| Line 15: | Line 15: | ||
'''<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> | ||
Revision as of 08:40, 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 ). $
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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.} $
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