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Unformatted text preview: STAT 443: Assignment 1 SOLUTIONS Only mark the parts of the assignments which are indicated in this mark scheme. The total mark for this assignment is 63. Please indicate to the students where they are losing marks and put your initials at the top of each paper that you marked in case there are questions. 1 Probability and Statistics Theory 1. [21 Marks] Let Z t be iid N (0 , 1) sequence and define X t = Z t t even Z 2 t 1 1 √ 2 t odd . (a) [4 marks] Compute E ( X t ) for all t If t is even the mean is E ( Z t ) = 0 [1], if odd we note that 1 = V ar ( Z t ) = E ( Z 2 t ) E ( Z t ) 2 = E ( Z 2 t ) [2] so that E ( Z 2 t 1 ) 1 √ 2 = 1 1 √ 2 = 0 . [1] (b) [5 marks] Write down the moment generating function for Z t and hence com pute the first four noncentral moments of Z t . MGF for standard normal is exp ( 1 2 t 2 ) [1] so the first four derivatives are te 1 / 2 t 2 ,e 1 / 2 t 2 + t 2 e 1 / 2 t 2 , 3 te 1 / 2 t 2 + t 3 e 1 / 2 t 2 , 3 e 1 / 2 t 2 + 6 t 2 e 1 / 2 t 2 + t 4 e 1 / 2 t 2 [2], So setting t = 0 gives moments , 1 , , 3 [2] (c) [3 marks] Compute V ar ( X t ) for all t If t is even the variance is 1 [1] If odd we the variance is E " Z 2 t 1 1 √ 2 2 # = 1 2 ( E Z 4 t 2 E Z 2 t + 1 ) = 3 2 + 1 2 = 1 [2] 1 (d) [3 marks] Compute Cov ( X 2 t ,X 2 t +1 ) Since all r.v. have mean zero we need to compute E Z 2 t Z 2 2 t 1 √ 2 = 1 √ 2 E Z 3 2 t E [ Z 2 t ] = 0 [3] (e) [6 marks] Show that X t is a WN (0 , 1) sequence but not an iid sequence. The autocovariance for any lag strictly greater than 1 will be zero since it involves independent random variables [1]. From (d) we see that the lag 1 autocovariance is also zero [1] and we have already seen that the mean (0) and variance (1) are independent of t . [1] So have a WN (0 , 1) process. [1] Its not iid since the distribution of X t for t odd is not normal (for example support is on ( 1 √ 2 , ∞ ) not (∞ , ∞ ) . ) [2, for any valid reason for difference] 2. [12 marks] Suppose that the ndimensional vector X has a multivariate normal dis...
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 Winter '09
 YuliaGel
 Forecasting, Normal Distribution, Standard Deviation, Variance, Probability theory, Xt, zt

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