Problem 11

A stochastic process is defined by

$$X_t = \cos 2 \pi F t$$

where the frequency $F$ is uniformly distributed over the interval $[0,f_0]$.

1. Find the mean and correlation function of $X_t$.
2. Show that this process is non-stationary.

Now suppose we redefine the process $X_t$ to be

$$X_t = \cos(2 \pi F t + \Theta)$$

where $F$ and $\Theta$ are statistically independent random variables. $\Theta$ is uniformly disributed over $[-\pi,\pi]$ and $F$ is distributed as before.

1. Compute the mean and correlation function of $X_t$.
2. Is $X_t$ wide-sense stationary? Show your reasoning.
3. Find the first order density $p_{X_t}(x)$.