If I have a signal of the form $x\left(n\right)=Acos\left(nω+ϕ\right)$
where
$\omega \in \left[\omega -\lambda ,\omega \:+\lambda \:\right]$ is a uniform random variable and
$\phi $ is also a uniform random variable $\left[-\pi ,\pi \right]$.
I need to find the mean and the autocorrelation.
So we know the mean $E\left[x\left(n\right)\right]\:=\:AE\left[cos\left(n\omega +\phi \right)\right]$.
I think the $E\left[x\left(n\right)\right]\:=0$ regardless of the random variables just because of the periodicity of the cosine sequence.
Now I would also like to find the auto-correlation function
$R\left(n,k\right)=\:E\left[x\left(n\right)x\left(n+k\right)\right]=A^2E\left[cos\left(n\omega +\phi \right)cos\left(\left(n+k\right)\omega \:+\phi \:\right)\right]=\frac{A}{2}^2\left\{E\left[cos\left(2\left(n\omega +\phi \right)+k\omega \right)\right]+E\left[cos\left(k\omega \right)\right]\right\}=\frac{A}{2}^2cos\left(k\omega \right)$
I am not sure if am solving this the right way, can somebody please tell me if am doing something wrong? because I think the random variables must have a role in this