(New page: = ECE Ph.D. Qualifying Exam in "Communication, Networks, Signal, and Image Processing" (CS) = = Question 1, August 2011, Part 1 = :[[ECE...)
 
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&nbsp;<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue}\text{1. } \left( \text{25 pts} \right) \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>  
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&nbsp;<font color="#ff0000"><span style="font-size: 19px;"><math>\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.}
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</math></span></font>  
  
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'''<math>\color{blue}\left( \text{a} \right) \text{ Find the joint probability density function } f_{YZ}(y,z).</math>'''<br>
 
  
 
===== <math>\color{blue}\text{Solution 1:}</math>  =====
 
===== <math>\color{blue}\text{Solution 1:}</math>  =====
  
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<math> f_{YZ}\left (y,z \right )=\int_{-\infty}^{+\infty}f_{XYZ}\left(x,y,z \right )dx </math>&nbsp;
 
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math> =\frac{3z^{2}}{7\sqrt[]{2\pi}}e^{-zy}\int_{-\infty}^{+\infty}exp\left[-\frac{1}{2}\left(\frac{x-y}{z} \right )^{2} \right ]dx\cdot 1_{[0,\infty)}
 
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\left(y \right )\cdot1_{\left [1,2 \right ]}\left(z \right )</math><br>
 
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<math>\text{But}\int_{-\infty}^{+\infty}exp\left[-\frac{1}{2}\left(\frac{x-y}{z} \right )^{2} \right ]dx \text{looks like the Gaussian pdf, so} </math>
 
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<math> =\frac{3z^{2}}{7\sqrt[]{2\pi}}e^{-zy}
 
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\underset{\sqrt[]{2\pi}z}{\underbrace{\frac{7\sqrt[]{2\pi}z}{7\sqrt[]{2\pi}z}  \int_{-\infty}^{+\infty}exp\left[-\frac{1}{2}\left(\frac{x-y}{z} \right )^{2} \right ]dx}}\cdot 1_{[0,\infty)}
 
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\left(y \right )\cdot1_{\left [1,2 \right ]}\left(z \right )
 
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</math>
 
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<math>
 
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=\frac{3z^{2}}{7}e^{-zy}\cdot 1_{[0,\infty)}
 
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\left(y \right )\cdot1_{\left [1,2 \right ]}\left(z \right )
 
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</math>
 
  
 
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here put sol.2
 
here put sol.2
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<math>\color{blue}\left( \text{b} \right) \text{Find}
 
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f_{x}\left( x|y,z\right )
 
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</math><br>
 
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<math>\color{blue}\text{Solution 1:}</math>
 
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<font color="#ff0000"><span style="font-size: 17px;">'''<font face="serif"></font><math>
 
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= \frac{f_{XYZ}\left( x,y,z\right )}{f_{YZ}\left(y,z \right )}
 
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</math>'''</span></font><font color="#ff0000"><span style="font-size: 17px;">
 
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</span></font>
 
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'''<font face="serif"><math>
 
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= \frac{e^{-\frac{1}{2}\left(\frac{x-y}{z} \right )^{2}}}{\sqrt[]{2\pi}z}
 
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</math>&nbsp;&nbsp;</font>'''
 
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<math>\color{blue}\text{Solution 2:}</math><br>
 
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sol2 here
 
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<math>\color{blue}\left( \text{c} \right) \text{Find}
 
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f_{Z}\left( z\right )
 
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</math><br>
 
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<math>\color{blue}\text{Solution 1:}</math>
 
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<font color="#ff0000"><span style="font-size: 17px;">'''<font face="serif"></font><math>
 
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=\int_{0}^{+\infty}{f_{YZ}\left(y,z \right )dy}
 
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</math>'''</span></font><font color="#ff0000"><span style="font-size: 17px;">
 
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</span></font>
 
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'''<font face="serif"><math>
 
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=\frac{3z^{2}}{7}\cdot1_{\left[1,2 \right ]}(z)
 
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</math>&nbsp;&nbsp;</font>'''
 
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----
 
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<math>\color{blue}\text{Solution 2:}</math><br>
 
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sol2 here
 
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<math>\color{blue}\left( \text{d} \right) \text{Find}
 
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f_{Y}\left(y|z \right )
 
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</math><br>
 
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<math>\color{blue}\text{Solution 1:}</math>
 
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<font color="#ff0000"><span style="font-size: 17px;">'''<font face="serif"></font><math>
 
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=\frac{f_{YZ}\left(y,z \right )}{f_{Z}(z)}</math>'''</span></font><font color="#ff0000"><span style="font-size: 17px;">
 
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</span></font>
 
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'''<font face="serif"><math>
 
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=e^{-zy}z\cdot1_{\left[0,\infty \right )}(y)
 
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</math>&nbsp;&nbsp;</font>'''
 
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<math>\color{blue}\text{Solution 2:}</math><br>
 
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sol2 here
 
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<math>\color{blue}\left( \text{e} \right) \text{Find}
 
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f_{XY}\left(x,y|z \right )
 
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</math><br>
 
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<math>\color{blue}\text{Solution 1:}</math>
 
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<font color="#ff0000"><span style="font-size: 17px;">'''<font face="serif"></font><math>
 
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=\frac{f_{XYZ}\left(x,y,z \right )}{f_{Z}(z)}
 
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</math>'''</span></font><font color="#ff0000"><span style="font-size: 17px;">
 
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</span></font>
 
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'''<font face="serif"><math>
 
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=\frac{e^{-zy}}{\sqrt[]{2\pi}}e^{-\frac{1}{2}\left(\frac{x-y}{z} \right )^{2}}\cdot1_{\left[0,\infty \right )}(y)
 
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</math>&nbsp;&nbsp;</font>'''
 
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<math>\color{blue}\text{Solution 2:}</math><br>
 
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sol2 here
 
 
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Revision as of 13:27, 27 July 2012

ECE Ph.D. Qualifying Exam in "Communication, Networks, Signal, and Image Processing" (CS)

Question 1, August 2011, Part 1

Part 1,2]

 $ \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.} $


$ \color{blue}\text{Solution 1:} $

$ \color{blue}\text{Solution 2:} $

here put sol.2

"Communication, Networks, Signal, and Image Processing" (CS)- Question 1, August 2011

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