Doubt about wide sense stationary random process
The mean of $X(n)$ is always $\mu_X =0$, because the noise has zero mean. Thus we should check whether the autocorrelation corresponds to a WSS process.

$$\begin{align}
R_X(n_1,n_2)
&amp;=\mathbb{E}[X(n_1)X(n_2)]\\
&amp;=\mathbb{E}[u(n_1)e^{-kn_1}F(n_1)u(n_2)e^{-kn_2}F(n_2)]\\
&amp;=u(n_1)u(n_2)e^{-k(n_1+n_2)}\mathbb{E}[F(n_1)F(n_2)]\\
&amp;=u(n_1)u(n_2)e^{-k(n_1+n_2)}\delta(n_1-n_2)
\end{align}$$

We can evaluate the autocorrelation function in two different cases:


If $n_1-n_2\neq0$, then $$R_X(n_1,n_2)=0$$
If $n_1-n_2=0$, then $$R_X(n_1,n_2)=u(n_1)e^{-2kn_1}=u(n_2)e^{-2kn_2}$$


The second case shows that $R_X$ depends on the specific values of $n_1$ or $n_2$ and not just their difference, thus the process is not WSS.