Probability and Queuing Theory Qp

7/14/2019 Probability and Queuing Theory Qp

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Reg. No. : I I I I I I r I [ J ]I u e s t i ~ ; ; P a ~ - - ; r Code 315 24\

B.E.lB.Tech, DEGREE EXAMINATION, NOVEMBERIDECEMBER 2013Fourth Semester

Computer Science and EngineeringMA 2262/MA 44/MA 1252/10177 PQ 4011080250008 - PROBABILITY AND

QUEUEING THEORY(Common to Information Technology)

(Regulation ?008/201O)Time: Three hours Maximum: 100 marks

1.

2,

3.

4.5.6.7.8.

(Use of statistical tables may be permitted)Answer ALL questions.

PART (10 x 2= 20 marks)A coin is tossed 2 times, if X denotes the number of heads, find the probabilitydistribution of X,f the probability that a target is destroyed on anyone shot is 0.5, find the

probability that it would be destroyed on 6th attempt.If the joint pdf of the two-dimensional random variable ex Y is given byf(x, y = Kxy e- x 2 +yl, x> 0, y>O find the value ofK.State central limit theorem.Prove that first' order stationary random process has a constant mean.Prove that Poisson process is a Markov process.Define Markovian Queueing Models.Suppose that customers arrive at a Poisson rate of one per every 12 minutesand that the service time is exponential at a rate of one service per 8 minutes,(a) What is the average number of customers in.the system?(b) What is the average time ofa custqmerspends in' the ystem?

9. Define series queues,10. Define open network.

PART B - (5 x 16 = 80 marks)11.. (a) (i) Find the MGF of the random variable X having the pdf (8)

(ii)l x, for 0 < x


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b)

(ii) The TPM of a Markov chain {X,J, = 1, 2, S .. having three states

[0.1 0.5 OAl

1, 2 and : is P = 0.6 0.2. 0.2 and the initial distribution isO.S OA 0.3J

p O) = {0.7, 0.2, O.l} find1) p [X2= 3]2) P [X = 2, X2 = 3, Xl = S, Xo =

Or8)

i) A salesman territory consists of three cities A, Band C. He neversells in the same city on successive days. If he sells in city A, thenthe next day he sells in city B. However if he sells in either B or Cthe next day he is twice as likely as to sell in city A as in the othercity. In the long run how often does he sell in each the cities? 8)

(il) Suppose that customers arrive at a bank according to a Poissonprocess with mean rate of 3 per minute. Find the probability thatduring a timeinterval of two minutes .(1) Exactly 4 customers arrive(2) More than 4 customers arrive-3) Less than 4 customers arrive. 8)

14. a) (i) A T.V. repairman firtds that the time spend on his .job has anexponential distribution with mean SO minLltes. If he repair :sets inthe order in which they came in and if the. arrival of sets isapproximately Poisson with an average rate of 10 per 8 hour day,what is the repai rman s expected idle time each day? How many \jobs are ahead of average set just brought? (8)

(il) Consider a single server queueing system with Poisson input,exponential service times. Suppose the mean arrival Tate is3 calling units per hour, the expected service time is 0.25 hours andthe maximutn permissible number calli:ng units .ir:, the system istvc,o. Find the steady state probability distribution of the number ofcalling units in the system and the expected number of calling unitsin the system. 8)

(b) (i)Or

A telephone exchange has two long distance operators. It isobserved that, during the peak load long distance calls arrIve in aPoisson fashion at an average rate of 15 per hour. The length ofservice on these calls is approximately exponentially distributedwith mean length 5 minutes.

