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Test2-sol - St at 5’4 0 Winter 2003 STAT34O Winter 2008...

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Unformatted text preview: St at 5’4 0 Winter 2003 STAT34O - Winter 2008 - Test 2 Family (last) Name: Given (first) Name: ID #79: User Name: Marks Available: 30 Please circle the ecrreet seeticun number belcw: Name Seeticun Leetnre Time R. Metzger 1 TR 10:00—11:20 R. Metzger 2 WF 1:30—2:20 Part ef yenr mark will be assigned fer the clarity ef yenr selutien. An answer Withent justificatien is werth 0 marks. The enly permitted aid is a math faculty appreved ealculater. Page 1 ef 5 St at 5’4 0 Winter 2003 1. Dr. Hap Haggard has created a new ceniputer Virue called Helter—Skelter. ln eeeence thie Virue eende randeni nurnbere at cernputere until they craeh. (a) Dr. Hap Haeeard ehewe yeu a eet ef randern nurnbere generated by Helter—Skelter. The numbere fellew: ...5, 6, 7, 3, 9, U, 1, 2, 3, 4, 5, 6, 7, 3, 9, U, 1, 2, 3, 4, 5, 6, 7, 3, 9, U, 1, 2... 1t ie ebvieue that Dr. Hap hae ueed an LCG te generate the randern numbere. Hie LOG, in fact, wae 23,, : (622,14 + c)n1ed 10. I [4 Marks] Find a. and c. an : (amn_1 —|— c) nied1U $2 = (1125 + c) Ined 10 6 : (5a—l—c)rned 10 (1) 7 = (661 + c)n1ed 10 (2) 7—6 : ((6a+c)— (5a+c))rned10 1 = arned 1U rTherefere a = 1, replacing in (1) 6 = (5 + c) med 10 1 = crned 10 rTherefere c = 1 (er any e = 102 + 1 Where 2 ie a pee. integer) I [2 Marks] 1f thie LCG fellewe the theerern in claee, we’d eay that the maximal peried ie 9. What ie the peried, and Why deee thie LCG net fellew frern the theerern? The peried ie 10, and the LCG deee net fellew it (the theereni) becauee 10 ie net prirne. I [4 Marke] On page 5 yeu will eee a figure ef the randern nurnbere in 2 dinieneiene (Fig. 1). Calculate the Obeerved Teet Statietic, ueing the indicated areae in the figure, te teet Whether the randeni nurnbere (the ceerdinatee) are uniferrnly epread eut in 2 dirneneiene. 06661966 6 6 11 16 25 ’i'T 4 — 21' 1 Expected 66 4 = 11.16 66 I = 12.96 12.96 12.96 12.96 are _ 5 (e, — 6,)2 _ (6 — 12.96)2 + (6 — 1:21.16)2 + (11 — 12.96)2 _ 1:1 6, _ 12.96 11.16 12.96 +(16 — 12.96)2 + (25 — 12.96)2 12.96 12.96 21.88 Page 2 ef 5 St at 5’4 0 Winter 2003 (b) Medeled similarly tn the Mentreal Preblem: He wants te see hew leng it takes befere hie master eemputer erashes. The master eemputer is eemputer number 1 and crashes when all ef the eenneeting eemputers, {2, ...,n}, have erashed. The V1I‘US will kill each eenneeting eemputer i E {2, ..., n} in T, minutes where f, (t) = (lni) (i4) and t ii} U. A diagram (Fig. 2) ean be feund en page 5. I [5 Marks] Determine F(t), the distributien ef the time befere the very first breakdewn. Leave yeur answer as a funetien ef n! and t. Yeu may NOT use the next part ef questien 1a te get this part ef the preblem. t PM) = (1112') J‘s—tat D | H , H H. I MI :4. H m I—‘r | Jim Hi. i “a..____.«-" || || |—1 |—'* | | ”E “if - c-l- 9;; C93 _- 4+ 23 3'3 | H_-r -:+ l, | | |_i | r—a E. h._.# .1, I [6 Marks] The number ef eemputers (master and nenmasters) is n, where n E {2,3, ...,6}. The I‘ll I 1 jeint distributien at t and n is f(t,n) : (”l—511W with marginal fin) : 3. Use mente earle integratien te estimate the E(T{N : 3). Yeu must use the fellewing unifermly generated randem 1 numbers: 1,1. Let y = L :‘;~ t = l — 1 t+ 1 y 1 ——dy = dt y2 E(TlN=3) = ftfltln) alt: ft- dt 0 [:1 10(3) 2 ft-3—*1n(3)dt U 1 {1 1) 1 _ __ 1 : ——1 -3 "y ln3-—dy Elly ) l l 132 l 1 ll 1 1 _ —_ 1 U 3-(0)13-— ——1 -3 1/“;1 13 (l “(l 1 T (1/4 ) “(l (1/4)2 a" m ((3) - 3-i3) ln (3) - 16) 2 g [1977 Page 3 sf 5 St at 5’4 0 Winter 2003 (c) Dr. Hap Hassard has decided that the Repair Prehlem might be a better medel te use since it allcevs repairs. He assumes that anti—viral seftware werks en any ene cf the n. cempnters. The hreakdcevn times cf the ccmpnters are assumed tc be Ti m EXP(1) and the repair times cf the ccmpnters are assumed tc be Ry w EXP(2) Where i E {2, 3, ...,n} (Recall: ccmpnter 1 is the master ccmpnter and is never directly attacked by the virus). Define the state tc be the number cf hrc-ken ccmputers. Let m be this number. Nete m = n. — 1. I [1 Marks] In this prehlem, define and previde netatien fer the rate cf death. A death rate is the rate at vrhich WEE ge frem state m te m — 1. In this case a cempnter gets fixed. The rate is pmvhere i is the state. I [2 Marks] Determine the prepertien cf time that the system is in state 2., given that there are 3 8 nenmaster cempnters in the system and aim = —. 27 (Afldln-Ae—l) m] — mew-m: 32 72-4 2 2 9 7’72 4 I [2 Marks] 1f the prcpcticn cf time the system is in state 1 is in = g, in What state will the system he in, in the leng run? Why might this make Dr. Haphassard happy? E (N) = Zuni II c: xiv—“s. be “Jim x..___,.«' _|_ {fill-ll- _|_ be m melee x..___,.«' _|_ c: m H. | {film | :5ch I be film x..___,/ | | |_'L Only ene cc-mpnter (his) is vrerking I [4 Marks] Give the system state equaticn (i.e. rate cf leaving : rate cf entering) when there are I: ccmputers that have hrcken dcevn. Me : 2+2+...+2=2k he 1+1+...+1=(n—k) age (pk + Ag) 2 ak+1pgfl+1 + ak_1)x;fi_1 (general fermula) We (its: + As) = ve+1se+1 + mafia—1 (T1 -|— 313)ka = 2013 -|- 1)?T,LH_1 -|— (H — ll”; — 1)’n',zc_1 Page 4 cf 5 St at 5’4 0 Winter 2003 Eircle Eircle [lentre Eentre Eircle: Eircle: Radius=| Rafiius=l IE Coordinates H Ennrdmates I Ellnnr‘dinatas Eflnurdinates , Jr” / 25 Ennrdinates Hf EiPEIB: f Eir'cle: Radiu3:| ' Radius=l EirEIE . Circle [lentre Equara [If Length 2 BMW Fig. 1 2 .I 3 l Ir n Fig. 2 GRADE :($) Page 5 0f 5 ...
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