Suppose $Z(t)={\Xi \pounds}_{k=1}^{n}X{e}^{j({\pi \x9d\x9c\x94}_{0}t+{\pi \x9d\x9a\xbd}_{k})}$, $t\beta \x88\x88R$ where ${\pi \x9d\x9c\x94}_{0}$ is a constant, n is a fixed positive integer, ${X}_{1},...,{X}_{n},\gamma \x80\x80{\pi \x9d\x9a\xbd}_{1},...,{\pi \x9d\x9a\xbd}_{n}$ are mutually independent random variables, and $E{X}_{k}=0,D{X}_{k}={{\rm O}\x83}_{k}^{2},\pi \x9d\x9a\xbd$ , $U[0,2{\rm O}\x80],k=1,2,...,n$ . Find the mean function and correlation function of $\{Z(t),\text{\Beta}t\beta \x88\x88R\}$ .

I have tried to solve it.

For mean function,

${m}_{Z}(s)=E\{{Z}_{s}\}=E\{{X}_{s}\}+iE\{{Y}_{t}\}$

$=E\{{\Xi \pounds}_{k=1}^{s}X{e}^{j({\pi \x9d\x9c\x94}_{0}t+{\pi \x9d\x9a\xbd}_{k})}\}$

For correlation function,

${R}_{Z}(s,u)=E\{{Z}_{s},{Z}_{u}\}$

$=E\{Y(s)Y(u)\}$

$=E\{{\Xi \pounds}_{k=1}^{s}X{e}^{j({\pi \x9d\x9c\x94}_{0}t+{\pi \x9d\x9a\xbd}_{k})}{\Xi \pounds}_{k=1}^{u}X{e}^{j({\pi \x9d\x9c\x94}_{0}t+{\pi \x9d\x9a\xbd}_{k})}\}$

$=E\{{\Xi \pounds}_{k=1}^{s}{\Xi \pounds}_{k=1}^{u}X{e}^{j({\pi \x9d\x9c\x94}_{0}t+{\pi \x9d\x9a\xbd}_{k})}X{e}^{j({\pi \x9d\x9c\x94}_{0}t+{\pi \x9d\x9a\xbd}_{k})}\}$

I am stuck here. How to move from here ahead?

I have tried to solve it.

For mean function,

${m}_{Z}(s)=E\{{Z}_{s}\}=E\{{X}_{s}\}+iE\{{Y}_{t}\}$

$=E\{{\Xi \pounds}_{k=1}^{s}X{e}^{j({\pi \x9d\x9c\x94}_{0}t+{\pi \x9d\x9a\xbd}_{k})}\}$

For correlation function,

${R}_{Z}(s,u)=E\{{Z}_{s},{Z}_{u}\}$

$=E\{Y(s)Y(u)\}$

$=E\{{\Xi \pounds}_{k=1}^{s}X{e}^{j({\pi \x9d\x9c\x94}_{0}t+{\pi \x9d\x9a\xbd}_{k})}{\Xi \pounds}_{k=1}^{u}X{e}^{j({\pi \x9d\x9c\x94}_{0}t+{\pi \x9d\x9a\xbd}_{k})}\}$

$=E\{{\Xi \pounds}_{k=1}^{s}{\Xi \pounds}_{k=1}^{u}X{e}^{j({\pi \x9d\x9c\x94}_{0}t+{\pi \x9d\x9a\xbd}_{k})}X{e}^{j({\pi \x9d\x9c\x94}_{0}t+{\pi \x9d\x9a\xbd}_{k})}\}$

I am stuck here. How to move from here ahead?