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15. (a)

Find;1) The probability a subscriber will have to wait for long distancecall during the peak hours of the deW2) If the subscribers v:iJ wait and arc serviced in turn, what is

the expected waiting time? 8)(ii) Customers arrive at a :oa:es counter manned by a single person

according to a Poisson process with mean rate of 20 per hour. Thetime required to serve a customer has an exponential distributionwith a mean of 100 seconds. Find the average waiting time of a

(i)customer. (8)A car wash facility operates with only one bay. Cars arriveaccording to a Poisson distribution with mean of 4 cars per hourand may wait in the factory's parking lot if the bay is busy. Theparking lot is large enough to accommodate any number of cars. fthe .service time for a car has uniform distribution between 8 and12 mihutes Find:1) The average number of cars waiting in the parking lot and2) The average waiting time of a car in the parking lot. 8)

(il) There are two salesmen in a ration shop one incharge of billing andreceiving payment and the other incharge of weighing anddelivering the items. Due to limited availability of space, only onecustomer is allowed to enter the shop, that too when the billingclerk is free. The customer who has finished his billing job has towait there until the delivery section becomes free. If customersarrive in acccrdance with a Poisson process at rate 1 and the servicetimes of two clerks are independent and have exponential rate of3 and 2 find(1) The proportion of customers who enter the ration shop.2) The average number of customers in the shop-3) The average amount of time that an entering customer spends. in the shop. 8)

Orb) (i) Derive Pollaczek - Khintchi:le formula. 8)

(ii) A one man barber shop takes exactly 25 minutes to complete onehair-cut. f customers arrive at the barber shop in a Poisson fashionat an average rate of one every 40 minutes, how long on the averagea customer spends in tJ:_e shop? Also find the average time acustomer must wait for serv;.ce. 8)

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60Reg.No:1 TTITl- I 1

Question Paper Code: 2 524JB.E. B.Tech. DEGREE EXAMINATION, M YIJUNE 2013

Fourth SemesterComputer Science and Engineering

MA 22621MA 441MA 1252/10177 PQ 4011080250008 - PROBABILITY ANDQUEUEING THEORY

(Common to Informati on Technology)(Regulation 2008/2010)

Time: Three hours Maximum: 100 marks

1.

2.3.4.5.

6.

7.8.

9

Answer ALL questions.PART (10 x 2 = 20 marks)

If X and Y are two independent random variables with variances 2 and 3,find the variance of 3X + 4Y .State memory less property of exponential distribution.f he joint pdf of X, Y) is given by f x,y) = 2, in 0 OS x < y OS 1 , find E(X).

State Central limit theorem.Define wide sense stationary process.

f the transition probability matrix (tpm) of a Markov chain is ~ . , find the 2 2

steady state distribution of the chain,What are the characteristics Qf a q u e u ~ n g system?What is the probability that a customer has to wait more than 15 minutes toget his service completed ina MIMII queuing system, if = 6 per hour andj l = 10 per hour?State Pollaczek-Khinchine formula.

10. Define closed network of a queuing system.

11. (a) . i)

(ii)

b) i)

(li)

61PART B - 5 x 16 = 80 marks)

A continuous random variable has the pdf f x) = hx4 ,-1 < x < O.Find the value ofh and also {X > -i jX J} . 8)Find the moment generating function of Uniform distribution.Hence find its mean and variance. 8)

OrFind the moment generating function and rth moment for thedistribution whose pdf is f x) = Ke- x ,0 OS x OS w. Hence find the meanand variance. 8)In a large con signment. of electric bulbs, 10 percent are defective.A random sample of 20 is taken for inspection. Find the probabilitythat 1) all are good bulbs 2) at most there are 3 defective bulbs3) exactly there are 3 defective bulbs. 8)

12. (a) i) The joint probability density function of a two-dimensionalrandom variable X,Y) is f(x,y)=. .(6-x-y),0


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82b) i) Find the nature of the states of the Markov chain with the tpm

[0 1 ]P = t t 8)1 0(ii) Prove that the difference of two independent Poisson processes is

not a Poisson process. (4)(iii) Prove that the Poisson process is a Markov Process. 4)

14. a) (i) Derive 1) L average number of customers in the system 2) Lq,average number of customers in the queue for the queuing model(MIM/1): (NIFIFO). (8)

(ii) There are three typists in an office. Each typist can type an averageof 6 letters per hour. If letters arrive for being typed at the rate of15 letters per hour, what fraction of time all the typists will bebusy? What is the average number of letters waitbg to be typed?(Assume Poisson arrival.s and exponential service times) 8)Or

(b) Customers arrive at a one man barber shop according :0 a Poissonprocess with a mean inter arrival time of 20 minutes. Customers spendan average of 15 minutes in the barber chair. The service time isexponentially distributed. f an hour is used as a unit oftime, theni) What is the probability that a customer need not wait for a haircut?

