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==Question== | ==Question== | ||
| − | + | '''Part 1. (20 pts)''' | |
| + | |||
| + | State and prove the Tchebycheff Inequality. | ||
| + | |||
| + | :'''Click [[ECE_PhD_QE_CNSIP_Jan_2000_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2000_Problem1.1|answers and discussions]]''' | ||
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| − | + | '''Part 2.''' | |
| + | '''(a) (7 pts)''' | ||
| + | Let <math class="inline">A</math> and <math class="inline">B</math> be statistically independent events in the same probability space. Are <math class="inline">A</math> and <math class="inline">B^{C}</math> 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. | ||
| + | |||
| + | :'''Click [[ECE_PhD_QE_CNSIP_Jan_2000_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2000_Problem1.2|answers and discussions]]''' | ||
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| − | == | + | '''Part 3.''' |
| − | + | ||
| + | '''3. (20 pts)''' | ||
| + | |||
| + | Let the <math class="inline">\mathbf{X}_{1},\mathbf{X}_{2},\cdots</math> be a sequence of random variables that converge in mean square to the random variable <math class="inline">\mathbf{X}</math> . Does the sequence also converge to <math class="inline">\mathbf{X}</math> in probability? (A simple yes or no answer is not acceptable, you must derive the result.) | ||
| + | |||
| + | :'''Click [[ECE_PhD_QE_CNSIP_Jan_2000_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2000_Problem1.3|answers and discussions]]''' | ||
| + | ---- | ||
| + | '''Part 4.''' | ||
| + | |||
| + | '''4. (20 pts)''' | ||
| + | |||
| + | Let <math class="inline">\mathbf{X}_{t}</math> be a band-limited white noise strictly stationary random process with bandwidth 10 KHz. It is also known that <math class="inline">\mathbf{X}_{t}</math> is uniformly distributed between <math class="inline">\pm5</math> volts. Find: | ||
| + | |||
| + | '''(a) (10 pts)''' | ||
| + | |||
| + | Let <math class="inline">\mathbf{Y}_{t}=\left(\mathbf{X}_{t}\right)^{2}</math> . Find the mean square value of <math class="inline">\mathbf{Y}_{t}</math> . | ||
| + | |||
| + | '''(b) (10 pts)''' | ||
| + | |||
| + | Let <math class="inline">\mathbf{X}_{t}</math> be the input to a linear shift-invariant system with transfer function: | ||
| + | <br> | ||
| + | <math class="inline">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}</math> | ||
| + | <br> | ||
| + | |||
| + | Find the mean and variance of the output. | ||
| + | |||
| + | :'''Click [[ECE_PhD_QE_CNSIP_Jan_2000_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2000_Problem1.4|answers and discussions]]''' | ||
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[[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]] | [[ECE_PhD_Qualifying_Exams|Back to ECE Qualifying Exams (QE) page]] | ||
Revision as of 08:12, 27 June 2012
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.
- Click here to view student answers and discussions
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.
- Click here to view student answers and discussions
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.)
- Click here to view student answers and discussions
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.
- Click here to view student answers and discussions
