Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

August 2011

## Question

Part 1. 25 pts

$\color{blue}\text{ Let } \mathbf{X}\text{, }\mathbf{Y}\text{, and } \mathbf{Z} \text{ be three jointly distributed random variables with joint pdf } f_{XYZ}\left ( x,y,z \right )= \frac{3z^{2}}{7\sqrt[]{2\pi}}e^{-zy} exp \left [ -\frac{1}{2}\left ( \frac{x-y}{z}\right )^{2} \right ] \cdot 1_{\left[0,\infty \right )}\left(y \right )\cdot1_{\left[1,2 \right]} \left ( z \right)$

$\color{blue}\left( \text{a} \right) \text{Find the joint probability density function } f_{YZ}(y,z).$

$\color{blue}\left( \text{b} \right) \text{Find } f_{x}\left( x|y,z\right ).$

$\color{blue}\left( \text{c} \right) \text{Find } f_{Z}\left( z\right ).$

$\color{blue}\left( \text{d} \right) \text{Find } f_{Y}\left(y|z \right ).$

$\color{blue}\left( \text{e} \right) \text{Find } f_{XY}\left(x,y|z \right ).$

Part 2. 25 pts

$\color{blue} \text{Show that if a continuous-time Gaussian random process } \mathbf{X}(t) \text{ is wide-sense stationary, it is also strict-sense stationary.}$

Part 3. 25 pts

Show that the sum of two jointly distributed Gaussian random variables that are not necessarily statistically independent is a Gaussian random variable.

Part 4. 25 pts

Assume that $\mathbf{X}(t)$ is a zero-mean continuous-time Gaussian white noise process with autocorrelation function

$R_{\mathbf{XX}}(t_1,t_2)=\delta(t_1-t_2).$

Let $\mathbf{Y}(t)$ be a new random process ontained by passing $\mathbf{X}(t)$ through a linear time-invariant system with impulse response $h(t)$ whose Fourier transform $H(\omega)$ has the ideal low-pass characteristic

$H(\omega) = \begin{cases} 1, & \mbox{if } |\omega|\leq\Omega,\\ 0, & \mbox{elsewhere,} \end{cases}$

where $\Omega>0$.

a) Find the mean of $\mathbf{Y}(t)$.

b) Find the autocorrelation function of $\mathbf{Y}(t)$.

c) Find the joint pdf of $\mathbf{Y}(t_1)$ and $\mathbf{Y}(t_2)$ for any two arbitrary sample time $t_1$ and $t_2$.

d) What is the minimum time difference $t_1-t_2$ such that $\mathbf{Y}(t_1)$ and $\mathbf{Y}(t_2)$ are statistically independent?