I have a hard time trying to solve this exercise :
Suppose that we have the CTMC $X(t)$ at the stationnary state such that $$ Q_X = \begin{bmatrix} -\lambda & \lambda \\ \lambda & -\lambda \end{bmatrix} $$ We build the process $Y(t)$ like that $$Y(t) = \begin{cases} 0 &\text{ if } \; X(t) = 0 \\ A &\text{ if } \; X(t) = 1 \end{cases} $$ where $A \sim \mathcal{N}(0,1)$.
a) Find the density functionf of $Y$ : $f_{Y(t)}(y)$ for all $y \in\mathbb{R}$.
b) Find the $\mathbb{E}[Y(t)]$.
c) Find the aucorrelation function of $Y$ : $R_Y(\tau)$.
I think that b) could be solve with the density of $Y$ but how can we deduce the density of $Y$ ? Does anyone have a hint ?
Also I don't see how to solve c) even with the density. In fact the problem is really confusing for me, I'm not sure that I understand the problem well : if $X$ at time $t$ is at state $0$ then $Y$ takes value $0$, but if $X$ at time $t$ is at state $1$ then $Y$ is distributed with a normal distribution ?
Any help is welcome, thanks !