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[[Category:CNSIP]] | [[Category:CNSIP]] | ||
[[Category:problem solving]] | [[Category:problem solving]] | ||
| − | [[Category: | + | [[Category:random variables]] |
| + | [[Category:probability]] | ||
| − | = [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | + | <center> |
| + | <font size= 4> | ||
| + | [[ECE_PhD_Qualifying_Exams|ECE Ph.D. Qualifying Exam]] | ||
| + | </font size> | ||
| + | |||
| + | <font size= 4> | ||
| + | Communication, Networking, Signal and Image Processing (CS) | ||
| + | |||
| + | Question 1: Probability and Random Processes | ||
| + | </font size> | ||
| + | |||
| + | August 2011 | ||
| + | </center> | ||
| + | ---- | ||
---- | ---- | ||
==Question== | ==Question== | ||
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| − | <font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue}\text{ Let X, Y, and Z be three jointly distributed random variables with joint pdf} f_{XYZ}\left ( x,y,z \right )= \frac{3z^{2}}{7\sqrt[]{2\pi}}e^{-zy} exp \left [ -\frac{1}{2}\left ( \frac{x-y}{z}\right )^{2} \right ] \cdot 1_{\left[0,\infty \right )}\left(y \right )\cdot1_{\left[1,2 \right]} \left ( z \right) </math></span></font> | + | <font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue}\text{ Let } \mathbf{X}\text{, }\mathbf{Y}\text{, and } \mathbf{Z} \text{ be three jointly distributed random variables with joint pdf } f_{XYZ}\left ( x,y,z \right )= \frac{3z^{2}}{7\sqrt[]{2\pi}}e^{-zy} exp \left [ -\frac{1}{2}\left ( \frac{x-y}{z}\right )^{2} \right ] \cdot 1_{\left[0,\infty \right )}\left(y \right )\cdot1_{\left[1,2 \right]} \left ( z \right) </math></span></font> |
| − | '''<math>\color{blue}\left( \text{a} \right) \text{ Find the joint probability density function } f_{YZ}(y,z).</math>'''<br> | + | '''<math>\color{blue}\left( \text{a} \right) \text{Find the joint probability density function } f_{YZ}(y,z).</math>'''<br> |
| − | <math>\color{blue}\left( \text{b} \right) \text{Find} | + | <math>\color{blue}\left( \text{b} \right) \text{Find } |
f_{x}\left( x|y,z\right ). | f_{x}\left( x|y,z\right ). | ||
</math><br> | </math><br> | ||
| − | <math>\color{blue}\left( \text{c} \right) \text{Find} | + | <math>\color{blue}\left( \text{c} \right) \text{Find } |
f_{Z}\left( z\right ). | f_{Z}\left( z\right ). | ||
</math><br> | </math><br> | ||
| − | <math>\color{blue}\left( \text{d} \right) \text{Find} | + | <math>\color{blue}\left( \text{d} \right) \text{Find } |
f_{Y}\left(y|z \right ). | f_{Y}\left(y|z \right ). | ||
</math><br> | </math><br> | ||
| − | <math>\color{blue}\left( \text{e} \right) \text{Find} | + | <math>\color{blue}\left( \text{e} \right) \text{Find } |
f_{XY}\left(x,y|z \right ). | f_{XY}\left(x,y|z \right ). | ||
</math><br> | </math><br> | ||
| Line 42: | Line 56: | ||
:'''Click [[ECE-QE_CS1-2011_solusion-2|here]] to view student [[ECE-QE_CS1-2011_solusion-2|answers and discussions]]''' | :'''Click [[ECE-QE_CS1-2011_solusion-2|here]] to view student [[ECE-QE_CS1-2011_solusion-2|answers and discussions]]''' | ||
| + | ---- | ||
| + | '''Part 3.''' 25 pts | ||
| + | |||
| + | Show that the sum of two jointly distributed Gaussian random variables that are not necessarily statistically independent is a Gaussian random variable. | ||
| + | |||
| + | :'''Click [[ECE-QE_CS1-2011_solusion-3|here]] to view student [[ECE-QE_CS1-2011_solusion-3|answers and discussions]]''' | ||
| + | ---- | ||
| + | '''Part 4.''' 25 pts | ||
| + | |||
| + | |||
| + | Assume that <math>\mathbf{X}(t)</math> is a zero-mean continuous-time Gaussian white noise process with autocorrelation function | ||
| + | |||
