I have this problem. Consider a random process X(t) defined by $X(t) = 2\cos(2 \pi t+ \phi)$. $\phi$ is a random variable with probability density function $f_\phi (\varphi ) = \left\{\begin{matrix} \frac{4}{\pi}, \quad |\varphi | \leq \frac{\pi}{8} \\ 0, \quad \mathrm{otherwise}. \end{matrix}\right.$
Find the mean and autocorrelation functions.
- Normally when I have a problem like this, the information about the values for which the phase is changing randomly, e.g., from $-\pi$ to $\pi$. And then i use the formula (the random process is $\cos(..)$ in this example) $\int_{-\pi}^{\pi} \cos(...) \frac{1}{2 \pi} dx$ and afterward i calculate the autocorrelation by calculating $E[\cos(t_1...)\cos(t_2\dots)]$. In this problem, the density function is given instead. I know that i should check for if the mean is constant for all $t$. Can I say the mean is $\frac{1}{2 \pi}$ for all $t$? And if that is correct, how should I calculate the autocrorrealtion?