(ii) \Vhat is the expected number of customer in the barber shop and inthe queue?

(iii) How much time can a customer expect to spe:J.d in the barber shop?Iv) Find the average time that a customer spend in the queue.v) Estimate the fraction of the day that the ctlstomer wili be idle?

(vi) What is the probability that there will be 6 or more customers:)(vil) Estimate the percentage of customers who have to wait prior to

getting into the barber s chair. 16)15. (a) Automatic car wash facility operates with only one bay. Cars arriveaccording to a Poisson process at the rate of 4 cars per hour and may wait

in the facility s parking lot if the bay is busy. The service time for all carsis constant and equal to 10 minutes. Determine L Lq ,W, and Wq 16)Or3 21524

. ,

63(b) Consider a system of two servers where customers from outside the

system arrive at sever 1 at a Poisson rate 4 and at server 2 at a Poissontate 5. The oervice rates for server 1 and 2 are 8 and 10 respectively.A customer.upon completion of s(;rvice at server 1 is likely to go to server2 or leave the system; whereas a departure from server 2 will go25 percent of the time to server 1 and will depart the system otherwise.Determine the limiting probabilities, L, and 16)

->

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Reg. No. : I 1 1 1-] 1

Question Paper Code: 487B.E.lB.Tech. DEGREE EXAlVIINATION, NOVEMBERIDECEMBER 2012.

Fourth SemesterComputer Science and Engineering

MA 2262 1S14041MA 441MA 1252 10177 PQ 401 0S025000S - PROBABILITY ANDQUEUING THEORY

(Common to Information Technology)(Regula ion 200S)

Time: Three hours Maximum: 100 marks

1.

2.

3.4.

5.

Statistical Tables may be permitted.Answer ALL questions.

PART lO x 2 = 20 marks)A continuous random variable X that can assume any value between x = 2 andx =5 has a density function given by f x) = k l x). Find P X < 4).

Identify the random variable and name the. distribution it follows, from thefollowing statement:A realtor claims that only 30% of the houses in a certain neighbourhood, are

appraised at less than { 20 lak,hs. A random sample of 10 houses from thatneighbourhood is selected and appraised to check the realtor s claim isacceptable are not . 'When will the two regression lines be (a) at right angles b) coincident?A small college has 90 male and 30 female professors. An ad-hoc committee of5 is selected at random to unite the vision and mission of the college. fX and Yare the numb er of men and women in the committee, respectively, what is thejoint probability mass function of X and Y? 'f N t) is the Poisson process, then what can you say about the time we will

wait for the first event. to occur? and the time we will wait for the nth event tooccur?

86,1\

.ik

6.

7.

S

9.

Is Poisson process stationery? Justify.

What is the probability that a customer has to wait more than 15 minutes toget his service completed in a MIMII queuing system, if ), = 6 per hour andf 1 = 10 per hour?

Give a real life situation in which (a) customers are considered for service withlast in first out queue discipline b) a system with infinite number of servers.Consider a tandem queue with 2 independent Markovian servers. Thesituation at server 1 is just as in an MIMl model. What will be the type ofqueue in server 2? Why?

10. State Jackson s theorem for an open network.

PART B - 5 x 16 = SO marks)11. (a) i) f the random variable X takes the values 1, 2, 3 and 4 such that

2P X =1) =3P X =2) =P X = 3) =5P X =4), then find theprobability distribution and cumulative distribution function of X.

(S)(ii) Find the MGF of the binomial distribution and hence find its mean.

