ECE Ph.D. Qualifying Exam: COMMUNICATIONS, NETWORKING, SIGNAL AND IMAGE PROESSING (CS)- Question 1, 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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