Communication, Networking, Signal and Image Processing (CS)

Question 1: Probability and Random Processes

January 2001

## Question

Part 1. (20 pts)

State and prove the Tchebycheff Inequality.

Part 2.

(a) (7 pts)

Let $A$ and $B$ be statistically independent events in the same probability space. Are $A$ and $B^{C}$ independent? (You must prove your result).

(b) (7 pts)

Can two events be statistically independent and mutually exclusive? (You must derive the conditions on A and B for this to be true or not.)

(c) (6 pts)'

State the Axioms of Probability.

Part 3.

3. (20 pts)

Let the $\mathbf{X}_{1},\mathbf{X}_{2},\cdots$ be a sequence of random variables that converge in mean square to the random variable $\mathbf{X}$ . Does the sequence also converge to $\mathbf{X}$ in probability? (A simple yes or no answer is not acceptable, you must derive the result.)

Part 4.

4. (20 pts)

Let $\mathbf{X}_{t}$ be a band-limited white noise strictly stationary random process with bandwidth 10 KHz. It is also known that $\mathbf{X}_{t}$ is uniformly distributed between $\pm5$ volts. Find:

(a) (10 pts)

Let $\mathbf{Y}_{t}=\left(\mathbf{X}_{t}\right)^{2}$ . Find the mean square value of $\mathbf{Y}_{t}$ .

(b) (10 pts)

Let $\mathbf{X}_{t}$ be the input to a linear shift-invariant system with transfer function:
$H\left(f\right)=\begin{cases} \begin{array}{lll} 1 \text{ for }\left|f\right|\leq5\text{ KHz}\\ 0.5 \text{ for }5\text{ KHz}\leq\left|f\right|\leq50\text{ KHz}\\ 0 \text{ elsewhere. } \end{array}\end{cases}$

Find the mean and variance of the output.

Part 5. (20 pts)

Let a linear discrete parameter shift-invariant system have the following difference equation: $y\left(n\right)=0.7y\left(n-1\right)+x\left(n\right)$ where $x\left(n\right)$ in the input and $y\left(n\right)$ is the output. Now suppose this system has as its input the discrete parameter random process $\mathbf{X}_{n}$ . You may assume that the input process is zero-mean i.i.d.

(a) (5 pts) Is the input wide-sense stationary (show your work)?

(b) (5 pts) Is the output process wide-sense stationary (show your work)?

(c) (5 pts) Find the autocorrelation function of the input process.

(d) (5 pts) Find the autocorrelation function, in closed form, for the output process. 