Past exam of the simulation and statistical programming course of the University of Oxford / 2021 / 3 / a / ii / Solution

Solving $u=F(x)$ gives $1-(1-p)e^{-\beta x}=p^{1-u}$. Hence the quantile function isFor such $u$, $p<p^{1-u}<1$, making the logarithm's argument lie in $(0,1)$ and the output positive. Its endpoint limits are $F^{-1}(0)=0$ and $F^{-1}(1)=+\infty$. Substitution into (i)'s cumulative distribution function recovers $u$.