hw6 - UCSD ECE153 Prof Young-Han Kim Handout#26 Thursday...

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UCSD ECE153 Handout #26 Prof. Young-Han Kim Thursday, May 19, 2011 Homework Set #6 Due: Thursday, May 26, 2011 1. Covariance matrices. Which of the following matrices can be a covariance matrix? Justify your answer either by constructing a random vector X , as a function of the i.i.d zero mean unit variance random variables Z 1 , Z 2 , and Z 3 , with the given covariance matrix, or by establishing a contradiction. (a) bracketleftbigg 1 2 0 2 bracketrightbigg (b) bracketleftbigg 2 1 1 2 bracketrightbigg (c) 1 1 1 1 2 2 1 2 3 (d) 1 1 2 1 2 3 2 3 3 2. Gaussian random vector. Given a Gaussian random vector X ∼ N ( μ , Σ), where μ = (1 5 2) T and Σ = 1 1 0 1 4 0 0 0 9 . (a) Find the pdfs of i. X 1 , ii. X 2 + X 3 , iii. 2 X 1 + X 2 + X 3 , iv. X 3 given ( X 1 , X 2 ), and v. ( X 2 , X 3 ) given X 1 . (b) What is P { 2 X 1 + X 2 - X 3 < 0 } ? Express your answer using the Q function. (c) Find the joint pdf on Y = A X , where A = bracketleftbigg 2 1 1 1 - 1 1 bracketrightbigg . 3. Gaussian Markov chain. Let X, Y, and Z be jointly Gaussian random variables with zero mean and unit variance, i.e., E ( X ) = E ( Y ) = E ( Z ) = 0 and E ( X 2 ) = E ( Y 2 ) = E ( Z 2 ) = 1. Let ρ X,Y denote the correlation coefficient between X and Y , and let ρ Y,Z denote the correlation coefficient between Y and Z . Suppose that X and Z are conditionally independent given Y . (a) Find ρ X,Z in terms of ρ X,Y and ρ Y,Z .
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