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# T1 - STA4005B Time Series(2007-2008 Tutorial 1(15/1...

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STA4005B Time Series (2007-2008) Tutorial 1 (15/1, 16/1) (Time & Venue: T9, BMS LT; W5, MMW 704) Def : Suppose X , Y and Z are random variables and a , b , c and d are constant. i) ) ( ) ( ) ( )]} ( )][ ( {[ ) , ( Y E X E XY E Y E Y X E X E Y X Cov = = ii) )) , ( ) , ( ( ) , ( ) , ( Y X abCov bY aX Cov Y X bdCov dY c bX a Cov = = + + ) , ( ) , ( ) , ( ) , ( ) , ( X Y Cov Y X Cov Z Y Cov Z X Cov Z Y X Cov = + = + iii) ) , ( )] ( [ ) ( 2 X X Cov X E X E X Var = = ) ( ) , ( 2 ) ( ) ( 2 2 Y Var b Y X abCov X Var a bY aX Var + + = + iv) ) ( ) ( ) , ( ) , ( Y Var X Var Y X Cov Y X Corr = 1) Suppose 9 ) ( , 9 ) ( , 4 ) ( , 4 ) ( = = = = Y Var Y E X Var X E and 25 . 0 ) , ( = Y X Corr Find a) b) ) 2 ( Y X Var ) 2 , 3 ( Y X Y Cov c) ) 2 , 2 ( Y X Y X Corr + 1

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Strictly Stationary . Def: A stochastic process { } t Z is said to be strictly stationary if the joint distribution of is the same as the joint distribution of for all choices of time points and all choices of time lag k . { n t t t Z Z Z , , , 2 1 K k t k t k t n Z Z Z , , , 2 1 K n t t t , , , 2 1 K ) , , , ( ) , , , ( 2 1 , , , 2 1 , , , 2 1 2 1 n k t k t k t n t t t x x x F x x x F n n K K K K = } Weakly Stationary . Def: A stochastic process { } t Z is said to be weakly (or second order) stationary if 1. t µ is constant for all time point t ( does not depend on t ). ) ( t Z E 2. k k k t t k t t Z Z Cov γ γ γ = = = + + , 0 , ) , ( for all time point t and lag k (the autocovariance function is a
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T1 - STA4005B Time Series(2007-2008 Tutorial 1(15/1...

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