Given

$$ Y(t) = A X(t) \cos(\omega t + \phi) $$

with $X(t)$ is zero-mean WSS (wide-sense stationary) process, $\phi$ ~ Unif$(0,2\pi)$. Suppose $X(t)$ and $\phi$ are independent random variables. I want to compute the mean and autocorrelation functions of $Y(t)$.

My attempt is (using independence of $X(t)$ and $\phi$, and $E(X(t)) = 0$)

$$ E(Y(t))= E(A X(t) \cos(\omega t + \phi)) = A \cdot E(X(t)) \cdot E(\cos(\omega t + \phi)) = 0$$

Is this a correct use of independence? If not, why?

Given

$$ Y(t) = A X(t) \cos(\omega t + \phi) $$

with $X(t)$ is zero-mean WSS (wide-sense stationary) process, $\phi$ ~ Unif$(0,2\pi)$. Suppose $X(t)$ and $\phi$ are independent random variables. I want to compute the mean and autocorrelation functions of $Y(t)$.

My attempt is (using independence of $X(t)$ and $\phi$, and $E(X(t)) = 0$)

$$ E(Y(t))= E(A X(t) \cos(\omega t + \phi)) = A \cdot E(X(t)) \cdot E(\cos(\omega t + \phi)) = 0$$

Similarly we have

$$ R_Y(t,t+\tau)= E(A X(t) \cos(\omega t + \phi) \cdot A X(t+\tau) \cos(\omega (t + \tau) + \phi)) = 0$$

Is this a correct use of independence? If not, why?