| Line 11: | Line 11: | ||
| − | <font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue} \text{ | + | <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> |
| − | + | '''<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} | |
| + | f_{x}\left( x|y,z\right ). | ||
| + | </math><br> | ||
| − | + | <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} | ||
| + | f_{XY}\left(x,y|z \right ). | ||
| + | </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]]''' | ||
| Line 26: | Line 37: | ||
| − | <font color="#ff0000"><span style="font-size: 19px;"><math>\color{blue} \text{ | + | <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> | |
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
Revision as of 09:36, 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
