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| + | [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | ||
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| + | Communication, Networking, Signal and Image Processing (CS) | ||
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| + | Question 1: Probability and Random Processes | ||
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| + | January 2001 | ||
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==Question== | ==Question== | ||
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State and prove the Tchebycheff Inequality. | State and prove the Tchebycheff Inequality. | ||
| − | :'''Click [[ | + | :'''Click [[ECE_PhD_QE_CNSIP_Jan_2001_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2001_Problem1.1|answers and discussions]]''' |
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'''Part 2.''' | '''Part 2.''' | ||
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State the Axioms of Probability. | State the Axioms of Probability. | ||
| − | :'''Click [[ | + | :'''Click [[ECE_PhD_QE_CNSIP_Jan_2001_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2001_Problem1.2|answers and discussions]]''' |
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'''Part 3.''' | '''Part 3.''' | ||
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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.) | 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 [[ | + | :'''Click [[ECE_PhD_QE_CNSIP_Jan_2001_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2001_Problem1.3|answers and discussions]]''' |
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'''Part 4.''' | '''Part 4.''' | ||
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Find the mean and variance of the output. | Find the mean and variance of the output. | ||
| − | :'''Click [[ | + | :'''Click [[ECE_PhD_QE_CNSIP_Jan_2001_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2001_Problem1.4|answers and discussions]]''' |
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'''Part 5.''' (20 pts) | '''Part 5.''' (20 pts) | ||
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'''(d) (5 pts)''' Find the autocorrelation function, in closed form, for the output process. | '''(d) (5 pts)''' Find the autocorrelation function, in closed form, for the output process. | ||
| − | :'''Click [[ | + | :'''Click [[ECE_PhD_QE_CNSIP_Jan_2001_Problem1.5|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2001_Problem1.5|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]] | ||
Latest revision as of 09:19, 13 September 2013
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.
- 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
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.
- Click here to view student answers and discussions
