Consider the sinusoidal process $\langle X(t), t \geq 0 \rangle$, where $X(t) = \xi \cos \omega t$, $\omega$ is a constant, and $\xi {\sim} U(0, 1)$. Find the probability density $f(t, x)$ for $t = \frac{\pi}{4 \omega}, \frac{\pi}{2 \omega}, \frac{3 \pi}{4 \omega}, \frac{\pi}{\omega}$, $\{X(t)\}$ is ().

Options:
A. $$f\Big(\frac{\pi}{4\omega},x\Big)=\begin{cases}0.6,&x\in\Big(0,\frac{\sqrt{2}}{4}\Big)\\\\0,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{2\omega},x\Big)=1.1$$
$$f\Big(\frac{3\pi}{4\omega},x\Big)=\begin{cases}0.6\:,&x\in\Big(-\frac{\sqrt{2}}{4},0\Big)\\\\0\:,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{\omega},x\Big)=\begin{cases}1.1,&x\in(-1.2,0)\\0,&\text{otherwise}\end{cases}$$
B. $$f\Big(\frac{\pi}{4\omega},x\Big)=\begin{cases}1.3,&x\in\Big(0,\frac{\sqrt{2}}{6}\Big)\\\\0,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{2\omega},x\Big)=0.4$$
$$f\Big(\frac{3\pi}{4\omega},x\Big)=\begin{cases}1.3\:,&x\in\Big(-\frac{\sqrt{2}}{6},0\Big)\\\\0\:,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{\omega},x\Big)=\begin{cases}0.4,&x\in(-0.7,0)\\0,&\text{otherwise}\end{cases}$$
C. $$f\Big(\frac{\pi}{4\omega},x\Big)=\begin{cases}\sqrt{2},&x\in\Big(0,\frac{\sqrt{2}}{2}\Big)\\\\0,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{2\omega},x\Big)=1$$
$$f\Big(\frac{3\pi}{4\omega},x\Big)=\begin{cases}\sqrt{2}\:,&x\in\Big(-\frac{\sqrt{2}}{2},0\Big)\\\\0\:,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{\omega},x\Big)=\begin{cases}1,&x\in(-1,0)\\0,&\text{otherwise}\end{cases}$$
D. $$f\Big(\frac{\pi}{4\omega},x\Big)=\begin{cases}0.8,&x\in\Big(0,\frac{1}{3}\Big)\\\\0,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{2\omega},x\Big)=1.2$$
$$f\Big(\frac{3\pi}{4\omega},x\Big)=\begin{cases}0.8\:,&x\in\Big(-\frac{1}{3},0\Big)\\\\0\:,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{\omega},x\Big)=\begin{cases}1.2,&x\in(-0.8,0)\\0,&\text{otherwise}\end{cases}$$
E. $$f\Big(\frac{\pi}{4\omega},x\Big)=\begin{cases}1.5,&x\in\Big(0,\frac{3}{4}\Big)\\\\0,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{2\omega},x\Big)=0.5$$
$$f\Big(\frac{3\pi}{4\omega},x\Big)=\begin{cases}1.5\:,&x\in\Big(-\frac{3}{4},0\Big)\\\\0\:,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{\omega},x\Big)=\begin{cases}0.5,&x\in(-1,0)\\0,&\text{otherwise}\end{cases}$$
F. $$f\Big(\frac{\pi}{4\omega},x\Big)=\begin{cases}1,&x\in\Big(0,\frac{0.5}{\sqrt{2}}\Big)\\\\0,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{2\omega},x\Big)=0.9$$
$$f\Big(\frac{3\pi}{4\omega},x\Big)=\begin{cases}1\:,&x\in\Big(-\frac{0.5}{\sqrt{2}},0\Big)\\\\0\:,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{\omega},x\Big)=\begin{cases}0.9,&x\in(-1,0.5)\\0,&\text{otherwise}\end{cases}$$
G. $$f\Big(\frac{\pi}{4\omega},x\Big)=\begin{cases}2,&x\in\Big(0,\frac{\sqrt{2}}{3}\Big)\\\\0,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{2\omega},x\Big)=0$$
$$f\Big(\frac{3\pi}{4\omega},x\Big)=\begin{cases}2\:,&x\in\Big(-\frac{\sqrt{2}}{3},0\Big)\\\\0\:,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{\omega},x\Big)=\begin{cases}0,&x\in(0,1)\\1,&\text{otherwise}\end{cases}$$
H. $$f\Big(\frac{\pi}{4\omega},x\Big)=\begin{cases}1.2,&x\in\Big(0,\frac{\sqrt{2}}{8}\Big)\\\\0,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{2\omega},x\Big)=0.3$$
$$f\Big(\frac{3\pi}{4\omega},x\Big)=\begin{cases}1.2\:,&x\in\Big(-\frac{\sqrt{2}}{8},0\Big)\\\\0\:,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{\omega},x\Big)=\begin{cases}0.3,&x\in(-1.8,0)\\0,&\text{otherwise}\end{cases}$$
I. $$f\Big(\frac{\pi}{4\omega},x\Big)=\begin{cases}1.7,&x\in\Big(0,\frac{\sqrt{2}}{5}\Big)\\\\0,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{2\omega},x\Big)=0.2$$
$$f\Big(\frac{3\pi}{4\omega},x\Big)=\begin{cases}1.7\:,&x\in\Big(-\frac{\sqrt{2}}{5},0\Big)\\\\0\:,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{\omega},x\Big)=\begin{cases}0.2,&x\in(-1.5,0)\\0,&\text{otherwise}\end{cases}$$
J. $$f\Big(\frac{\pi}{4\omega},x\Big)=\begin{cases}0.9,&x\in\Big(0,\frac{0.4}{\sqrt{2}}\Big)\\\\0,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{2\omega},x\Big)=0.7$$
$$f\Big(\frac{3\pi}{4\omega},x\Big)=\begin{cases}0.9\:,&x\in\Big(-\frac{0.4}{\sqrt{2}},0\Big)\\\\0\:,&\text{otherwise}\end{cases}$$
$$f\Big(\frac{\pi}{\omega},x\Big)=\begin{cases}0.7,&x\in(-0.9,0)\\0,&\text{otherwise}\end{cases}$$
Answer: C