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.

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**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.

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**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.)

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**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.

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**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.

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