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1 ENGLISH TRNSLTION EXM MOT V5 EKSMEN I: MOT STOKSTISKE PROSESSER VRIGHET: 4 TIMER DTO: februar 25 TILLTTE HJELPEMIDLER: Kalkulator; Tabeller og formler i statistikk (Tapir forlag): Rottman: Matematisk formelsamling. OPPGVESETTET ESTR V 4 OPPGVER P 3 SIDER Problem Slippery roads are surveyed. In the winter time there is an inspection at 5 M each morning to make decisions on how the surface of the road should be treated (sanding, salting etc.) The condition of the road is divided into 3 categories: : Not slippery : Snow in the road, somewhat slippery 2: Ice in the road, very slippery To describe the development from one day to another they use a Markov model with a transition matrix P = :7 :2 : :4 :5 : : :6 : :3 a) omment on how this matrix characterizes the condition of the road from one day to another. Find P (X n+ = jx n = 2), P (X n+2 = jx n = ) and P (X n+2 = X n+ = jx n = ), where n denotes an arbitrary day b) Formulate the equilibrium equations for and solve these. In the long run, how much of the time is the road not slippery? c) The condition of the road is measured in friction units, where friction units are for state 2, 6 for state and for state. In terms of these units nd the expected friction. Given that it is not slippery today, what is the expected friction tomorrow? ompare the two answers and comment. d) Given that it is not slippery today, nd the probability that it will be very slippery in the course of the three next days. If you have trouble in getting an explicit answer, at least set up a matrix expression, and explain how you would go about nding the probability asked for.

2 e) In fact the condition of the road does not only depend on the condition of the road yesterday but also on the condition the day before yesterday. Show how you can model this as 9-state Markov chain. Moreover, set up the transition matrix for this chain when it is informed that the same state 2 days in a row gives this state for the next day as well with a probability of.5, and with a probabuility of.25 for each of the two other possible states. Furthermore, for two days in a row with dierent states, the state the next day will be equal to last day's state with probability.5 and equal to the rst day's state with a probability of.3. Problem 2 To obtain a drivers licence the candidates in the town of Timbuktu have to go through a theory test. There are 3 dierent outcomes of the test, namely acceptable, almost acceptable and bad. If a condidate does not get the acceptable grade, he has to wait before he is allowed to take the test again (with new questions). If this person had almost acceptable as his grade, he has to wait for week until he is allowed to take the test again. If he gets the bad grade, he must wait for 2 weeks. Our candidate has a probability of.5 for getting the acceptable grade,.3 for almost acceptable and.2 for bad. He is very lazy and does not study if he does not pass. He just tries again. What is the expected number of times he has to try this exam before he passes? What is the expected time until he passes? Problem 3 large medical center has 3 practicing doctors. It is not possible to reserve a time for appointment. The patients are treated on a rst come rst serve basis. The patients are arriving according to a Poisson process with an arrivel rate of 6 per hour. The time of consultation with the doctor is exponentially distributed with an expectation of 5 minutes. a) Let N(t) be the number of patients that arrive in the time period [; t), where the time unit is one hour. Find P (N() = 3), P (N(2) = 9jN() = 3), E(S jn() = 4), where S is the time until the -th patient arrives. b) Let T be the consultation time for an arbitrary patient with the doctor. Let the time unit be one minute and nd P (T 5), P (T 25jT 5). Moreover, if all of the doctors are busy, nd the distribution for the time T min until the rst doctor becomes avaiable. t last nd P (T min > 3) and E(T min ). c) If all of the doctors are busy and in addition you have 8 patients in front of you in the line, what is the expected time until you can be served by a doctor? What is the probability that all of the three patients that are with the doctors when you arrive, have left when you are getting served? What is the probability that your consultation

3 with the doctor will be nished before the consultation will be nished for the person that are just ahead of you in the line? d) Let the state n in this system denote that there are n patients at the medical center (included those that are treated and those in line.let the time unit be one minute and explain why the arrival rate of the system is n =, n = ; : : :, and when it comes to the departure rate n, we have =, 5 2 = 2, 5 n = 3 for n 3. Set up the 5 balance equations (you are not required to solve them), if all patients that arrive at the medical center when there are patients in the line, just leave the center.

