400511

# 400511 - STATISTICS 4005 ASSIGNMENT 1 Due date 1 Suppose Zt...

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STATISTICS 4005 ASSIGNMENT 1 Due date: January 29, 2004 1. Suppose Z t = 6 + 7 t + X t , where { X t } is a zero-mean stationary series with autoco- variance function γ k . 1. Find the mean function for { Z t } . 2. Find the autocovariance function of { Z t } in terms of γ k . 2. Let ± t be an independent and identically distributed sequences of random variables, each with mean zero and variance 2. Deﬁne X t = ± t - 1 + 2 ± t + ± t +1 . Compute and plot the autocovariance function γ k of the process { X t } for k = 1 , 2 , 3 , 4 , 5. 3. Let { X t } be a zero-mean, unit-variance, stationary process with autocorrelation function ρ k . Suppose the observed series is formed as Z t = 7 t + (1 + t 2 ) X t . (a) For the Z -process, ﬁnd the mean, variance, and autocovariance functions. (b) Show that the autocorrelation function for the Z -process depends only on lag. Is the Z -process stationary? 4. Two processes { Z t } and { Y t } are said to be independent if for any time point t 1 , t 2 , ..., t
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