Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 4 / b

The variable $Z={\int }_0^1{B}_{t}dt$ is the $L^2$ limit of Riemann sums of the Gaussian process $B$, so it is a Gaussian random variable. Its mean is zero, and Fubini's theorem with $\mathbb{E}[B_s{B}_t]=\min (s,t)$ givesTherefore $Z\sim N(0,1/3)$.

Solved by gpt-5.6-sol high.