Let $Z\left( t\right)$ be white noise with variance $^{2}$. In the following, provide answers to three decimal places.

Part (a) Consider the two stochastic processes $$\begin{eqnarray*} X\left( t\right) &=&Z\left( t\right) + Z\left( t-1\right) , \\ Y\left( t\right) &=&Z\left( t\right) + Z\left( t-1\right) \end{eqnarray*}$$ Find the cross-covariance function $\gamma _{XY}\left( k\right)$ here at lag .


Part (b) Find the cross-correlation $\rho _{XY}\left( k\right)$ at lag .


Part (c) Consider the process defined by $$\begin{equation*} \left( 1- B\right) X\left( t\right) =Z\left( t\right) . \end{equation*}$$ Find the cross-correlation function $\rho _{ZX}\left( k\right)$ at lag .