Quiz_5_Solutions

# Quiz_5_Solutions - STAT 333 Quiz #5 Nov. 30, 2009 4:35 —...

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Unformatted text preview: STAT 333 Quiz #5 Nov. 30, 2009 4:35 — 5:20 pm Name: S I.D: UWUserid: Mark / 20 1. Suppose that trafﬁc at a certain point along Westmount Road can be described by a Poisson process at rate A = 2 per minute and that 60% of the vehicles are cars, 30% are trucks, and 10% are buses. Furthermore, it is assumed that vehicles are independent of one another. [3] (a) What is the joint probability that exactly 3 trucks and exactly 2 buses pass by this point in a 5-minute span of time? [8—6 N7- refresgn‘f‘ '62: numéer O70 7Lrucl<s faSSI'Iy é '6‘Iefoin7L évfime—f. ' [El’ IVS Vﬁffaerfl— {ﬁei hum Aer 670 buses/0435119 loo/n7" 5/1 {I'me We know @a’f‘ N7- is 535011 (DI/:S'ﬁiébnl—QJ lax/7% mean 2-031“ = 0.6“? ave N569) is lac/'55:»: dis+riéu+eal W41 mean 2 - 01+ -_— 0.2+. 726(3) F(N7‘(5)=3)N3[5):Z) = Jae 7L0 Mo’s/Dena/ence :: .gc—jéi. CHI/2’ . 3! ' .2! = 73-642: 0.0‘+/Z [3] (b) Calculate the probability that 2 trucks will pass by this point before two vehicles that are not trucks pass by there. Lei— Tc/ena'zLe "tack/’59 N denote "non~—7me/<ll(£g. car or bus). FOSS/He Camélha’ll'rlonr .f C l} ‘ (2.) TN T (3) N T T Based Oh'H’lese COMLina+ian5> FéZ-frucks fuss by [ye-Pore Z non—’h‘uck Vehicle: Foss by) = (0.3)2‘+ 260.3)4607) =2 (AZ/é [3] (c) What is the probability that the length of time between the. ﬁrst and third arrivals of any vehicle passing this point exceeds 45 seconds? 711C {-ime’ T balk/ea Ve/pic/e arrIVa/s 1.: 6,62me Z) wan-meal Me com Ea): /— (2%! = /-— (/+Zx)€ZX,X>/O. .77ms, FCT> 45sec0nds) : /’(7‘> % mmﬁes) :(/+Zo%)e~2'% = 572 63/2 20.5578 [3] (d) Given that 5 cars passed by the point in question over a 4—minute time period, What is the variance of the total number of vehicles that passed by that point during these 4 minutes? Le‘li‘ Nﬁé) Felprggen'f ﬁe numLer Veil/ales Fags/‘3 é), rave/00,571" “firm: refrerem“ ‘Hze nanéer 07c Can: fassiv 5); "éheFap'77L.é), '{v‘me L66 N” ('5) refresernL 'Hze. huméer a‘lz1 "Non—car: 0/0435?) 5/5515 [no/if!“ 5/ {line'é NO'lLe ‘Hm’i‘ N(+)=/Vc[‘t'-)+M,[+)) WAere A/N A4003,” (2. a I 7710,, l/ar (N(‘f)/Nc€‘f)=5) = l/arKA/CM) +/V~(’7‘)/Nc(ﬁ‘) =5) = Wr(5+A/~(4H/A/C(ﬁ‘)= 5) : l/al’CA/NZLH/Nc64):5> V = Var (NA/(4H) due 7’17 Male/Deﬁance = 2-0.4(41 [4] (e) Suppose instead that = Z 25/5 for OStglO, /\=A(t)= 3—0—3392 for 10<tg50, (60—t)/5 for 50<t~\$ so. Calculate P(N (60) — N (40) = 30), where N (t) represents the total number of vehicles in the interval [0, t] I We hzea/ 710 ca/cu/a'iLe/V m (GO)——-n—, (40): (if)? (Holt I ' V I c 50 _30 z (,0 g; [3— *2” )Ja’l‘l“ 7L; 90" o/‘Z’ 5 ’ l =300>~zofz We +5J33ec/e _‘ 30-20é3/y: + 5-2—2 02 . = 30— 5-5 + /0 : 205A4 «205/ 0 0 724(5) /’(N[éO)—N(4o)=30)= 6 «(25); c,» 0. 0557 _ 30! ' [4] (f) Corresponding to the same assumptions as in part (e), let 6'1 denote the arrival time of the ﬁrst vehicle to pass by the point in question. Find the (conditional) probability density function of 81 given N (4) = 1. DE'IOine. 1779. Conn/[7L1'Ona/ =—‘ ='/) 7%,- OS'XSAA 77'8"; gm: M0. = ﬂamm- [0x3 acme/n Q43) ’DCNH’W) FCNC‘H=/) - _ P( = = ._ W by;ngere"ie,ﬂ— PK/VK‘H—ﬂ) mama-:3 X . a 3 Now; mc’thﬁmaie = 7% a": 7’5;— ﬁaa m(‘i‘)~m6v)=£:’)(+)J-{- =§XL MO“ 723 .