(S)Or

(b) i) If the probability that an applicant for a driver s licence will passthe road test on any given trial is O.S, what is the probability thathe will finally pass the test (1) on the 4th trial 2) in fewer than4 trials? (S)

(ii) The number of monthly breakdowns of a computer is a randomvariable having a Poisson distribution with mean equal to I.S. Findthe probability that this computer will function for a month(1) without a breakdown (2) with only one breakdown. S)

12. (a) i) Obtain the equations of the lines of regression from the followingdata: S)

X: 1 2 3 4 5Y: 9 S 10 12 11

2

613

714

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(ii)

(b) (i)

(ii)

13. (a) (i)

The joint pdf of random variable X and Yis given by

x, y = {).x/;o . Osxsysotherwise.(1) Determine the value of ,1..(2) Find the marginal probability density function of X. (8)

OrLet X X 2 , XlOO be independent identically distributed'random variables with j..I=2P(192 < Xl + X2 + ... + X lOo < 210).

and 2:;- 4 Find(8)

The regression equation of X and Y is 3 y 6x + 108 = O. If he meanvalue of Y is 44 and the variance of X were th of the variance of16 .Y. Find the mean value of X and the correlation coefficient. (8)A fair die is tossed repeatedly. The maximum of the first noutcomes is denoted by Xn. Is {Xn: n = 1; 2, ...} a Markov chain?Why or why not? If it is a Markov chain, calculate its transitionprobability matrix. Specify the classes. (8)

(ii) An observer at a lake notices that when fish are caught, only 1 outof 9 trout is caught after another trout, with no other fish between,whereas 10 out of 11 non rout are caught following non-trout, withno trout between. Assuming that all fish are equally likely to becaught, what fraction of fish in the lake is trout? (8)

(b) (i)Or

The following is the transition probability matrix of a Markov chainwith state space {I, 2, 3, 4, 5}. Specify the classes and determinewhich classes are transient and which are recurrent. (8)

(0 0 0 0 1

pol 1/3 0 2/3 00 1/2 0 1/20 0 00 2/5 0

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g (

, i

(ii) For an English course, there are four popular textbooksthe market. The English department of an institution ailows iffaculty tc teach only from t.hese textbooks. Each year, Prof. RoseMary 0 Donoghue adopts t.he same book she was using t.heprevious year with prJbability 0.61. The probabilities of herchanging to any of the other 3 books are equaL Find the proportionof years Prof. 0 Donoghue uses each book. (8)

14. (a) Obtain the steady state probaCilities of birth-death process. Also draw

15.

(b)

the transition graph. (16)Or

At a port there are 6 unloading berths and 4 unloading crews. VVhen allthe berths are full, arriving ships are diverted to an overflow facility20 kms down the river. Tankers arrive according to Poisson process witha mean of 1 every 2 hrs. It takes for an unloading crew, on the average,1011's to unload a tanker, the unloading time following an exponentialdistribution. Find(i) how may tankers are at the port on the average?(ii) how long does a tanker spend at the port on the average?(iii) what is the average arrival rate at the overflow facility?Derive Pollaczek-Khinchin formula for M/G/1 queue.

Or

(16)(16)

Cb (i) Consider a two stage tandem queue with external arrival rate A tonode 0 . Let o and j..Il be the service rates of the exponentialservers at node 0 and 1 respectively. Arrival process is Poisson.Model this system using a Markov chain and obtain the balanceequations. (8)

(ii) Consider two servers. An average of 8 customers per hour arrive. from outside at server 1 and an average of 17 customers per hourarrive from outside at server 2. Inter' arrival times are exponential.Server 1 can serve at an exponential rate of 20 customers per hourand server 2 can serve at an exponential rate of 30 customers perhour. After completing service at station 1, half the customers leavethe system and half go to server 2. After completing service atstation 2, 3/4 of the customer complete service and 114 return toserver 1. Find the expec:ed no. of customers at each server. Findthe average time a customer spends in the system. 8)

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1..1 0o elsewhere.2. f a random variable has the moment generating function given by

M x(t) = _2_ , determine the ~ r i n c e of X.2- t3 The regression equations of X on Yand Yon X are respectively 5x - Y = 22 and

64 x - 45 Y = 24 . Find the means of X and Y4. State Central limit theorem.5. Define Wide sense stationary process.6. f the initial state probability distribution of a Markov chain is p(O) = [ ]

and the transition probability matrix of the chain is [ 0 1 ] find the1/2 1/2probability distribution of the chain after 2 steps.

