Solution
= Solution
Both variables are centered. The <Itô isometry> in its bilinear form gives
$$
\operatorname{Cov}(B_t,\beta_t)=\mathbb E\!\left[\left(\int_0^t\operatorname{sign}(\beta_s)d\beta_s\right)\left(\int_0^t1\,d\beta_s\right)\right]=\int_0^t\mathbb E[\operatorname{sign}(\beta_s)]ds=0,
$$
where the last equality follows because a centered <Gaussian distribution> is a <symmetric probability distribution>. Thus $B_t$ and $\beta_t$ are <uncorrelated random variables>.