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Visit us http://s2sgateway.com for all mini and final year projects. Visit us http://tneducation.zoomshare.com for all anna university question papers B.E/B.Tech. DEGREE EXAMINATION,MAY/JUNE 2007 Fourth Semester Computer Science and Engineering MA1252-PROBABILITY AND QUEUING THEORY Answer all questions (PART A-10*2=20marks) 1.A lot of integrated circuit chips consists of 10 good,4 with minor defects and 2 with major defects. Two chips are randomly chosen from the lot. What is the probability that atleast one chip is good. 2.A continuous random variable X has a p.d.f f(x)=k(1+x),2=x=5. Find P(x<4). 3.One percent of jobs arriving at a computer system need to wait until weekends for scheduling, owing to core-size limitations. Find the probability that among a sample of 200 jobs, there are no jobs that have to wait until weekends. 4.A fast food chain finds that the average time customers have to wait for service is 45 seconds. If the waiting time can be treated as an exponential random variable, what is the probability that a customer will have to wait more than 5 minutes given that already he waited for 2 minutes? 5.The joint p.d.f of 2 random variables X and Y is f(x,y)=cx(x-y),o<2,-x =0 , elsewhere. Evaluate C. 6.Let (x,y) be a two dimensional random variable. Define covariance of (x,y). If x and y are independent what will be the covariance of (x,y)? 7.Define a Markov chain and give example. 8.If the transaction probability matrix of a Markov Chain is , find the limiting distribution of the chain. 9.In the usual notation of an M/M/1 queueing system, if ?=12 per hour and µ=24 per hour find the average number of customers in the system. 10.Write Pollaczek-Khintchine formula and explain the notations (Part B-5*16=80 marks) 11(a).(i).A binary communication channel carries data as one of 2 types of signal denoted by 0 and 1. Due to noise, a transmitted 0 is sometimes received as a 1 and a transmitted 1 is sometimes received as a 0. For a given channel assume a probability of 0.94 that a transmitted 0 is correctly received as a 0 and a probability of 0.91 that a transmitted 1 ia received as a 1. Further assume a probability of 0.45 of transmitted a 0. If a signal is sent determine (1) a 1 is received (2) a 1 was transmitted given that 1 was received (3) a 0 was transmitted given that a 0 was received (4) an error occurs. 11(a).(ii).In a continuous distribution, the probability density is given by f(x)=kx(2-x), 0<2.> Or 11.(b)(i).The cumulative distribution function (c d f) of a random variable x is given by 0 x<0 x2 0=x<1/2 F(x)=1-3/35(3-x)2 1/2=x<3 1 x=3 Find the p.d.f of x and evaluate P(1*1<1)> 11.(b)(ii).Find the moment generating function of the geometric random variable with p.d.f. f(x)=pqy-1 , x=1,2,3…… and hence obtain its mean and variance. 12(a).(i).The number of monthly break down of a computer is a poisson distribution with mean equal to 1.8. Find the probability, that this computer will function for a month (1) without a break down (2) with only one breakdown and (3) with atleast one breakdown. 12(a).(ii).The lifetime x in hours of a component is modeled by a Weibull distribution with ß=2. Starting with a large number of components, it is observed that 15% of the components that have lasted 90 hours fail before 100 hours. Find the parameter a. Or 12(b).(i).The marks obtained by a number of students in a certain subject are approximately normally distributed with mean 65 and standard deviation 5. If 3 students are selected at random from this group, what is the probability that atleast one of them would have scored above 75? [Given that the area between z=0 and z=2 under the standard normal curve is 0.4772]. 12(b).(ii).Write the p.d.f of Gamma distribution. Find the MGF, mean and variance. 13(a).(i).The joint density function of the random variable (x,y) is given by f(x)= 8xy 0<1,0 0 elsewhere Find the (1).Marginal density of y (2).conditional density of x given y=y and (3).P(x<1/2) 13(a).(11)Calculate the correlation coefficient for the following data: x: 65 66 67 67 68 69 70 72 y: 67 68 65 68 72 72 69 71 or 13(b).(i).If x and y are independent exponential random variables each with parameter 1, find the p.d.f of u=x-y. 13(b).(ii).State central limit Theorem in Lindberg-leby’s form. A random sample of size 100 is taken from a population whose mean is 60 and variance is 400. Using central limit theorem with what probability can we assert that the mean of the sample will not differ from µ=60 by more than 4? Area under the standard normal curve from 0 to 2 is 0.4772. 14.(a).(i).At the receiver of an AM radio, the received signal contains a cosine carrier signal at the Carrier frequency ?0 with a random phase ?, that is uniformly distributed over (0,2p). The received carrier signal is x(t)=Acos(?0t + ?). Show that the process is second order stationary. 14.(a).(ii).Queries presented in a computer data base are following a Poisson process of rate ?=6 queries per minute . An experiment consists of monitoring the data base for ‘m’ minutes and recording N(m) the number of queries presented. 1).What is the probability that no queries in a one minute interval? 2).What is the probability that exactly 6 queries arriving in one minute interval? 3).What is the probability of less than 3 queries arriving in a half minute interval? Or 14(b).(i).Assume that a computer system ia in any one of the three states: busy, idle and under repair respectively, denoted by 0,1,2. Observing its state at 2P.M each day, we get the transition probability matrix as p= .Find out the 3rd step transition prob matrix. Determine the limiting probabilities. 14(b).(ii).Obtain the steady state or long run probabilities for the population size of a birth death process. 15(a).(i).Arrivals at a telephone booth are considered to be Poisson with an average time of 12 min between one arrival and the next. The length of a Phone call is distributed exponentially with mean 4 minutes. (1).What is the average number of customers in the system? (2).What fraction of the day the phone will be in use? (3).What is the probability that an arriving customer have to wait? 15(a).(ii).There are three typists in an office. Each typist can type an average of 6 letters per hour. If letters arrive for being typed at the rate of 15 letters per hour. (1).What is the probability that no letters are there in the system? (2).What is the probability that all the typists are busy? Or 15(b).(i).Explain an M/M/1, finite capacity queueing model and obtain expressions for the steady state probabilities for the system size. 15(b).(ii).Patients arrive at a clinic according to Poisson distribution at a rate of 30 patients per hour. The waiting room does not accommodate more than 14 patients. Examination time per patient is exponential with mean rate of 20 per hour. (1).What is the probability that an arriving patient will not wait? (2).What is the effective arrival rate?