Cse2a



Geethanjali College of Engineering and TechnologyCheeryal (V), Keesara (M), Ranga Reddy District – 501 301 (T.S)PROBABILITY AND STATISTICSCOURSE FILE22383752533650DEPARTMENT OFCOMPUTER SCIENCE & ENGINEERING(2016-2017)Faculty In charge HOD-CSEA. RAMESH, N. NAGI REDDY Dr. S Nagender Kumar ContentsS.NoTopicPage. No.1Cover Page32Syllabus copy43Vision of the Department54Mission of the Department55PEOs and POs66Course objectives and outcomes77Course mapping with POs88Brief notes on the importance of the course and how it fits into the curriculum99Prerequisites if any1010Instructional Learning Outcomes1011Class Time Table1112Individual time Table1513Lecture schedule with methodology being used/adopted1614Detailed notes2815Additional topics2816University Question papers of previous years2917Question Bank4518Assignment Questions5619Unit wise Quiz Questions and long answer questions6120Tutorial problems6621Known gaps ,if any and inclusion of the same in lecture schedule6922Discussion topics , if any7023References, Journals, websites and E-links if any7024Quality Measurement Sheets71ACourse End Survey71BTeaching Evaluation7125Student List7126?Group-Wise students list for discussion topic71Course coordinator Program Coordinator HOD GEETHANJALI COLLEGE OF ENGINEERING AND TECHNOLOGYDEPARTMENT OF SCIENCE AND HUMANITIESName of the Subject : probability & statistics JNTU CODE: 113AN Programme : UG Branch: CSE- A,B,C&DVersion No : 01Year : I I Year Updated on : 04-05-2016Semester: I No. of pages : 71Prepared by : 1) Name : Mr. A. Ramesh 2) Design : Asst. Professor 3)Sign : 4) Date :04-05-2016Verified by : 1) Name :Dr.V.S.Triveni 2) Sign : 3) Design : Professor 4) Date : 10-05-2016* For Q.C Only.1) Name :2) Sign : 3) Design :4) Date :Approved by : (HOD ) 1) Name : Dr. G. Neeraja Rani 2) Sign : 3) Date : 2.SyllabusJAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABADII Year B.Tech CSE-I Sem L T/P/D C 4 -/-/- 4Syllabus:Single Random variables and probability distributions: Random variables – Discrete and continuous. Probability distributions, mass function/ density function of a probability distribution . Mathematical Expectation, Moment about origin, Central moments Moment generating function of probability distribution. Binomial, Poisson & normal distributions and their properties. Moment generating functions of the above three distributions, and hence finding the mean and variance.UNIT-IIMultiple Random variables, Correlation & Regression: Joint probability distributions- Joint probability mass / density function, Marginal probability mass / density functions, Covariance of two random variables, Correlation -Coefficient of correlation, The rank correlation.Regression- Regression Coefficient, The lines of regression and multiple correlation & regression.UNIT-IIISampling Distributions and Testing of HypothesisSampling: Definitions of population, sampling, statistic, parameter. Types of sampling, Expected values of Sample mean and variance, sampling distribution, Standard error, Sampling distribution of means and sampling distribution of variance.Parameter estimations – likelihood estimate, interval estimations.Testing of hypothesis: Null hypothesis, Alternate hypothesis, type I, & type II errors – critical region, confidence interval, Level of significance. One sided test, two sided test.Large sample tests:(i) Test of Equality of means of two samples equality of sample mean and population mean (cases of known variance & unknown variance, equal and unequal variances)(ii) Tests of significance of difference between sample S.D and population S.D.(iii) Tests of significance difference between sample proportion and population proportion & difference between two sample proportions.Small sample tests:Student t-distribution, its properties; Test of significance difference between sample mean and population mean, difference between means of two small samples. F- distribution and it’s properties. Test of equality of two population variances. Chi-square distribution , it’s properties, Chi-square test of goodness of fit.UNIT-IVQueuing Theory: Structure of a queuing system, Operating Characteristics of queuing system, Transient and steady states, Terminology of Queuing systems, Arrival and service processes- Pure Birth-Death process. Deterministic queuing models- M/M/1 Model of infinite queue, M/M/1 model of finite queue.UNIT-VStochastic processes: Introduction to Stochastic Processes –Classification of Random processes, Methods of description of random processes, Stationary and non-stationary random process, Average values of single random process and two or more random processes. Markov process, Markov chain, classification of states –Examples of Markov Chains, Stochastic Matrix.TEXT BOOKS:1) Higher Engineering Mathematics by Dr. B.S. Grewal, Khanna Publishers2) Probability and Statistics for Engineers and Scientists by Sheldon M.Ross, Academic Press3) Operations Research by S.D. Sarma,REFERENCE BOOKS:1. Mathematics for Engineers by K.B.Datta and M.A S.Srinivas,Cengage Publications2. Probability and Statistics by T.K.V.Iyengar & B.Krishna Gandhi Et3. Fundamentals of Mathematical Statistics by S C Gupta and V.K.Kapoor4. Probability and Statistics for Engineers and Scientists by Jay l.Devore.3.Vision of the Department:To produce globally competent and socially responsible computer science engineers contributing to the advancement of engineering and technology which involves creativity and innovation by providing excellent learning environment with world class facilities.4.Mission of the Department:To be a center of excellence in instruction, innovation in research and scholarship, and service to the stake holders, the profession, and the public.To prepare graduates to enter a rapidly changing field as a competent computer science engineer.To prepare graduate capable in all phases of software development, possess a firm understanding of hardware technologies, have the strong mathematical background necessary for scientific computing, and be sufficiently well versed in general theory to allow growth within the discipline as it advances.To prepare graduates to assume leadership roles by possessing good communication skills, the ability to work effectively as team members, and an appreciation for their social and ethical responsibility in a global setting.5.Program Educational Objectives (PEOs): To provide graduates with a good foundation in mathematics, sciences and engineering fundamentals required to solve engineering problems that will facilitate them to find employment in industry and / or to pursue postgraduate studies with an appreciation for lifelong learning.To provide graduates with analytical and