sol2 - Solution of Assignment 2 Problem 2.3 a The process X...

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Unformatted text preview: Solution of Assignment 2 Problem 2.3 a. The process X t = Z t + 0 . 3 Z t- 1- . 4 Z t- 2 where Z t ∼ WN (0 , 1) is stationary. Note that μ X ( t ) = E [ X t ] = 0; and γ X ( h ) = Cov ( X t + h ; X t ) = Cov ( Z t + h + 0 . 3 Z t + h- 1- . 4 Z t + h- 2 ; Z t + 0 . 3 Z t- 1- . 4 Z t- 2 ) = 1 . 25 if h = 0 . 18 if | h | = 1- . 4 if | h | = 2 otherwise. b.The process Y t = ˜ Z t- 1 . 2 ˜ Z t- 1- 1 . 6 ˜ Z t- 2 where ˜ Z t ∼ WN (0 , . 25) is stationary. Note that μ Y ( t ) = E [ Y t ] = 0; and γ Y ( h ) = Cov ( Y t + h ; Y t ) = Cov ( ˜ Z t + h- 1 . 2 ˜ Z t + h- 1- 1 . 6 ˜ Z t + h- 2 ; ˜ Z t- 1 . 2 ˜ Z t- 1- 1 . 6 ˜ Z t- 2 ) = 1 . 25 if h = 0 . 18 if | h | = 1- . 4 if | h | = 2 otherwise. The two processes ( X t and Y t ) have the same ACVF function. Problem 2.4 a. Let Z be a random variable with mean 0 and variance 1. We define X t = (- 1) t Z ; t = 0 , ± 1 ,... Then, X t is a stationary process and its ACVF γ X ( h ) = κ ( h ). In fact, μ X ( t ) = (- 1) t E [ Z ] = 0 and γ X ( t + h ; t ) = Cov ( Xt + h ; Xt ) = Cov ((- 1) t + h Z ; (- 1) t Z ) = (- 1) h Cov ( Z ; Z ) = (- 1) | h | 1 b. Let A,B,C,D,Z be uncorrelated random variables with mean 0 and variance 1. We define X t = Acos ( πt 2 ) + Bsin...
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sol2 - Solution of Assignment 2 Problem 2.3 a The process X...

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