Solution
= Solution
For times $0\leq t_1<\cdots<t_m$, the <vector> $(W_{t_1},\ldots,W_{t_m})$ is a linear transformation of the independent Gaussian increments
$$
W_{t_1}, W_{t_2}-W_{t_1},\ldots,W_{t_m}-W_{t_{m-1}},
$$
so Brownian motion is a Gaussian process. Its mean is zero, and for $s\leq t$,
$$
\boxed{\mathbb E(W_sW_t)=\mathbb E\{W_s(W_s+W_t-W_s)\}=s.}
$$