Random process $X(t) = 10 \cos(Wt + A)$.

Question

I am doing some exercises based on random process, but I can't find a way out on this:

Let $X(t) = 10 \cos(Wt + A)$, where W is a Gaussian aleatory variable with parameters $N(10,2)$ and $A$ is uniform under $(0, 2\pi)$. Assuming that $W$ and $A$ are independent, determine if the process is WSS and/or ergodic.

Well, I need to calculate $E[X(t)]$ and the autocorrelation function, but how can I use the information that $W$ and $A$ are independent?

Thanks in advance!

Answer 1

I would write it out explicitly. For example: $$ E[X_t] = \int_{w=-\infty}^\infty \int_{a=0}^{2\pi} 10\cos(wt+a) \frac{1}{2\pi} \frac1{2\sqrt{2\pi}} \exp(-(w-10)^2/4) \, da \, dw .$$ If you do the integral over $a$ first, this one is rather obvious. I think the autocorrelation function should be only slightly harder to compute.