Quiz_4_Solutions

Quiz_4_Solutions - STAT 333 Quiz #4 Nov. 10, 2009 4:35 —...

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Unformatted text preview: STAT 333 Quiz #4 Nov. 10, 2009 4:35 — 5:20 pm Name: S I.D: UWUserid: Mark / 20 1. Laurel and Hardy enter a bank simultaneously — Laurel to make a withdrawal and Hardy to purchase some mutual funds. Suppose the amount of time it takes to make a withdrawal is exponentially distributed with a mean of 10 minutes, and the amount of time required to purchase mutual funds is exponentially distributed with a mean of 25 minutes. Suppose that Laurel and Hardy are each served by independent tellers immediately upon entering the bank. [3] (a) 80% of the time, the time it takes to make a withdrawal will be at least how many minutes? Let X a’zno7Lfi 19m /eU7L/1 070 19m; ‘7‘?) make. a wiviéalmwal ~x/o 7778b) X’VEx/Danah'l'l'a/[2-1 ,l/Io) an/ 30 = e ) X7/0 VVC 7‘a JelLerm/‘ne f Sue/7 fia‘l‘" = c— /”’= 0.8 “f/a = /n 0.? 14:. 405702;: 223/ 777’“; 5* fluke: .aIIL’casf‘ $237 mini:ch +0 make a wh‘fic/rawa/ (3’0 % a7°7i$e7ihra [3] (b) What is the probability that Hardy finishes service before Laurel? Let Y c/emfi 7%; Anjfl 070 19mg 7Lb/ourcfiase mu+ua/ #um/s. 7—5217) Y” Byonerfl‘ia/Q = [/25) . Ms / FKHm—é, 'Fl'nislies 527%]! Laure/) m =§/2—+‘/“ = -$—‘- [3] (c) On average, how long does it take for the first service to finish? Ad T ale”°7Le 7916 film: 760 2972 #irs't" service camF/e'h‘on. 7291/, 7—: min Yffvaronen’fial (7%- + 254-: i126 = 33%,) 777415, 7/,50 = $71.2 7/43 mindfu- I [3] (d) It is observed that after 7 minutes Laurel is still being served by his teller. What is the probability that Laurel’s service time will be at most 13 minutes long? We Wis/r +0 a/a7l'ermine. /D(X\</3/X>7) = F(X\< 7+é/X>7) =fCX\<é)é/IfieMeMo es: frof rb/ =/__ej6/Io =/ _ 8—0.6 1’ O. ‘7‘5/2 2. For a branching process having X0 = 1, ‘calculate the probability that the population will eventually die out when the number of offspring (produced by any one individual) has probability distribution given by: [4] (a) P0 = 2/5, P1: 3/10, P2 = 1/5, P3 = 1/10 3 Fills‘t‘) We aka/fie”: = CI)(%)+ (z)(’/5)+(3)(%,) = / (Since/AS //~ We. can COHC/uJe 7424+ 770—: [4] (b) P0 = 1/8, P1 = 3/8, P2 = 3/8, P3 = 1/8 3 Fir-sf, we ca/cuéfll'd/u: =C/)(%)-I—(Z)[3/g)-I—(3)(%)= 77/ Slncf/M 7 l) 77; i5 'fliedsmé//es+ F03l+ive 50/u7LI.0n $795649 in = g, 773/; 4 V 77;: %+ %77°+%7/22‘+ %7[3 7234—3732—5773 +/ =0 (770—1)sz4— 47/34) =0 TE=I —Z:\/5— _/4_N_0. . : ;c-Farmuld~, owinea) vfa $uaolrn 77m, We see 7974+ 770 =— ~2+1f§ 2 0.236. STAT 333 Quiz #4 Nov. 16, 2009 4:35 — 5:20 pm Name: S LD: UWUserid: Mark / 20 1. Thelma and Louise enter a bank simultaneously ~ Thelma to make a withdrawal and Louise to purchase some mutual funds. Suppose the amount of time it takes to make a withdrawal is exponentially distributed with a mean of 8 minutes, and the amount of time required to purchase mutual funds is exponentially distributed with a mean of 18 minutes. Suppose that Thelma and Louise are each served by independent tellers immediately upon entering the bank. [3] (a) 60% of the time, the time it takes to make a withdrawal will be at least how many minutes? Lef /\’ alena'l‘e ‘yle /ev7% of, 7Limc 7‘22 make a WI'7q7z/rawa/. 721cm X’V EX/acmemL/a/ (Q = V?) aha/so PKX>X) = aux/8) X20 We wfsA 7L0 O/EéI’mine ‘LL sac/7 7%a7L FCX>f) = 0.4: 6—178 = aé "f/s =/nO.é '6' = “8570.63 777(6) hL +a/(es_ 47L /eas+ 087m1'nu7l‘es 1L0 make a WI'7%/ra wa/ 60 % mafia 7‘7!!!- [3] (b) What is the probability that Thelma finishes service before Louise? 16+ l/ Jena 5 '6‘): kry‘fli 07a 7L/me "/22 Furcfiase ma7zua/ 744ml: .7716”) Y’V EXFonen‘hfi/éfl: l/l? ), P(7l12.lma 'pinishes loaf-lore [DU/3‘8) = T“ — __._‘t_ = i— [3] (C) On average, how long does it take for the first service to finish? 1.8+ T O/QH07LG. {be {II/77¢. f0 7%: #if'sflSElfi/I'ce camf/gfigrh 72k") 7—: "7"” YENEXFonafiL/a/(‘TgL + 7% = 97:4 = .4:— I 771113) = ,3/72 =' gag 5538Minu+é5 [3] (d) It is observed that after 13 minutes Louise is still being served by her teller. What is the probability that Louise’s service time will be at most 19 minutes long? We Wish +0 cia‘f'ermine. =F(y\</3+G/Y>/3) =/0(Y\<é) é 797a menu»— 4555‘ = /_fl 5 6/0] Fro/every 3 0.2835 2. For a branching process having X0 = 1, calculate the probability that the population will eventually die out when the number of offspring (produced by any one individual) ' has probability distribution given by: (a) P0=1/8, P1=3/8, P2=1/4, P3 Firs‘l) We caiculn‘i’g/u =. = J=D (Sich 7/) 7/7; is IL/ve SMa//es+/005}’7l‘ive 50/147901: 5a7LLs'P/iv . 3 , m=zmw d :0 77: 73+ %=7/3+%;7/22+ flf/f 277i+275£57c +/ =0 (742—0 (2m2+4m—/)=O 7/3=/ rive —-/i- 1/2: K $7etl'hea/ Via ZuaJra'l‘I'c grinqu 77m) Wasee‘i'l’la'i' 7):: — 1+ 52: r: 0,225. [4} (b) P0 = 3/5, P1 = 3/20, P2 = 1/12, P3 = 1/6 Fins-7: We ea/cd/m‘e/M = =C/)(%a)+QX%z)+(3)[%) 4—; 4%0 Sl'nce/M< /) we Can conc/uc/e. 7990171“ 7F:/. ...
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This note was uploaded on 07/12/2010 for the course STAT 333 taught by Professor Chisholm during the Fall '08 term at Waterloo.

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Quiz_4_Solutions - STAT 333 Quiz #4 Nov. 10, 2009 4:35 —...

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