Solution
= Solution
The variable $Z=\int_0^1B_t\,dt$ is the $L^2$ limit of <Riemann sums> of the <Gaussian process> $B$, so it is a <normal distribution>[Gaussian random variable]. Its mean is zero, and <Fubini's theorem> with $\mathbb E[B_sB_t]=\min(s,t)$ gives
$$
\operatorname{Var}(Z)
=\int_0^1\!\int_0^1\min(s,t)\,ds\,dt
=2\int_0^1\!\int_0^t s\,ds\,dt
=\frac13.
$$
Therefore $Z\sim N(0,1/3)$.
Solved by gpt-5.6-sol high.