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==Question== | ==Question== | ||
| − | + | 1 (33 points) | |
| − | + | Let <math class="inline">\mathbf{X}</math> and <math class="inline">\mathbf{Y}</math> be two joinly distributed random variables having joint pdf | |
| + | |||
| + | <math class="inline">f_{\mathbf{XY}}\left(x,y\right)=\left\{ \begin{array}{lll} | ||
| + | 1, & & \text{ for }0\leq x\leq1\text{ and }0\leq y\leq1\\ | ||
| + | 0, & & \text{ elsewhere. } | ||
| + | \end{array}\right.</math> | ||
| + | |||
| + | (a) | ||
| + | |||
| + | Are <math class="inline">\mathbf{X}</math> and <math class="inline">\mathbf{Y}</math> statistically independent? Justify your answer. | ||
| + | |||
| + | |||
| + | (b) | ||
| + | |||
| + | Let <math class="inline">\mathbf{Z}</math> be a new random variable defined as <math class="inline">\mathbf{Z}=\mathbf{X}+\mathbf{Y}</math> . Find the cdf of <math class="inline">\mathbf{Z}</math> . | ||
| + | |||
| + | (c) | ||
| + | |||
| + | Find the variance of <math class="inline">\mathbf{Z}</math> . | ||
:'''Click [[ECE_PhD_QE_CNSIP_Jan_2006_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2006_Problem1.1|answers and discussions]]''' | :'''Click [[ECE_PhD_QE_CNSIP_Jan_2006_Problem1.1|here]] to view student [[ECE_PhD_QE_CNSIP_Jan_2006_Problem1.1|answers and discussions]]''' | ||
Revision as of 09:21, 10 March 2015
Communication, Networking, Signal and Image Processing (CS)
Question 1: Probability and Random Processes
January 2006
Question
1 (33 points)
Let $ \mathbf{X} $ and $ \mathbf{Y} $ be two joinly distributed random variables having joint pdf
$ f_{\mathbf{XY}}\left(x,y\right)=\left\{ \begin{array}{lll} 1, & & \text{ for }0\leq x\leq1\text{ and }0\leq y\leq1\\ 0, & & \text{ elsewhere. } \end{array}\right. $
(a)
Are $ \mathbf{X} $ and $ \mathbf{Y} $ statistically independent? Justify your answer.
(b)
Let $ \mathbf{Z} $ be a new random variable defined as $ \mathbf{Z}=\mathbf{X}+\mathbf{Y} $ . Find the cdf of $ \mathbf{Z} $ .
(c)
Find the variance of $ \mathbf{Z} $ .
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Part 2.
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Part 3.
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Part 4.
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