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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 } \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 ). $

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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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