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# RP-HW2 - Kyung Hee University Department of Electronics and...

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1 Kyung Hee University Department of Electronics and Radio Engineering C1002900 Random Processing Homework 2 Spring 2010 Professor Hyundong Shin Issued: April 14, 2010 Due: April 28, 2010 Reading: Course textbook Chapter 7 HW 2.1 Consider the following 3 3 matrices: . A B C D E F G 10 3 1 10 5 2 10 5 2 10 5 2 2 5 0 5 3 3 5 3 3 5 3 1 1 0 2 2 3 2 2 3 2 2 1 2 10 5 2 10 0 0 2 1 1 5 3 1 0 3 0 1 2 1 2 1 2 0 0 2 1 1 2 Your answers to the following questions may consist of more than one of the above matrices or none of them. Justify your answers. (a) Which of the above could be the covariance matrix of some random vector? (b) Which of the above could be the cross-covariance matrix of two random vectors? (c) Which of the above could be the covariance matrix of a random vector in which one com- ponent is a linear combination of the other two components? (d) Which of the above could be the covariance matrix of a random vector with statistically independent components? Must a random vector with such a covariance matrix have sta- tistically independent components? HW 2.2 For each of the following statements, determine whether it is TRUE or FALSE. Make sure you give a sufficient but brief justification of your answers. Note that each statement is independent of the others, so you cannot use the assumptions of one in the others. (a) If random vectors X and Y are jointly Gaussian and | E E X Y X , then X and Y are statistically independent.

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2 (b) Given two random variables X and Y , if | E X Y E X , then for all functions g , E Xg Y E X E g Y . (c) Given two random variables X and Y , if E Xg Y E X E g Y for all functions g , then | E X Y E X . HW 2.3 We wish to estimate a random variable X by some constant ˆ x . There are many ways to meas- ure how good an estimate ˆ x is. Here you will derive an important property of the minimum mean square error estimation (MMSE) . Define the mean square estimation error by   ˆ ˆ ( ) e x E X x 2 .
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