Difference between revisions of "ECE-QE CS1-2011" - Rhea

Revision as of 21:46, 29 March 2015

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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Part 3. 25 pts

 $ \color{blue} \text{Show that the sum of two jointly distributed Gaussian random variables that are not necessarily statistically independent is a Gaussian random variable.} $

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Part 4. 25 pts

 $ \color{blue} \text{Assume that } \mathbf{X}(t) \text{ is a zero-mean continuous-time Gaussian white noise process with autocorrelation function} $

               $ \color{blue} R_{\mathbf{XX}}(t_1,t_2)=\delta(t_1-t_2). $

 $ \color{blue} \text{Let } \mathbf{Y}(t) \text{ be a new random process ontained by passing } \mathbf{Y}(t) \text{ through alinear time-invariant system with impulse response } h(t) \text{ whose Fourier transform} H(\omega) \text{ has the ideal low-pass characteristic} $

               $ \color{blue} H(\omega) = \begin{cases} 1, & \mbox{if } |\omega|<\Omega,\\ 0, & \mbox{elsewhere,} \end{cases} $

 $ \color{blue} \text{where } \Omega>0. $

 $ \color{blue} \text{a) Find the mean of } \mathbf{Y}(t). $

 $ \color{blue} \text{b) Find the autocorrelation function of } \mathbf{Y}(t). $

 $ \color{blue} \text{c) Find the joint pdf of } \mathbf{Y}(t_1) \text{ and } \mathbf{Y}(t_2) \text{ for any two arbitrary sample time } t_1 \text{ and } t_2. $

 $ \color{blue} \text{d) What is the minimum time difference } t_1-t_2 \text{ such that } \mathbf{Y}(t_1) \text{ and } \mathbf{Y}(t_2) \text{ are statistically independent?} $

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