Past exam of the mathematics course of the University of Cambridge / 2025 / ia / Paper 2 / 12F / b / Solution

For $0<x,y<1$, set $a=x^{-1/n}-1$ and $b=y^{-1/n}-1$. Conditional on $S$,Using part (a),Differentiation givesPutting $y=1$ yields $F_{X_1}(x)=x$, so $X_1$ is uniform on $(0,1)$.