> - 760 2 —I.6_+x§/D ' Thus, Goo: z _x_z_ MW e "60.4) /6 J HEnce/ ‘Hie desired [Ea/V8115); \96<)=GZY)= %— 7494 0<X<4/‘ STAT 333 Quiz #5 Nov. 30, 2009 4:35 — 5:20 pm Name: S LD: UWUserid: Mark / 20 1. Suppose that trafﬁc at a certain point along Columbia Street can be described by a Poisson process at rate A = 2 per minute and that 50% of the vehicles are cars, 30% are trucks, and 20% are buses. Furthermore, it is assumed that vehicles are independent of one another. » [3] (a) What is the joint probability that exactly 2 trucks and exactly 3 buSes pass by this point in a 4—minute Span of time? [.e‘l‘ Ma‘) refresen‘f 1972 number 070 ﬂack: «5-5)ng "five ain‘t é/ time t L619 A4366) rCfVCSeﬂL 1916 Huméer' 07A 545‘?! fit—951'!) 519k fC/Fﬂl‘ d), "H "166. We, know #41“ A666) is Ina/550,7 c/jsﬁiéu'fﬁq/ while mean 2' 0.3% =: 0.6% M9 N366“) is /U0/'5‘50n algﬁiéuf‘eo/ 140.7% mean 2' 0.2‘6 == 0/7479, 7;!145‘) F(Nr(4)=Z)NBCLI-)=3)=F(NTH) =2)P0VB(’7‘)=3) due +0 in deFeMdence = Meme-4)". Z ! 3! = Amara": 0.03é0/ [3] (b) Calculate the probability that 2 buses will pass by this point before two vehicles that are not buses pass by there. Left 5 Gland/‘9. “Aux” ﬂag N alﬂho+e "harp—éasl/6:8.Caror7Lruck), POSS/ﬁle CombinaﬂLiong: CI) (2) ENE C3) N88 Based on 'Hlese combinarhohs) FCZ buses fuss by be'IQoreth-éas WM“ fwd?) -—— (azﬂ— 260.2)‘Co, 53> ' = (2/04 [3] (c) What is the probability that the length of time between the ﬁrst and third arrivals of any vehicle passing this point exceeds 40 seconds? 726 fibre 7— éa+Wecn /"/+anq/ ehia/e arrivals is GAMMQ (.2) Z) dis+ribu+eal wi+h c479 5Q): /— 523% (2x141: /~(/+zx2e”2’;x>0. 756(5) 405ec0n¥9) = T> Z/QMI'hu/Es) = (/+2c %) 672% = 7/3 ef‘Va 9: 0.47/57 [3] (d) Given that 4 trucks passed by the point in question over a 5—minute time period, What is the variance of the total number of vehicles that passed by that point during these 5 minutes? L676 Farresen'f' 'Hze. numbe» 07a VeAio/es [pas-5 it? ‘éIYC/Dolhf £21 {137,676. L66 A469) represen—F'ch humécr- 07C llhan—TLraa/x’s 30a 55/27 A/ﬁe {fine /v0+e {7m— A/(H: Maw/tare) w/ym Mew/item 62-07%). Wm, Var (Nay/v7 (3:4) = Mir (NT(5)+A/N (5)//vr(5)=4) - = Var (4f +M,(5)/A/T(5)=4) = Var due 7L0 Ibo/efena/ence y: ,a. 07(5) : 7 I t/lO for O S t S 20, Mt) = 3 — (‘ﬁﬂf for 20 < t g 40, (60 — t)/10 for 40 < t g 60. [4] (e) Suppose instead that A: Calculate P(1V:(60) —. N (30) = 45), where N (t) represents the total number of vehicles in the interval [0, t]. _ 4 » We ma +0 (la/came m(éoj—mcao>=36‘h'z+)d+ 3o 0 :__ 9" _-(-—3az {4,—5’ 3k; (la 0 Oli— /O / ﬂack/0; We +1/0fozﬁdt :1 30 e. fig/weal 2. =30~J953+ZO : mic/3 ' .‘J‘f'O Thus; F(N(4o)~A/(3o)=45) = 8 A36]? 45 45/ :3 0.0574 [4] (f) Corresponding to the same assumptions as in part (e), let 51 denote the arrival time of the ﬁrst vehicle to pass by the point in question. Find the (conditional) probability density function of \$1 given N (6) = 1. De'Pihe {he Cona’fJ-I'ona/ CCJ‘ID €60: 106V l 7772") GCX)= P(5!\<X2N(é)=/) : FC/everﬁl‘l'n [#0)ij Oevehtin(xjéj) P(N(é)= I) RCA/(aw) PTA/(é): I) incremen'is 2 7 ' é ’ z 6 Now) m&)=J;),\$/l(+)0/{'= 571,”; g; m Inca—meg: 26% = 5—; =/.a°_ ~21; )< (a x720) (ca—"8+ X1 3 :2 W : ~“_‘ \< S 7;!“ ) 8_/'? (/I r gé ) O X G HQHCE) ‘Hze C/ﬁSirea/faclja i\$jiv&é// 700” O<X<éf ...
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## Quiz_5_Solutions - STAT 333 Quiz #5 Nov. 30, 2009 4:35 —...

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