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# hw2 - y and z are not necessarily statistically independent...

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EE5620 Stochastic Processes 1 EE5620 - Homework #2 Due: Oct. 13, 2006, in class Reading assignments: Section 4.1–Section 4.5 and Section 5.1–Section 5.6 of the textbook. Problem assignments: 1. Problem 4.19 in textbook. 2. Problem 4.21 in textbook. 3. Problem 5.9 in textbook. 4. Problem 5.16 in textbook. 5. Problem 2.13 in Gallager’s note. 6. Let x , y , and z be collectively jointly Gaussian random vectors. That is the elements of the random vector [ x T , y T , z T ] T are jointly Gaussian. (a) If y and z are statistically independent, show that E [ x | y , z ] = E [ x | y ] + E [ x | z ] - m x , where m x = E [ x ] . (b) If
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Unformatted text preview: y and z are not necessarily statistically independent, show that E [ x | y , z ] = E [ x | y , ˆ z ] , where ˆ z = z-E [ z | y ] . 7. Let w = [ X,Y,Z ] T be a zero mean jointly Gaussian random vector with covariance matrix K w =   4 2 1 2 2 1 1 1 1   . (a) Find α such that X-αY and Y are independent. (b) What is the conditional expectation E [ X 2 | Y ]? (c) Find the probability density function (pdf) for S = X + 2 Y . (d) Find the conditional density f X | Y,Z ( Y,Z ) ....
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