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mid-term soln

mid-term soln - STA4005B Time Series 2007-2008 Mid-Term...

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Unformatted text preview: STA4005B Time Series 2007-2008 Mid-Term Solution 1a) A stochastic process { is said to be weakly stationary if } t Z t μ is constant for all time point t and k k k t t k t t Z Z Cov γ γ γ = = = + + , , ) , ( for all time point t and lag k . 1b) A stochastic process { 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 . } t Z 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 1c) , t t X Z 3 3 + = 3 3 − = t t Z X 5 . ) 1 ( ) 1 ( = = = − = t t X P X P 5 . ) 1 3 3 ( ) 1 3 3 ( = = − = − = − t t Z P Z P 5 . ) 6 ( ) ( = = = = t t Z P Z P ⎪ ⎩ ⎪ ⎨ ⎧ ≥ < ≤ < = ≤ 6 , 1 6 , 5 . , ) ( z z z z Z P t ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ ≥ ≥ < ≤ ≥ ≥ < ≤ < ≤ < ≤ < < = ≤ ≤ 6 6 1 6 6 6 6 5 . 6 6 25 . ) , ( 2 1 2 1 2 1 2 1 2 1 2 1 2 1 z and z if z and z or z and z if z and z if z or z if z Z z Z P t t 2a)...
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mid-term soln - STA4005B Time Series 2007-2008 Mid-Term...

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