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| − | = [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | + | |
| + | <center> | ||
| + | <font size= 4> | ||
| + | [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | ||
| + | </font size> | ||
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| + | <font size= 4> | ||
| + | Communication, Networking, Signal and Image Processing (CS) | ||
| + | |||
| + | Question 1: Probability and Random Processes | ||
| + | </font size> | ||
| + | |||
| + | August 2000 | ||
| + | </center> | ||
| + | ---- | ||
---- | ---- | ||
==Question== | ==Question== | ||
| − | + | '''Part 1.''' | |
| + | |||
| + | a) The Laplacian density function is given by <math class="inline">f\left(x\right)=\frac{A}{2}e^{-A\left|x\right|}\text{ where }A>0.</math> Determine its characteristic function. | ||
| + | |||
| + | b) Determine a bound on the probability that a RV is within two standard deviations of its mean. | ||
| + | |||
| + | :'''Click [[ECE_PhD_QE_CNSIP_2000_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_2000_Problem1.1|answers and discussions]]''' | ||
---- | ---- | ||
| − | + | '''Part 2.''' | |
| + | <math class="inline">\mathbf{X}\left(t\right)</math> is a WSS process with its psd zero outside the interval <math class="inline">\left[-\omega_{max},\ \omega_{max}\right]</math> . If <math class="inline">R\left(\tau\right)</math> is the autocorrelation function of <math class="inline">\mathbf{X}\left(t\right)</math> , prove the following: <math class="inline">R\left(0\right)-R\left(\tau\right)\leq\frac{1}{2}\omega_{max}^{2}\tau^{2}R\left(0\right).</math> (Hint: <math class="inline">\left|\sin\theta\right|\leq\left|\theta\right|</math> ). | ||
| + | :'''Click [[ECE_PhD_QE_CNSIP_2000_Problem1.2|here]] to view student [[ECE_PhD_QE_CNSIP_2000_Problem1.2|answers and discussions]]''' | ||
| + | ---- | ||
| + | '''Part 3.''' | ||
| + | Inquiries arrive at a recorded message device according to a Poisson process of rate 15 inquiries per minute. Find the probability that in a 1-minute period, 3 inquiries arrive during the first 10 seconds and 2 inquiries arrive during the last 15 seconds. | ||
| + | :'''Click [[ECE_PhD_QE_CNSIP_2000_Problem1.3|here]] to view student [[ECE_PhD_QE_CNSIP_2000_Problem1.3|answers and discussions]]''' | ||
---- | ---- | ||
| − | == | + | '''Part 4.''' |
| − | + | ||
| + | A RV is given by <math class="inline">\mathbf{Z}=\sum_{n=0}^{8}\mathbf{X}_{n}</math> where <math class="inline">\mathbf{X}_{n}</math> 's are i.i.d. RVs with characteristic function given by <math class="inline">\Phi_{\mathbf{X}}\left(\omega\right)=\frac{1}{1-j\omega/2}.</math> | ||
| + | |||
| + | '''(a)''' Determine the characteristic function of <math class="inline">\mathbf{Z}</math> . | ||
| + | |||
| + | '''(b)''' Determine the pdf of <math class="inline">\mathbf{Z}</math> . You can leave your answer in integral form. | ||
| + | :'''Click [[ECE_PhD_QE_CNSIP_2000_Problem1.4|here]] to view student [[ECE_PhD_QE_CNSIP_2000_Problem1.4|answers and discussions]]''' | ||
---- | ---- | ||
[[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
August 2000
Question
Part 1.
a) The Laplacian density function is given by $ f\left(x\right)=\frac{A}{2}e^{-A\left|x\right|}\text{ where }A>0. $ Determine its characteristic function.
b) Determine a bound on the probability that a RV is within two standard deviations of its mean.
- Click here to view student answers and discussions
Part 2.
$ \mathbf{X}\left(t\right) $ is a WSS process with its psd zero outside the interval $ \left[-\omega_{max},\ \omega_{max}\right] $ . If $ R\left(\tau\right) $ is the autocorrelation function of $ \mathbf{X}\left(t\right) $ , prove the following: $ R\left(0\right)-R\left(\tau\right)\leq\frac{1}{2}\omega_{max}^{2}\tau^{2}R\left(0\right). $ (Hint: $ \left|\sin\theta\right|\leq\left|\theta\right| $ ).
- Click here to view student answers and discussions
Part 3.
Inquiries arrive at a recorded message device according to a Poisson process of rate 15 inquiries per minute. Find the probability that in a 1-minute period, 3 inquiries arrive during the first 10 seconds and 2 inquiries arrive during the last 15 seconds.
- Click here to view student answers and discussions
Part 4.
A RV is given by $ \mathbf{Z}=\sum_{n=0}^{8}\mathbf{X}_{n} $ where $ \mathbf{X}_{n} $ 's are i.i.d. RVs with characteristic function given by $ \Phi_{\mathbf{X}}\left(\omega\right)=\frac{1}{1-j\omega/2}. $
(a) Determine the characteristic function of $ \mathbf{Z} $ .
(b) Determine the pdf of $ \mathbf{Z} $ . You can leave your answer in integral form.
- Click here to view student answers and discussions