7. State Little s formula for a (M/M/1): (GD/N /(0) queuing model.8. Define steady state and transient state in Queuing theory.9. When a MIGl1 queuing model will become a classic MIMl1 queuing model?10. State Pollaczek-Khintchine formula for the average number of customers in aMIG I 1 queuing model.

11. (a)

(b)

118PART B - (5 x 16 =80 marks)

(i) A random variable X has the following probability function:X: 0 1 2 3 4 5 6 7P X) : 0 k 2h 21< 3k k2 2 k2 7h2+h(1) Find the value of k.(2) Evaluate P(X < 6 , P(X 6 .

(3) f P (X,,; c) >. . find the minimum value of c.2 8)(ii) Find the moment generating function of an exponential random

variable and hence find its mean and variance. (8)Or(i) If X is a Poisson variate such thatP(X = 2) = 9 P(X = 4) +90 P(X =6 . Find(1) Mean and E(X2)(2) P(X 2). 8)

(ii) In a certain city, the daily consumption of electric power in millionsof kilowatt hours can be treated as a random variable havingGamma distribution with parameters l =. . and v = 3 . If the power2plant of this city has a daily capacity of 12 millions kilowatt-hotH s,what is the probability that this power supply will be inadequate onany given day? 8)

12. (a) (i) Let X and Y be two random variables having the joint probability,function x , y) = hex + 2y) where x and y can assume only theinteger values 0, 1 and 2. Find the marginal and conditionaldistributions. 8)(ii) Two random variables X and Y have the joint probability densityfunction

X ,y )= {C 4-X - y , 0,,;x ;2, 0,,;y,,;20, elsewhere.

Find cov (X, Y) . 8)Or

(b) (i) Two dimensional random variable (X, Y) have the joint probabilitydensity functionx , y ) = 8xy, 0 < x < y < 1

= 0, elsewhere.(1) Find p X ~ n Y


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13 a) (i)

(ii)

(b) (i)

(ii)

14. a) (i)

(ii)

lI

11 7The process {X(t)} whose p r o b b i l ~ t y distribution undcr:ertaincondition is given by

(at)"-P[X(t) = n.] I n = 1 2 31 + at)"+at n = O.1+at

Show that {X(t)} is not stationary. 8)A salesman territory consists of three cities A Band C. He neversells in the same city on successive days. If he sells i.n city-A, thenthe next day he sells in city-B. However if he sells in either city-B orcity-C, the next day he is twice as likely to sell in city-A as in theother city. In the long run how often does he sell in each of thecities? 8)

OrThe transition probability matrix of a Markov chain {X,,},

[0.1 0.5 0.4]-

n = 1, 2, 3,. having three states 1, 2 and 3 is P = 0.6 0.2 0.20.3 0.4 0.3

and the initial distribution is p(O) = (0,7,0.2,0.1). Find1) P[X 2 = 3]

(2) P [X3 = 2, X 2 =3, Xl =3, Xo =2]. (8)Suppose that customers arrive at a bank according to Poissonprocess with mean rate of 3 per minute. Find the probability thatduring a time interval of two minutes1) exactly 4 customers arrive(2) greater than 4 customers arrive(3) fewer than 4 customers arrive. 8)A T.V. repairman finds that the time spent on his job has anexponential distribution with mean 30 minutes. f he repair sets inthe order in which they came in and if the arrival of i3ets isapproximately Poisson with an average rate of 10 ,Jer 8 hour day.Find(1) the repairman s expected idle time each day2) how many jobs are ahead of average set just bought? 8)

A supermarket has 2 girls running up sales at the counters. f theservice time for each customer is exponential with mean 4 minutesand if people arrive in Poisson fashion at the rate of 10 per hour,find the following:1)

(2)3)

What is the probability of having to wait for service?What is the expected percentage of idle time for each girl?What is the expected length of customer s waiting time?

