Problem 14

Let $X_t(\omega)$ be a stochastic process defined on $\Omega = \{\omega_1,\ldots,\omega_4\}$ having probabibility assignments $Pr\{\omega_i\}=\frac{1}{4}$ for i=1,2,3,4. The sample functions are

$$\begin{array} {ll} X_t(\omega_1) = t & X_t(\omega_2) = -t \\... ...ega_3) = \cos 2\pi t & X_t(\omega_4) = -\cos 2 \pi t\end{array}$$

1.

Compute the joint probability $Pr\{X_0(\omega)=1,X_1(\omega)=1\}$.

2.

Compute the conditional probability $Pr\{X_1(\omega)=1\vert X_0(\omega)=0\}$.

3.

Compute the mean and autocorrelation function of $X_t(\omega)$.

4.

Is this process stationary? wide-sense stationary?