4 Fasit, med forbehold om feil Oppgave a) The rst line in P gives the probabilities for not slippery, soewhat slippery and very slippery tomorrow given that it is not slippery today. P 2 = :7 :2 : :4 :5 : :6 : :3 :7 :2 :8 :4 :5 : = :6 : :3 :63 :25 :2 :54 :34 :2 :64 :2 :5 From this it follows that P (X n+ = jx n = 2) = :, P (X n+2 = jx n = ) = :63, P (X n+2 = X n+ = jx n = ) = P (X n+2 = jx n+ = )P (X n+ = jx n = ) = :5 :2 = :: b) Equilibrium equations: :7 + :4 + :6 2 = :2 + :5 + : 2 = : + : + :3 2 = = Solutions: = 34=56, = 5=56, 2 = 7=56. Not slippery 34/56 ot time. c) Expected condition in the equlibrium: = E(Glatthetsgrad i morgenjikke glatt idag) = :7+6 :2+ : = 2:9 < ; which makes sense since one expects less slippery given not much slippery today.. d) bsorpsjonsteknikken brukes: (P ) 2 = (P ) 3 = :7 :2 : :4 :5 : : : : :7 :2 : :4 :5 : : : : P = :7 :2 : :4 :5 : : : : :7 :2 : :4 :5 : = : : : :57 :24 :9 :48 :33 :9 = : : : :57 :24 :9 :48 :33 :9 : : : x x :27 x x x : : : Det vil si den skte sannsynlighet blir.27. (Den kan ogsa nnes pa mer tungvinte mater.)

5 e) Tilstander (idag,igar): =, =, 2 = 2, 3 =, 4 =, 5 = 2, 6 = 2, 7 = 2, 8 = 22. P = :5 :25 :25 :5 :3 :2 :5 :2 :3 :3 :5 :2 :25 :5 :25 :2 :5 :3 :3 :2 :5 :2 :3 :5 :25 :25 :5 Kanskje lite sannsynlig at ikke glatt to dager pa rad gir.25 sannsynlighet for snfre og.25 sannsynlighet for isfre imorgen. Oppgave 2 La N vre antall ganger han gar opp. First step analyse gir E(N) = :5 + :3[ + E(N)] + :2[ + E(N)] = + :3E(N) + :2E(N); dvs. E(N) = 2: La T vre tid brukt i uker: E(T ) = :5 + :3( + E(T )) + :2(2 + E(T )); dvs. E(T ) = 7 5 : Oppgave 3 a) Vi har P (N() = 3) = 63 3! e 6 = :89; P (N(2) = 9jN() = 3) = P (N() = 6) = :6; b) Vi har E(S jn() = 4) = = 2: P (T 5) = e 5 5 = e = :368; P (T 25jT 5) = P (T ) = e 5 = :53: T min er eksponensialfordelt med = =E(T min ) = 3=5. Derfor P (T min > 3) = e 9 5 = :549; E(Tmin ) = 5 3 = 5min:

6 c) U: Tid fr du slipper til. Vi har E(U) = 9 5 = 45min. P (lle tre ferdigbhandlet) = P (En igjen) P (To igjen)! 3 2! = = :923: P (Du ferdigbehandlet fr person foran) = : d) nkomstraten n = svarer til en konstant ankomstrate pa 6 pr time. vgangsratene svarer til situasjonen at det er henholdsvis, 2 eller 3 leger i arbeid. alanselikninger: Tilstand : P = 5 P Tilstand : ( + 5 )P = P P 2 Tilstand 2: ( )P 2 = P P 3 Tilstand 3: ( )P 3 = P P Tilstand 3: 3 5 P 3 = P 2