Let $ .., X_{-1}, X_0, X_{1}, ..$ and $.., Y_{-1}, Y_0, Y_{1}, ..$ be two i.i.d processes(individually).
We construct the processes Z, W by fair coin flips where the result is $R \in \{0, 1\}$.
For Z, the coin is flipped once and either X or Y is selected. So, $Z_n = RX_n + (1-R)Y_n$
For W, the coin is flipped at every time n. Hence, $W_n = R_nX_n + (1-R_n)Y_n$, where $R_n$ is the result of the coin flip at time n.
Now I'm being asked if $Z$ and $W$ are i.i.d and stationary processes.
My initial thought was that since Z is determined after the coin flip, it is either $X$ or $Y$ during the process. Thus, i.i.d and stationary. Also, $W$ alternates between $X$ and $Y$ hence it is nor i.i.d and stationary.
However, the answer says the opposite. What is wrong in my reasoning ?