quiz_1sol - 7 a) x ( h ) = cov ( X t ,X t + h ) = cov ( Z t...

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Quiz One Solution 1 5 12 X t = X t - X t - 12 = b t + S t + ± t - b t - 12 - S t - 12 - ± t - 12 = 12 b + ± t - ± t - 12 (1) 2 5 X t = X t - X t - 1 = a + bt + ct 2 - ( a + b ( t - 1) + c ( t - 1) 2 ) = b + 2 tc - c (2) 5 2 X t = 5 X t -5 X t - 1 = b + 2 tc - c - ( a + b ( t - 1) + c ( t - 1) 2 - ( a + b ( t - 2) + c ( t - 2) 2 ) ) = 2 c (3) 3 γ 1 = N - 1 t =1 ( X t - X )( X t +1 - X ) N t =1 ( X t - X ) 2 (4) γ k = N - k t =1 ( X t - X )( X t + k - X ) N t =1 ( X t - X ) 2 (5) Where X = N t =1 X t N (6) 4 γ x ( h ) = cov ( X t ,X t + h ) = cov ( X t - h ,X t ) = γ x ( - h ) (7) ρ x ( k ) = cov ( X t ,X t + k ) = γ x ( k ) γ x (0) (8) ρ x (0) = γ x (0) γ x (0) ) = 1 (9) 1
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5 A process is called second-order stationary if its mean is constant and its acv.f. depends only on the lag, so that E ( X t ) = μ (10) cov ( X t ,X t + h ) = γ ( h ) (11) 6 a) γ z ( h ) = cov ( Z t ,Z t + h ) = ± σ 2 : h = 0 0 : h 1 (12) ρ z ( h ) = γ z ( h ) γ z (0) = ± 1 : h = 0 0 : h 1 (13) b) E ( Z t ) = 0 (14) γ z = ± σ 2 : h = 0 0 : h 1 (15) its weakly stationary.
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Unformatted text preview: 7 a) x ( h ) = cov ( X t ,X t + h ) = cov ( Z t ,Z t + h ) + cov ( Z t ,Z t + h-1 + cov ( Z t-1 ,Z t + h ) + 2 cov ( Z t-1 ,Z t + h-1 ) (16) x ( h ) = = 0 : 2 z + 2 2 z = 1 : 2 z 2 : (17) x ( h ) = x ( h ) x (0) = 1 : h = 0 1+ 2 : h = 1 : h 2 (18) b) its weakly stationary 2...
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This note was uploaded on 02/28/2012 for the course AMS 316 taught by Professor Xing during the Fall '09 term at SUNY Stony Brook.

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quiz_1sol - 7 a) x ( h ) = cov ( X t ,X t + h ) = cov ( Z t...

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