UNIVERSITI SAINS MALAYSIA. First Semester Examination 2015/2016 Academic Session. December 2015/January 2016
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1 UNIVERSITI SAINS MALAYSIA Frst Semester Examnaton 215/216 Academc Sesson December 215/January 216 MST 562 STOCHASTIC PROCESSES [Proses Stoast] Duraton : 3 hours [Masa : 3 jam] Please chec that ths examnaton paper conssts of SEVEN pages of prnted materal before you begn the examnaton [Sla pastan bahawa ertas pepersaan n mengandung TUJUH mua surat yang berceta sebelum anda memulaan pepersaan n] Instructons:Answer all eght [8] questons [Arahan: Jawab semua lapan [8] soalan] In the event of any dscrepances, the Englsh verson shall be used [Seranya terdapat sebarang percanggahan pada soalan pepersaan, vers Bahasa Inggers hendalah dguna paa] 2-
2 - 2-1 A post offce s manned by cler 1 and cler 2 When A enters the post offce, he fnds that B s beng served by cler 1 and C s beng served by cler 2 A s told that he wll be served as soon as ether B or C leaves Suppose that the servce tme 1 dstrbuton of cler s exponental wth mean, =1, 2 Show that the probablty that cler 2 who s servng C, completes servce 1 after cler 1 who s servng B, s () What s the probablty that, of the three customers, A s the last to leave the post offce? [15 mars] 1 Sebuah pejabat pos dendal oleh eran 1 dan eran 2 Apabla A masu pejabat pos tersebut, belau mendapat bahawa B sedang dlayan oleh eran 1 dan C sedang dlayan oleh eran 2 A dbertahu bahawa belau aan dlayan seba sahaja B atau C beredar Andaan masa layan bag eran tertabur secara 1 esponen dengan mn, =1, 2 () Tunjuan bahawa ebarangalan eran 2 yang melayan C, selesa 1 layanannya selepas eran 1 yang melayan B, alah Apaah ebarangalan bahawa antara etga-tga pelanggan, A alah yang terahr mennggalan pejabat pos tersebut? [15 marah] 2 A box contans 2 balls, some of whch are whte and others are blac At each stage of the process, a con havng probablty p, < p < 1, of landng heads s flpped If a head appears, then a ball s chosen from the box and s replaced wth a whte ball; f a tal appears then a ball s chosen from the box and s replaced wth a blac ball Let Xn denote the number of whte balls n the box after the n-th stage Is, n X n a Marov chan? Explan () () Determne the classes of the system and ther perods Are the classes transent or recurrent? Compute the transton probabltes Pj (v) Fnd the proporton of tme the system s n each state [25 mars] 3-
3 - 3-2 Sebuah ota mengandung 2 bj bola yang seblangannya berwarna puth dan selannya htam Pada setap tahap proses, seepng dut sylng yang berebarangalan p, < p < 1, untu muncul epala, dlambung Ja epala muncul, sebj bola deluaran dar ota tersebut dan dgantan dengan sebj bola puth; ja bunga muncul, sebj bola deluaran dan dgantan dengan sebj bola htam Andaan Xn mewal blangan bola puth d dalam ota tersebut selepas tahap e-n Adaah, n X n suatu ranta Marov? Terangan () () (v) Tentuan elas-elas bag system n dan tempohnya? Adaah elas-elas tersebut fana atau berulang? Htung ebarangalan peralhan Pj Dapatan adaran masa sstem berada dalam setap eadaan 3 Consder a branchng process X [25 marah],,1,2,, where Xn denotes the sze of the n th generaton and X 1 The number of offsprng from a member, Z, has probablty generatng functon: Z 1 2 G( s) E s (2 5s 3 s ) 1 Fnd G () s and G () s Hence, show that the probablty functon of Z s: z P( Z = z) () Fnd P X2 () Calculate the extncton probablty, 3 Pertmbangan suatu proses bercabang [2 mars] X,,1,2,, yang mana Xn mewal saz generas e-n dan X 1 Blangan ana darpada seorang ahl, Z, mempunya fungs penjana ebarangalan: Z 1 2 G(s)= E s = ( 2+5s +3s ) 1 Dapatan G () s dan G( s) ebarangalan Z alah : z P( Z = z) () Dapatan P X2 Dengan tu, tunjuan bahawa fungs () Htung ebaranglan epupusan, [2 marah] 4-
