Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 201 / 5 / c / Solution

For fixed $s\in (0,1)$, uniqueness givesThe two sides of the comparison are independent and distributed as $\sqrt{s}|Z_1|$ and $\sqrt{1-s}|Z_2|$ for independent standard normal random variables. Rotational invariance of $(Z_1,Z_2)$ makes its angle uniform, soThe endpoint values follow by continuity. Thus the time of the Brownian maximum has the arcsine distribution.