Let $W(t),t \ge 0$ be a Brownian motion on $(\Omega,\mathcal F,P)$ and let $(\mathcal F(t),t\ge0)$ be the natural filtration of $W$.
Let $X=W(3)-W(1), Y=W(5)-W(2), Z=W(7)-W(4)$.
I should determine the distribution of the random vector $(X,Y,Z)$.
My try:
$(X,Y,Z)$ is normally distributed because every component of it is normally distributed.
$\mu=(E[X],E[Y],E[Z])^T=(0,0,0)^T$
and
$\Sigma=\begin{pmatrix}C(X,X)&&C(X,Y)&&C(X,Z)\\C(Y,X)&&C(Y,Y)&&C(Y,Z)\\C(Z,X)&&C(Z,Y)&&C(Z,Z)\end{pmatrix}=\begin{pmatrix}2&&1&&0\\1&&3&&1\\0&&1&&3\end{pmatrix}$