Let $X(t) = \begin{bmatrix} cos(t) + N(t)\\ sin(t) + S(t)\\ \end{bmatrix} $ (where $N(t)$ is a gaussian process and S(t) is a Poisson's process).
Let $Y(t)=\begin{bmatrix} 2 & 4\\ 6 & 8\\ \end{bmatrix} X(t)$. Evaluate $E[Y(t)]$ and $R_y(t_1,t_2)$.
This is an extra exercise, but out of curiosity I would like to know how to solve it.
First I evaluated $Y(t)$
$Y(t)=\begin{bmatrix} 2cos(t) + 4sin(t) +2N(t)+4S(t)\\ 6cos(t) + 8sin(t) +6N(t)+8S(t) \\ \end{bmatrix}$
My question is how do I evaluate $E[Y(t)]$. Should I evaluate $E[Y_1(t)]$ and $E[Y_2(t)]$ separately?
Also what does $R_y(t_1,t_2)$? Is that the covariance function (cause this exercise has different notation from what I've seen)?
