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# A1 - Z t and Y t are independent stationary process then...

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STATISTICS 4005B ASSIGNMENT 1 Due date: January 25, 2008 1. Suppose E ( X ) = 4 , V ar ( X ) = 4 , E ( Y ) = 9 , V ar ( Y ) = 9, and Corr ( X, Y ) = 1 4 . Find: (a) V ar ( X + Y ), (b) Cov ( X, X + Y ), (c) Corr ( X + Y, 2 X - Y ). 2. Suppose Z t = 8 + 3 t + X t , where { X t } is a stationary series with E ( X t ) = 2 and autocovariance function γ k . (a) Find the mean function for { Z t } . (b) Find the autocovariance function of { Z t } . 3. 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 ). 3. Two processes { Z t } and { Y t } are said to be independent if for any time point t 1 , t 2 , ..., t n and s 1 , s 2 , ..., s m , the random variables { Z t 1 , Z t 2 , ..., Z t n } are independent of the random variables { Y s 1 , Y s 2 , ..., Y s m } . Show that if
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Unformatted text preview: { Z t } and { Y t } are independent stationary process, then the process W t = 2 Z t + Y t . is stationary. 4. Let { a t } be white noise with V ar ( a t ) = σ 2 a and let c be a constant with | c | < 1. The series { Z t } is then constructed as follows: Z 1 = a 1 ; Z t = cZ t-1 + a t for t > 1 (a) Show that E ( Z t ) = 0. (b) Show that V ar ( Z t ) = σ 2 a (1 + c 2 + c 4 + ... + c 2 t-2 ). (c) Show that Corr ( Z t ,Z t-k ) = c k [ V ar ( Z t-k ) V ar ( Z t ) ] 1 / 2 for k > ....
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