Differentiation givesand zero outside this interval. Since $B(1/a,1)=a$, this is exactly the beta distribution probability density function with parameters $(1/a,1)$. The power transform of a uniform random variable therefore satisfies $U^a\sim \mathrm{Beta}(1/a,1)$; the second parameter does not depend on $a$.