4 - 4-4 Consder a Posson process N(t) wth rate λ > For s³ t non-negatve and n a non-negatve nteger, determne and hence, determne E[ N( s) N( t )] () In case of s< t, fnd P[ N( s) = N( t) = n] and hence, determne () If S n denotes the tme of the n th arrval, compute E [ S4 N(1) 3 ] 4 Pertmbangan proses Posson N(t) dengan adar λ > [15 mars] Bag s t ta negatf dan n suatu nteger ta negatf, tentuan dan seterusnya, tentuan E[ N( s) N( t )] () Bag es s < t, dapatan P[ N( s) = N( t) = n] dan seterusnya, tentuan () Ja S n mewal masa etbaan yang e-n, htung E[ S4 N(1)= 3 ] [15 marah] 5 Operatons 1, 2 and 3 are performed n successon on a major pece of equpment Operaton, where 1, 2, 3, taes a random amount of tme T, whch s exponentally dstrbuted wth parameter All operaton tmes, T, are ndependent Let X t denote the operaton beng performed at tme t, wth tme t marng the start of the frst operaton If 1 5, 2 2 and 3 7, determne P n ( t) P X ( t) n [2 mars] 5 Operas 1, 2, dan 3 djalanan e atas suatu peralatan utama secara berturutan Operas, dengan 1, 2, 3, memaan masa selama suatu tempoh rawa T, yang tertabur secara esponen dengan parameter Semua tempoh masa operas, T, adalah ta bersandar Andaan X t mewal operas yang djalanan pada masa t, dengan t menandaan permulaan operas pertama Ja 1= 5, 2= 2 dan 3 7, tentuan P t P X t n n ( ) ( ) [2 marah] 5-
5 - 5-6 Consder a queueng system wth a sngle server Customers arrve accordng to a Posson dstrbuton at rate and jon the system wth probablty 1 ( j 1), where j s the number of customers already n the system Assume that servce tmes are exponental wth mean 1 If and P j denotes the lmtng probablty that j customers are n j the system, show that Pj P for j,1, 2, j! x x () Fnd a formula for P [Hnt: e x 1 ] 1! 2! () If 1 per day and 5 per day, calculate the lmtng probablty that there are, at most, 2 customers n the system [25 mars] 6 Pertmbangan suatu sstem glran dengan seorang pelayan Pelanggan tba mengut suatu taburan Posson dengan adar dan menyerta sstem dengan ebarangalan 1 ( j 1), dengan j sebaga blangan pelanggan yang telah berada dalam sstem Andaan masa layan tertabur secara esponen dengan mn 1 Ja dan P j mewal ebarangalan penghad bahawa j j pelanggan berada dalam sstem, tunjuan bahawa Pj P bag j j!,1,2, () Dapatan suatu rumus bag P [Petua: x x x e = ] 1! 2! () Ja 1 setap har dan 5 setap har, htung ebarangalan penghad bahawa sebanya-banyanya 2 pelanggan berada dalam sstem [25 marah] 7 Consder a brth and death process wth brth rates and death rates for Assume that X Let m be the expected tme to go from state to state + 1 By condtonng on the frst transton from state to state + 1, show that 1 m m 1, 1 () If ( 1) and, for, determne the expected tme to go from state 2 to state 4, n terms of and [2 mars] 6-
6 - 6-7 Pertmbangan suatu proses elahran dan ematan dengan adar elahran dan adar ematan bag Andaan bahawa X Andaan m alah masa yang djanga untu peralhan dar eadaan e eadaan + 1 Dengan bersyaratan epada peralhan pertama dar eadaan e eadaan 1 + 1, tunjuan bahawa m m 1, 1 () Ja ( 1) dan, bag, tentuan masa yang djanga untu peralhan dar eadaan 2 e eadaan 4, dalam sebutan dan [2 marah] 8 Potental customers arrve at a sngle-server ban n accordance wth a Posson process wth rate 3 A potental customer wll only enter the ban f the server s free when he arrves Suppose that the amount of tme spent n the ban by an enterng customer s a random varable havng a dstrbuton G wth mean 1 () Fnd the rate at whch customers enter the ban What proporton of potental customers actually enter the ban? Suppose that the amounts deposted n the ban by successve customers are ndependent random varables unformly dstrbuted over [3, 1] () Determne the rate at whch deposts accumulate at the ban [1 mars] 8 Baal-baal pelanggan tba d sebuah ban satu-pelayan menurut suatu proses Posson dengan adar =3 Seoran baal pelanggan aan hanya masu e dalam ban ja pelayan tda sbu semasa etbaannya Andaan amaun masa yang dhabsan d dalam ban oleh seorang pelanggan yang masu alah suatu pembolehubah rawa yang mempunya taburan G dengan mn 1 () Dapatan adar emasuan pelanggan e dalam ban Apaah adaran baal-baal pelanggan yang sebenarnya masu e dalam ban? Andaan amaun wang yang dsmpan d dalam ban oleh pelangganpelanggan yang berturutan alah suatu pembolehubah rawa yang tertabur secara seragam pada [3, 1] () Tentuan adar wang smpanan mengumpul d ban tersebut [1 marah] 7-
7 - 7 - APPENDIX 1 X ~ Posson ( λ), 2 X ~ Geometrc(p), p 1 3 X ~ Bnomal(n, p) p 1 4 X ~ exponental (λ) e x! x : P X x, x,1,2, x : 1 P X x p 1 p, x 1, 2, n P X x p q x n x x nx :,,1,2,, x : f x e, x 5 X ~ unform (a, b) : 1 f x, a x b b a 6 geometrc seres: ar a 1 r ; r 1 7 exponental power seres: e! ; GX s E s E X G 1 8 Probablty generatng functon: X X 1 P X n G X n! ( n) - ooo O ooo -
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