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A1 soln

# A1 soln - = L L L 1 2 2 2 2 2 2 2 2 2 − − − − = = k...

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STA4005B Assignment 1(Solution) 1a) 5 . 1 9 4 4 1 ) , ( = × × = Y X Cov 16 ) 5 . 1 ( 2 9 4 ) , ( 2 ) ( ) ( ) ( = + + = + + = + Y X Cov Y Var X Var Y X Var 1b) 5 . 5 5 . 1 4 ) , ( ) ( ) , ( = + = + = + Y X Cov X Var Y X X Cov 1c) ) ( ) , ( ) , ( 2 ) ( 2 ) 2 , ( Y Var Y X Cov Y X Cov X Var Y X Y X Cov + = + 5 . 0 9 5 . 1 ) 4 ( 2 = + = ) , ( ) 2 ( 2 ) ( ) ( 4 ) 2 ( Y X Cov Y Var X Var Y X Var + = 19 ) 5 . 1 ( 4 9 ) 4 ( 4 = + = 0287 . 0 19 16 5 . 0 ) 2 ( ) ( ) 2 , ( ) 2 , ( = × = + + = + Y X Var Y X Var Y X Y X Cov Y X Y X Corr 2a) t X E t Z E t t 3 10 ) ( 3 8 ) ( + = + + = 2b) s t s t s t s t X X Cov X s X t Cov Z Z Cov = = + + + + = γ ) , ( ) 3 8 , 3 8 ( ) , ( 3) ) 2 ( 6 5 4 X X X Var + + [ ] [ ] 5 . 10 5 . 0 ) 5 . 0 ( 2 ) 5 . 0 ( 2 2 1 4 1 ) , ( ) , ( 2 ) , ( 2 2 ) ( ) ( 4 ) ( 2 6 4 6 5 5 4 6 5 4 = + + × + + + = + + × + + + = X X Cov X X Cov X X Cov X Var X Var X Var 4) Proof: Since { } t Z and { } are independent stationary process, let t Y y k k t t z k k t t t t Y Y Cov Z Z Cov Y Y E Z Z E γ γ = = = = ) , ( ; ) , ( ; ) ( ; ) ( which all do not depend on t . So Y Z Y E Z E W E t t t + = + = 2 ) ( ) ( 2 ) ( , does not depend on t ; ) 2 , 2 ( ) , ( k t k t t t k t t Y Z Y Z Cov W W Cov + + = 4 ) , ( ) , ( 4 k y k z k t t k t t Y Y Cov Z Z Cov γ γ + = + = which also does not depend on t . Thus the process { } t W is stationary.

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5a) 1 1 2 2 2 2 1 a c a c a c ca a Z t t t t t t + + + + + = L 0 ) ( = t Z E 5b) ) ( t Z Var ) ( 1 1 2 2 2 2 1 a c a c a c ca a Var t t t t t + + + + + = L ) 1 ( 2 2 4 2 2 + + + + = t a c c c L σ 5c) ) 1 ( ) ( 2 2 2 4 2 2 + + + + = k t a k t c c c Z Var L σ ) , ( ) , ( 1 1 1 1 1 1 a c ca a a c a c ca a Cov Z Z Cov k t k t k t t k t k t t k t t + +
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Unformatted text preview: + + + + + + = L L L ) 1 )( ( ) ( 2 2 2 2 2 2 2 2 2 − − − − + + + + = + + + = k t k a k t k k a c c c c c c L L ) ( ) ( ) , ( ) , ( k t t k t t k t t Z Var Z Var Z Z Cov Z Z Corr − − − = 2 1 2 2 2 2 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = = + + + = − − − − − − t k t k k t t k t k k t t k t a k Z Var Z Var c Z Var Z Var Z Var c Z Var Z Var c c c L...
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A1 soln - = L L L 1 2 2 2 2 2 2 2 2 2 − − − − = = k...

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