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Testspring08solution

# Testspring08solution - University of Toronto Mississauga...

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Unformatted text preview: University of Toronto Mississauga STA457H — Time Series Analysis, Spring 2008 Term Test Wednesday, February 27, 2007 Last Name: So i k‘b'o n _ First Name: Student Number: Instructions: 0 Time: 50 minutes. 0 Aids: You are allowed to have a one sided 8.5" x 11" aid sheet and a non-programmable calculator. No other aids are allowed. 0 Show your work and answer in the space provided, in ink. Pencil may be used, but then remarks will not be allowed. Use back of pages for rough work. 0 If you do not understand a question, or are having some other difﬁculty, do not hesitate to ask your instructor for clariﬁcation. 0 There are 5 pages including this page. Please check that you are not missing any page. 0 Total point: 35. Good luck!!! © Question 1 (10 points) Let {2,} be a sequence of independent normal random variables, each with mean 0 and variance 0'2 and let a, b, and c be constants. Determine which, if any, of the following is a stationary process. For each stationary process specify the mean and autocovariance function. a) X, =a+bZt +cZ,_2. G E(X€): 5(0. 4-bit *C}£'23= Q'f %E(%t)+ CEC tfk’z) :— Q Confea‘t @ v04“): Vav<Q+bhivca~m)= b‘Vw<.m+await-3% (Maw am ﬁshedmmtu):edwbh* C%t'1,0\+b%t+5 + cam ) ->. '1 b COV ( it I itJ-S) + b'C COV (it I %-L+j-z) + LC.aV(Z’t-1)%t*s)+Czcu"(%t*1,%td => NS): (b‘+c2)CL ‘ '9’ S :0 a ‘L « - (D 0 i9 l\$l=A o: I a“; ” dec‘r‘b “acetic“ S n‘ b'C'G‘l I»? '5‘; ; 0 j ' ® 0 3* Ni 9.. ’) TM Process is St“ tic no. yj ‘ b) X, = Z1 cos(ct)+ 22 sin(ct) ® E(Xt)‘ EC%.'C\9§(Ct) +%1:sfn(ct\)’ Co; (ctlEfb)+S\h(ct)-E(h) =0 - CO'VJt‘iI/x‘é Q) V0“): VQVCL CochU—tii W‘KC—tn; C°\$ZCCf)‘Vquln)+ S\‘Az(ct)\farC£z) ‘* C Cos;l Cct) + ~ 1 € t (7 — \ MLCC )) ._.. C 1 .Co A; taut t \ACQ): Cir/(Kt; Xt-I-s) : OWL-i“ 'QDSCCt)+'%ISW‘(Ct)) %‘ GSC¢C£*S))+i‘ZSlnéchl’S); G) : C95(ct)v cojgccwspcav (3\,%.,)+ cachH‘sig (th+s))-cw(?;.,>c1) + “0 5“ (Ct)~C05 (CC-HSUQJ (12, b) + gm (ct)-§‘M(c(£+s)) Luz-(f: , h) Q’0 I * I C [MCCJCyCOSCC'Cfﬂn + ”New” “0“”) 5 CLCochtm - c-é) 61605(cg) TW ' F E. t \ \3 IS 0 C‘E‘K‘r“) ox £mng’eu'0m o S Onl‘g {:kvt V 55 V I a 5 l‘quGy j. Question 2 (7 points) Let {X,} and {Y,} be uncorrelated stationary sequences, i.e., X, and Y3 are uncorrelated for all r and s. a) Show that the autocovariance function of {X,+ Y,} is equal to the sum of the autocovariance functions of {X,} and {Y,}. ”Swag; Léé YX(5) Q'NUQ» Y‘ICS) LC he, GutOchxvfawta guwett‘om 0+ X any» y . Lat Wt: Xt +Yt_ we \JQntt" ‘(ﬂMr/l You (5) YU (S) : w _ ‘ C3V( ‘L;W-th)’C0\I( Xt*\/t’ Xt+\$,+v4:+}) : C°V(Xt X ) J t-‘S‘ +CDV( X .4- ::;Y-L~+5\ Cov(‘~{t\lxt+g)+ C0V(Yt,>’4;r5) : lx“) +rm. D Y . b) Is the process {X,+ Y,} stationary? Why? £ )CpA SVAce {Xi} «Hui {Vt} q“ lac-h; Static Why‘j I itself q) Mann Vaunlouncﬁ “WC COAS tﬁv‘ts 004.1, er auto Coﬁne [QLVOﬁ LRWo£io ‘ V1 IS ‘3 slumct {0‘4 o-(r g 0 “\b ‘ :> - (—- CD’UtC‘IAt Y (s) XLJY 5 V“)-+ ( ' ‘ t J, x ‘ Y7 S) ulna», n, c «Cuuc’c'bn 34 3 043 {mm ' 3 Park, 9") 2) {Xt-Fytg {S S‘EQ'ENHQVD‘ Question 3 (8 points) Show that the following two processes X, =Z,+¢9.Zt_li where {Z,}~WN(0,0'2) 1 2 2 Yt = a, +5124 where {3, } ~ WN(0, 6 0' ) where 0 < |6| <1, have the same autocovariance functions. {MA is °m MINA) process wi’ek {503A ﬁ.\$8~ TMWj—me, it's SiRt‘wAb y ' I ‘ ‘ S .L . [*3 I an MAO” Pmcess ou'JCk 90:) EA" 6 ‘ W‘VQ V(5£)=Ololr TkeVC‘G'OK 1 EM ACVF O‘F Vt .‘S ‘ ((SL 1 {‘45. Y 86 53‘53'ﬁjﬂsl H- \S‘SI , :3 0 )‘L \s|>! ((5) 1 | Y : 610 Zﬁ z. 7‘61 ‘ 5% ) f5; 0 (High) ’ 326 +01 \4 S=o 610‘ ° z a 35(4) (5+. 6 G 1'; = 901 air I314 ‘4 Question 4 (10 points) Find and plot the ACF p(s) for s = 0, 1, 2, 3, 4 of the following process X,= -0.8X,.2 + s, where a, is WN(O, 62). Tm; is a; HR(;) PWCerf mm (5,:0 mm”, C504 use Y..\A) eat/Matn'oﬂs ﬁ '9th HCVF; “((0): 70431;) +6" w): 'ogﬂ—U‘ L'oz, m) E) w) :0 MM = “0.3 uo) :7 ‘ ‘ W0) = (rogyC—oxww) +6 3‘ =~> Y(o) 43,“ rCo)‘—»6L :> _ CZ “‘3)’ 0.35 :3 HAV’O-ES’C =w;.;;ol 0-36 \(3)= OwYQQ) +(*o.‘5)‘Y(() :0 “W (may (-o.z,v.,)-\m) = 0811“ 0153‘, :> TR 6 mr‘"; {X1 {5 9631 ' |§ S: f(5): O . 0.5‘1 0 Hr 5:. '08 W S =;_ 0 IL 3‘3 0 0% ML 5 =4, -glg END! '93 ...
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