| + | <math>R_{\mathbf{XX}}(t_1,t_2)=\delta(t_1-t_2). | ||
| + | </math> | ||
| + | |||
| + | Let <math>\mathbf{Y}(t)</math> be a new random process ontained by passing <math>\mathbf{X}(t)</math> through a linear time-invariant system with impulse response <math>h(t)</math> whose Fourier transform <math>H(\omega)</math> has the ideal low-pass characteristic | ||
| + | |||
| + | <math>H(\omega) = | ||
| + | \begin{cases} | ||
| + | 1, & \mbox{if } |\omega|\leq\Omega,\\ | ||
| + | 0, & \mbox{elsewhere,} | ||
| + | \end{cases} | ||
| + | </math> | ||
| + | |||
| + | where <math>\Omega>0</math>. | ||
| + | |||
| + | a) Find the mean of <math>\mathbf{Y}(t)</math>. | ||
| + | |||
| + | b) Find the autocorrelation function of <math>\mathbf{Y}(t)</math>. | ||
| + | |||
| + | c) Find the joint pdf of <math>\mathbf{Y}(t_1)</math> and <math>\mathbf{Y}(t_2)</math> for any two arbitrary sample time <math>t_1</math> and <math>t_2</math>. | ||
| + | |||
| + | d) What is the minimum time difference <math>t_1-t_2</math> such that <math>\mathbf{Y}(t_1)</math> and <math>\mathbf{Y}(t_2)</math> are statistically independent? | ||
| + | |||
| + | :'''Click [[ECE-QE_CS1-2011_solusion-4|here]] to view student [[ECE-QE_CS1-2011_solusion-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 15:40, 30 March 2015
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
August 2011
Question
Part 1. 25 pts
$ \color{blue}\text{ Let } \mathbf{X}\text{, }\mathbf{Y}\text{, and } \mathbf{Z} \text{ be three jointly distributed random variables with joint pdf } f_{XYZ}\left ( x,y,z \right )= \frac{3z^{2}}{7\sqrt[]{2\pi}}e^{-zy} exp \left [ -\frac{1}{2}\left ( \frac{x-y}{z}\right )^{2} \right ] \cdot 1_{\left[0,\infty \right )}\left(y \right )\cdot1_{\left[1,2 \right]} \left ( z \right) $
$ \color{blue}\left( \text{a} \right) \text{Find the joint probability density function } f_{YZ}(y,z). $
$ \color{blue}\left( \text{b} \right) \text{Find } f_{x}\left( x|y,z\right ). $
$ \color{blue}\left( \text{c} \right) \text{Find } f_{Z}\left( z\right ). $
$ \color{blue}\left( \text{d} \right) \text{Find } f_{Y}\left(y|z \right ). $
$ \color{blue}\left( \text{e} \right) \text{Find } f_{XY}\left(x,y|z \right ). $
- Click here to view student answers and discussions
Part 2. 25 pts
$ \color{blue} \text{Show that if a continuous-time Gaussian random process } \mathbf{X}(t) \text{ is wide-sense stationary, it is also strict-sense stationary.} $
- Click here to view student answers and discussions
Part 3. 25 pts
Show that the sum of two jointly distributed Gaussian random variables that are not necessarily statistically independent is a Gaussian random variable.
- Click here to view student answers and discussions
Part 4. 25 pts
Assume that $ \mathbf{X}(t) $ is a zero-mean continuous-time Gaussian white noise process with autocorrelation function
$ R_{\mathbf{XX}}(t_1,t_2)=\delta(t_1-t_2). $
Let $ \mathbf{Y}(t) $ be a new random process ontained by passing $ \mathbf{X}(t) $ through a linear time-invariant system with impulse response $ h(t) $ whose Fourier transform $ H(\omega) $ has the ideal low-pass characteristic
$ H(\omega) = \begin{cases} 1, & \mbox{if } |\omega|\leq\Omega,\\ 0, & \mbox{elsewhere,} \end{cases} $
where $ \Omega>0 $.
a) Find the mean of $ \mathbf{Y}(t) $.
b) Find the autocorrelation function of $ \mathbf{Y}(t) $.
c) Find the joint pdf of $ \mathbf{Y}(t_1) $ and $ \mathbf{Y}(t_2) $ for any two arbitrary sample time $ t_1 $ and $ t_2 $.
d) What is the minimum time difference $ t_1-t_2 $ such that $ \mathbf{Y}(t_1) $ and $ \mathbf{Y}(t_2) $ are statistically independent?
- Click here to view student answers and discussions
