Let $X\left( t\right)$ and $Y\left( t\right)$ be stationary processes, with means $\mu _{X}$ and $\mu _{Y}$ respectively.
Part (a) Define the cross covariance function, $\gamma_{XY}(k)$, at lag $k$.


Part (b) Could $\gamma_{XY}(k)$ depend on t as well as k here?

A. No.
B. Yes.


Part (c) Suppose $$\begin{eqnarray*} X\left( t\right) &=&Z_{1}\left( t\right) +0.3Z_{1}\left( t-1\right) +0.2Z_{2}\left( t-1\right) \\ Y\left( t\right) &=&Z_{2}\left( t\right) +0.7Z_{1}\left( t-1\right) +0.4Z_{2}\left( t-1\right) \end{eqnarray*}$$ where $Z_{1}\left( t\right)$ and$Z_{2}\left( t\right)$ are independent white noise processes, each with mean zero and variance $\sigma ^{2}.$
Find $\gamma_{XY}(k)$ here.


Find $\gamma_{YX}(k)$ here.