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&nbsp;<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue} \text{ Consider the optimization problem, }</math></span></font>  
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&nbsp;<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>  
  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>\text{maximize} -x_{1}^{2}+x_{1}-x_{2}-x_{1}x_{2}</math>  
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'''<math>\color{blue}\left( \text{a} \right) \text{ Find the joint probability density function } f_{YZ}(y,z).</math>'''<br>  
  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>\text{subject to  }   x_{1}\geq0, x_{2}\geq0</math><font color="#ff0000" face="serif" size="4"><br></font>  
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<math>\color{blue}\left( \text{b} \right) \text{Find}  
 +
f_{x}\left( x|y,z\right ).
 +
</math><br>  
  
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'''<math>\color{blue}\left( \text{i} \right) \text{ Characterize feasible directions at the point } x^{*}=\left[ \begin{array}{c} \frac{1}{2} \\ 0 \end{array} \right]</math>'''<br>  
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<math>\color{blue}\left( \text{c} \right) \text{Find}  
 +
f_{Z}\left( z\right ).
 +
</math><br>
 +
 
 +
<math>\color{blue}\left( \text{d} \right) \text{Find}  
 +
f_{Y}\left(y|z \right ).
 +
</math><br>
 +
 
 +
<math>\color{blue}\left( \text{e} \right) \text{Find}
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f_{XY}\left(x,y|z \right ).
 +
</math><br>  
  
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<math>\color{blue}\left( \text{ii} \right) \text{Write down the second-order necessary condition for } x^{*} \text{. Does the point } x^{*} \text{ satisfy this condition?}</math><br>
 
  
 
:'''Click [[ECE-QE_CS1-2011_solusion-1|here]] to view student [[ECE-QE_CS1-2011_solusion-1|answers and discussions]]'''
 
:'''Click [[ECE-QE_CS1-2011_solusion-1|here]] to view student [[ECE-QE_CS1-2011_solusion-1|answers and discussions]]'''
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&nbsp;<font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue}  \text{ Use the simplex method to solve the problem, }</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.}
−
 
+
</math></span></font>  
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<span class="texhtml">maximize &nbsp; &nbsp; &nbsp; &nbsp;''x''<sub>1</sub> + ''x''<sub>2</sub></span>
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−
 
+
−
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;<math>\text{subject to  }   x_{1}-x_{2}\leq2</math><font color="#ff0000" face="serif" size="4"><br></font>'''&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<math>x_{1}+x_{2}\leq6</math>''' &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;
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−
 
+
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&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;<math>x_{1},-x_{2}\geq0.</math>  
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Revision as of 08:36, 27 July 2012


ECE Ph.D. Qualifying Exam in Communication Networks Signal and Image processing (CS), Question 1, August 2011

Question

Part 1. 25 pts


 $ \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) $

$ \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.


 $ \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

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