Question

2.4 Let (e) be a zero mean white noise process. Suppose that the observed process is Y = e, + 0,-1, where is either 3 or 1/3. (a) Find the autocorrelation function for {Y} both when 0 = 3 and when 0 = 1/3. (b) You should have discovered that the time series is stationary regardless of the value of and that the autocorrelation functions are the same for 0 = 3 and 0 = 1/3. For simplicity, suppose that the process mean is known to be zero and the variance of Y, is known to be 1. You observe the series (Y) for t = 1, 2,..., n and suppose that you can produce good estimates of the autocorrelations pour Do you think that you could determine which value of is correct (3 or 1/3) based on the estimate of px? Why or why not? 25 Sunnase V-52Y where is no

Answer 1

Solution Given that, for zero meou while noue, The giver equation is Y4=€400-, -- 0 ,9 - 2 (a) The auto correlation function fest VA} if 0 Cov(x - 1) [ct ( ( ( (-t('. .. from the above ear o ELY) = Elettoex, ) :tle]+00(led = 0+0.0 en portant pellen)-07 Now Substitute these values in cand cov (xx, xt-w). (14.44 m] E[1919.) Lant e l Eltr ant be lo cuanto antaan

= t[ll.ex_]of 14--1140€[elt-in] O toe [exped-1-h] - ③ for th=0 {q" ® becomes cou ( X4,X4W)- Elé)+ oelet-, +8€ 10494_)teele) we know that el is white nouse Therefore, Ele.es. J. E(et). Ele-1) = 0 and also Eléta valet). 02 #t. .: Cow (x+,xt-u)+0+0+0+2 = cov Cxt, Xt-4)= (1-0²)=²) for [h=1 ta" @ will becomes Cov( x41 84_W) = Elet:€4.-+06[e-et-J108(et @t.,) + ott [ex-pet-2] Now by applying noise formulas =covext, Yt-W) = 0+00+0+0 » Covl 4t, xt-a)=0047

for the 1] lan becomes cov (x41x4.).((('{, haotlemise beleet->) toelen-141-3) By white noire formulas => [cov (4, Mew) :0 Therefore, it is simply denoted as cov (x4, x4-u) = } Oct LOVU ihail 11:12,3 - Now, the auto correlation is devoted as corrch) - Corr(K), Coulxe, xt-w) Cov(x+17+) Covlytath) Var (44) var (74): low ('t. Iw) -(10)- Therefore, the auto correlation be ((1+0²) at no Lite 2 Corrch) I current inte i h = t1 (1+ i 1 +1,

iho COPY ) . (10%) Now put these value, 0-3 and '; to get the values when 0:3 Lurrah): 0.3 ) in:32,+3. олои 0:13 corr(h): ( 1 0.3 ) ;h-0 h = 11 ;u:#2, 3... that both Hence, by seeing the values, it has declared the o values are same for 3 and 1/2