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mid-term - STA 4005A Mid-term Examination(Total Marks 35...

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STA 4005A Mid-term Examination (Total Marks 35) October 25, 2007 Answer ALL questions: Time allowed: 1 1/2 hours. Let a t NID (0 , σ 2 a ). 1. Consider a MA(2) model: Z t = a t - 1 . 3 a t - 1 - 0 . 4 a t - 2 , σ 2 a = 5 . (a) (2 marks) Find E ( Z t ) and V ar ( Z t ). (b) (3 marks) Find the autocovariance function γ k , k = 1 , 2 , 3 , ... . (c) (2 marks) Find the autocorrelations ρ 1 and ρ 2 . 2. Consider the process Z t = 6 + X t where X t = 1 when t is odd and X t = 0 when t is even. (a) (3 marks) Find the joint distribution function of Z 1 , Z 2 and Z 3 . (b) (2 marks) Is { Z t } strictly stationary? If the answer is YES, prove it. If the answer is NO, explain why. 3. Suppose that a random process { Z t } ( t = 0 , ± 1 , ± 2 , ... ) is defined by Z t - αZ t - 1 = Y t - βY t - 2 where α, β are real constants and { Y t } is a process of zero mean uncorrelated random variables with variance σ 2 Y = 4. (a) (3 marks) Find the values of π i , i = 0 , 1 , 2 , 3 , ... if the model can be rewrit- ten as Z t = π 0 Y t + π 1 Y t - 1 + π 2 Y t - 2 + ... . (b) (2 marks) If α = 0 . 9 , β = 0 . 2 find the Cov ( Z t , Z t - 1 ). 4. (a) (3 marks) Suppose X t = 3 + Y t where the autocovariances of { Y t } are given by γ k = (0 . 5) k , k = 0 , 1 , 2 , 3 , ... . Find V ar ( X 4 + 2 X 5 + X 6 ). (b) (3 marks) Consider the model Z t = a t - 0 . 3 a t - 1 - 0 . 6 a t - 4 + 0 . 18 a t - 5 . Find the autocovariance function
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