Or8)

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(b) (i)

(ii)

15. (a) (i)

(ii)

b) (i)

(ii)

118Trains arrive at the every J.5 minutes and the service time is33 minutes. f the capacity of the yard is limited to 5 tra .ns,find the probability that the yard is empty and ehe average numberof trains in the system. given that the inter arrival time ,md servicetime are following exponential distribution. 8)There are three typists in an office. Each typist can type an averageof 6 letters per hour. If letters arrive for being typed at the rate of15 letters per hour, what fraction of times all the typists will bebusy? What is the average number of letters waiting to be typed? 8)Automatic car wash facility operates with only one bay. Cars arriveaccording to a Poisson distribution with a mean of 4 cars per hourand may wait in the facility s parking lot if the bay is busy. Theparking lot is large enough to accommodate any number of cars. fthe service time for all cars is constant and equal to 10 minutes,determine1) mean number of customers.in the system L,2) mean number of customers in the queue Lq

(3) mean waiting t i ~ E of a customer in the system W(4) mean waiting tiDE of a customer in the. queue 8)An average of 120 students arrive each hour (inter-arrival times areexponential) at the controller office to get their hall tickets. Tocomplete the process, a candidate must pass through threecounters. Each counter consists of a single server, service times ateach counter are expcnential with the following mean times :counter 1, 20 secondo; counter 2, 15 seconds and counter 3,12 seconds. On the average how many students will be present inthe controller s office. 8)

OrDerive the P-I( formu:a for the (M/Gil) : (GD/w/w) queueing modeland hence deduce that with the constant service time theP-K formula reduces to L = p ~ where f .1=_1_ and. 2(1-p) E(T)

1.p =_ (10)f 1 .For a open queueing network with three nodes 1, 2 and 3, letcustomers arrive froo outside the system to node j according to a.Poisson input process with parameters Ii and let Pij denote theproportion of cllstomers departing from faciiity i to facility j. Given(11 1 2 IS) = (1,4, 3) and

0 O.G 031Pij = 1 0.1 0 0.3J\0.4 0.4 ()

determine the average arrival rate ) j to the node j for j = 1, 2, 3.

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13Reg. No. :1 -1 - - - - - - . - - . - - - - r - - . - ~ ~ I - Ir Question Paper Code 6 431

B.E.IB.Tech. DEGREE EXAMINATION; MAY/JUNE 2012.Fourth Semester

Computer Science and EngineeringMA 12521MA 1252 A - PROBABILITY AND QUEUEING THEORY

(Regulation 2004)(Common to B.E. (Part-Time) Third Semester Regulation 2005)

Time : Three hours Maximum: 100 marks

1.

2.

3.

4.5.

6.

Answer ALL questions.PARTA (10 x 2 = 0 marks)

A lot contains 12 items, of which 4 are defective .Three. items are drawn atrandom from the lot, one after the other. Find the probability that all three arenon defective.Find the second central moment of following distribution:

x: 1 3 4 5[{xl: 0.4 0.1 0.2 0.3

The random variable X has the exponential distribution with parameter 2.Find the probability that X is not smaller than 3.State the memoryless property of Geometric distribution.f the joint probability density function of (X,Y) is given byf(x,y) =24y l -x) , 0:0; y O; x O; 1, find E[XYJ.Let X and Y be random variables with the joint distribution given below. Findthe distribution of X and Y

-3X

2 4

1 0.1 0.2 0.23 0.3 0.1 0.1

II 13 7

7. Define wide-sense stationary process. Give an example.8. What is a stochastic matrix? When is it said to be regular?

9. In the usual notation of a (M \ M \1) : CD / 00 / 00 queue system if A = :2 perminute and J L = per minute, find the average number of customers in the10 .system and in the queue.

