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1 JEPPIAAR ENGINEERING COLLEGE Jeppiaar Nagar, Rajiv Gandhi Salai, Chennai 119 DEPARTMENT OF INFORMATION TECHNOLOGY 2009-2010 Even Semester SEMESTER IV Unit wise 2 mark and 16 mark Questions with Answers (As per Anna University syllabus) Code No. Course Title Page Number MA 2262 Probability and Queuing Theory 2 CS 2255 Data Base Management Systems 23 CS2252 Microprocessors & Microcontrollers 71 CS 2253 Computer Organization and Architecture 100 CS 2254 Operating Systems 153 IT 2251 Software Engineering and Quality Assurance 283 2MA2262 PROBABILITY AND QUEUEING THEORY (Common to CSE & IT) SYLLABUS UNIT I RANDOM VARIABLES 9+3 Discrete and continuous random variables - Moments - Moment generating functions and their properties. Binomial, Poisson ,Geometric ,Negative binomial, Uniform, Exponential, Gamma, and Weibull distributions . UNIT II TWO DIMENSIONAL RANDOM VARIABLES 9+3 Joint distributions - Marginal and conditional distributions Covariance - Correlation and regression - Transformation of random variables - Central limit theorem. UNIT III MARKOV PROCESSES AND MARKOV CHAINS 9+3 Classification - Stationary process - Markov process - Markov chains - Transition probabilities - Limiting distributions-Poisson process UNIT IV QUEUEING THEORY 9+3 Markovian models Birth and Death Queuing models- Steady state results: Single and multiple server queuing models- queues with finite waiting rooms- Finite source models- Littles Formula UNIT V NON-MARKOVIAN QUEUES AND QUEUE NETWORKS 9+3 M/G/1 queue- Pollaczek- Khintchine formula, series queues- open and closed networks TUTORIAL 15 TOTAL : 60 TEXT BOOKS 1. O.C. Ibe, Fundamentals of Applied Probability and Random Processes, Elsevier, 1st Indian Reprint, 2007 (For units 1, 2 and 3). 2. D. Gross and C.M. Harris, Fundamentals of Queueing Theory, Wiley Student edition, 2004 (For units 4 and 5). BOOKS FOR REFERENCES 1. A.O. Allen, Probability, Statistics and Queueing Theory with Computer Applications, Elsevier, 2nd edition, 2005. 1. H.A. Taha, Operations Research, Pearson Education, Asia, 8th edition, 2007. 3. K.S. Trivedi, Probability and Statistics with Reliability, Queueing and Computer Science Applications, John Wiley and Sons, 2nd edition, 2002. 3 UNIT I UNIT I UNIT I UNIT I RANDOM VARIABLES PART A 1. If Var(x) = 4, find Var (3x+8), where X is a random variable. Solution: Var (ax+b) = a2 Var x Var (3x+8) = 32 Var x = 36 2. If a random variable X takes the values 1,2,3,4 such that 2P(X=1) = 3P(X=2) = P(X=3) = 5P(x=4). Find the probability distribution of X. Solution: Let P(X=3)= k, P(X=1) = k/2 P(X=2) = k/3 P(X=4) = k/5 613015 3 2= = + + +kkkk k P(X=1) = 15/61 P(X=2) = 10/61 P(X=3)= 30/61 P(X=4)= 6/61 3. A continuous random variable X has probability density function given by f(x) = 3x2 0,x,1. Find K such that P(X>K) = 0.05. Solution: ( ) 983 . 0 95 . 095 . 0 5 . 0 395 . 0 ) (31302= == == KK dx xK X PK 4. Find the cumulative distribution fuction F(x) corresponding to the p.d.f. f(x) = < < +xx,) 1 (12 4Solution: [ ]21tan1tan1) 1 (1) (112+ ==+= xxdxxx Fxx 5. If a RV X has the moment generating function Mx(t) = t 22 Determine the variance of X. Solution: ( )41) ( ) ( ) (21) (21) (2122) (2 221= == =||

\| ==X E X E x VarX E X Ettt Mx 6. In a binomial distribution the mean is 4 and variance is 3, Find P(X=0). Solution: np=4, npq=3 Hence q=3/4, p=1-q=1/4, Since np=4 , n=16. 01002 . 043) 0 9,... 2 , 1 , 0434116) (1616= ||

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\|== =X Px cq p nc x X Px xxx n xx 7. The moment generating function of a random variable X is given by Mx(t) = .) 1 ( 3 tee Find P(X=1). Solution: = 3 1494 . 0! 13) 1 (,... 3 , 2 , 1 ,!3) (1 33= = == =eX Pxxex fx 8. Find the moment generating function of uniform distribution. 5Solution:) (1) ( ) (a b te edxa bedx x f e t Mat btbatxtxx=== 9. What are the properties of Normal distribution? Solution: The normal curve is symmetrical when p=q or pq The normal curve is a single peaked curve The normal curve is asymptotic to x-axis as y decreases rapidly when x increases numerically. The mean, median and mode coincide and lower and upper quartiles are equidistant from the median The curve is completely specified by mean and standard deviation along with the value of yo 10. The life time of a component measured in hours is Weibull distribution with parameter . 5 . 0 , 2 . 0 = = Find the mean lifeti;me of the component. Solution: Mean = E(X)= |||

\|+ 111 The mean life of the component = hours 505 . 011 2 . 05 . 01= ||

\|+ 11. If X is binomially distributed with n=6 such that P(X=2)=9P(X=4), find E(x) and Var(x). Solution: 6C2 p2q4 = 9 (6C4 p4q2); q=3p ;p=1/4. E(X)=1.5 ; Var(X) = 9/8 13. If f(x) = kx2 , 0 )`> ||

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