Random process Mean function and Correlation function

Before finding the mean and variance function, one has to fix the notation of $Z(t)$, which should be $$ Z(t)=\sum_{k=1}^{n}X_ke^{j(𝜔_0t+\Phi_k)}. $$ In order to compute the mean of $Z(t)$, it suffices to compute the expectation of $X_ke^{j(𝜔_0t+\Phi_k)}$. To do so, use the independence between $X_k$ and $\Phi_k$ and the fact that $X_k$ is centered.

For the correlation function, the most technical part is to compute $\operatorname{Cov}\left(Z(s),Z(t)\right)$ which actually reduces to compute $$ \sum_{k=1}^{n}\sum_{\ell=1}^n\mathbb E\left[X_kX_\ell e^{j(\omega_0s+\Phi_k)}e^{j(\omega_0t+\Phi_\ell)}\right] $$ (when you have to sums, it is always preferable to sum over different indices). If $k\neq\ell$, the corresponding term vanishes.