10. Trains arrive at the yard every 15 minutes and the service time is 33 minutes.f he line capacity of the yard is limited to 4 trains, find the probability that

the yard is empty.PART B - 5 x 16 = 0 marks)

11. (a) (i) Two urns contain 4 white, 6 black balls and 4 white, 8 black ballsrespectively. One urn is selected at random and a ball is taken out.It turns out to be white. Find tne-probability tha,t it is from the firsturn.

(il) A random variable X has the following probability distribution :

Find:1) K

x: -2 -1 0 1 2 3p(x): 0.1 K 0.2 2K 0.3 3K

2) P(X < 2)3) the cumulative distribution ofX.

(4) the mean of X.Or

(b) (i) f a random variable X has the moment generating functionM x (t) =_3_ , obtain the standard deviation ofX. (6)3 t

(ii) If M x t) is the moment generating functions of a random variableX then find the moment generating function of U = X - E[XJ. (5)Jx

(iii) Th e probabili ty density function of the random variable X followsthe following probability law:

1 I x - e )(x) =- e x p _00 < x < 00e eFind the moment generating function of X. 5)

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l ~ H ~12. a) i) It is known that the probability of an item produced by a certain

machine will be defective is 0.05. If the p.Toduced items are sent tothe market in packets of 20,find the number of packets containingat least exactly and at most 2 defective items in a consignment of1000 packets.

ii) Find the moment generating function of Gamma distribution.Or

b) i) Find the mean and variance of Normal distribution with

13. a)

b)

parameters J and J. 10)ii) If X has an exponential distribution with parameter 1, find the

pdf of Y = IX 6)A fair coin is tossed three times. LetX equal 0 or 1 according as a head OTa tail occurs on the first toss and let Y equal the total number of headsthat occur.i) Find the distributions ofX and Y respectiveiy. 4ii) Find the joint distribution ofX and Y 6iii) Determine whether X and Yare independent. 2)iv) Find Cov X, Y) . (4)

Or

If X,Y) is a two-dimensional R V uniformly distributed over thetriangular region R bounded by y = 0, x = 3 and y = i x . Find3i) the marginal density functions ofX and Y.

ii) the variance ofX and Y.iii) the correlation coefficient betweenX and Y

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13814. a) i) Give n a Random Variable X with density ( x ) and another a n d o m

variable i J uniformly iistributed in -7[, 7[) and independent ofX and Y t)=acos Xt+i J), prove that yet)) is a wide sen:cestationary process.

:b)

15. (a)

b)

ii) The three state Markov chain is given by the transition probability

f\ If

ii)/

ii)

o 1 01. 1 1 1)matnx 6 2 3

2 13 3

Find the steady state dis tribution of the chain.Or

If XU)} is a Poisson process, prove thatP{X s) = r IX t) = n} = n 1 - where < tFind the mean, variance and autocorrelation of a Poisson process.A self-service store employs one cashier at its Cow1ter. Ninecustomers arrive on an average every 5 minutes while the cashiercan serve 10 customers in 5 minutes. Assuming Poisson distributionfor arrival rate and exponential distribution for service time, findll Average number of customers in the system2) Average time a C.lstomer waits before being served.Repairing a certain type of machine which breaks down in a givenfactory consists of 5 basic steps that must be performedsequentially. The time taken to perform each of the 5 steps is foundto have an exponential distribution with mean 5 minutes and is

independent of other steps. If these machines break down in aPoisson fashion at an average rate of two per hour and if there isonly one repairman, what is the average idle Lime for each machinethat has broken down?Or

Ships anive at a port at the rate of one in every 4 hours with exponentialdistribution of inter arrival times. The time a ship occupies a berth forunloading has exponenti21 distribution with an average of 10 hours. Ifthe average delay of ships waiting for berths is to be kept below 14 hours,how many berths should be provided at the port?

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Probability and Queuing Theory Qp

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