problem solving skills to design algorithms, other hardware / software systems, and inculcate professional ethics, inter-personal skills to work in a multi-cultural team.To facilitate graduates to get familiarized with the art software / hardware tools, imbibing creativity and innovation that would enable them to develop cutting-edge technologies of multi-disciplinary nature for societal development.PROGRAM OUTCOMES (PO)An ability to apply knowledge of mathematics, science and engineering to develop and analyze computing systems.an ability to analyze a problem and identify and define the computing requirements appropriate for its solution under given constraints.An ability to perform experiments to analyze and interpret data for different applications.An ability to design, implement and evaluate computer-based systems, processes, components or programs to meet desired needs within realistic constraints of time and space.An ability to use current techniques, skills and modern engineering tools necessary to practice as a CSE professional.An ability to recognize the importance of professional, ethical, legal, security and social issues and addressing these issues as a professional. An ability to analyze the local and global impact of systems /processes /applications /technologies on individuals, organizations,?society and environment.An ability to function in multidisciplinary teams.An ability to communicate effectively with a range of audiences.Demonstrate knowledge and understanding of the engineering, management and economic principles and apply them to manage projects as a member and leader in a team.A recognition of the need for and an ability to engage in life-long learning and continuing professional developmentKnowledge of contemporary issues.An ability to apply design and development principles in producing software systems of varying complexity using various project management tools.An ability to identify, formulate and solve innovative engineering problems.6.Course Objectives and Outcomes:Course Objectives: The aim of this course is,Understand a random variable that describes randomness or an uncertainty in certain realistic situation. It can be of either discrete or continuous type.In the discrete case, study of the binomial and the Poisson random variables and the Normal random variable for the continuous case predominantly describe important probability distributions. Important statistical properties for these random variables provide very good insight and are essential for industrial applications.Most of the random situations are described as functions of many single random variables. In this unit, the objective is to learn functions of many random variables through joint distributions.The types of sampling, Sampling distribution of means, Sampling distribution of variance, Estimations of statistical parameters, Testing of hypothesis of few unknown statistical parameters.The mechanism of queuing system, the characteristics of queue, the mean arrival and service rates, the expected queue length, the waiting line.The random processes, the classification of random processes, Markov chain, classification of states.Stochastic matrix (transition probability matrix), Limiting probabilities, Application of Markov chains.Course Outcomes:At the end of this course, the students are able to CO1: Explain the concept of random variables and probability distributions and applythe same to solve simple engineering problems.CO2: Explain the concept of correlation coefficient, regression coefficient and regression lines and apply the same to solve simple engineering problems to analyze the situations properly and to take a right decision to choose the right one.CO 3: Explain the concept of hypothesis testing for small as well as for large samples and apply the same to solve simple engineering problems.CO4: Explain the models of queuing system and apply the same to solve some real problems and simple engineering problems.CO5: Explain the concept of Markov Chains and Markov process and apply the same to solve simple engineering problems and also to find solutions to real world problems.7.Mapping of Course to PEOs and Pos:Course ComponentPOsPEOsMathematicsi,ii,iv,v,vii, viii,xi,xii, xiii,xivvPEO1, PEO2Course Mapping With PEOs And Pos:Mapping of Course with Programme Educational Objectives:S.NoCourse componentcodecourseSemesterPEO 1PEO 2PEO 31Mathematics113ANP&SI√√POs1234567891011121314MathematicsP&SCO 1 √√√√CO 2√√√CO 3√√CO 4√√√√CO 5√√√8.Brief notes on the Importance of the course and how to fits into the curriculum:Most of the random situations are described as functions of many single random variables. The importance is to learn functions of many random variables through joint distributions.The types of sampling, Sampling distribution of means ,Sampling distribution of variance, Estimations ofstatistical parameters, Testing of hypothesis of few unknown statistical parameters.The mechanism of queuing system ,The characteristics of queue, The mean arrival and service ratesThe expected queue length, The waiting lineThe random processes, The classification of random processes and Markov process.It improves the analytic and logical skills of the student. They will be in a position to guess the result of their project results.9.Prerequisites:The student should have the fundamental knowledge aboutSets and functionsPermutationsCombinationsProbability Statistics Intermediate mathematics.10.Instructional Learning outcomes:Unit wise Learning Outcomes:UNIT I: Single random variables and probability distributionsAfter the completion of this unit, the students should be able to:Explain the difference between discrete random variable and continuous random variablesApply the distributions in certain realistic situationsUNIT II: Multiple random variables, correlation & RegressionAfter the completion of this unit, the students should be able to:Explain the closeness of the relationship between the variables.Explain how to find the value of one variable, when the value of the other variable is given.Explain the difference between correlation coefficient and regression coefficient.UNIT III: Sampling Distributions and Testing of HypothesisAfter the completion of this unit, the students should be able to:Explain mean and proportions for large samples as well as for small samplesExplain the concept of confidence interval for large as well as for small samplesApply different types of tests for large and small samplesUNIT IV: Queuing TheoryAfter the completion of this unit, the students should be able to:Explain the terminology of queuing systemApply the models of queuing system in real life situationsHow best the Queuing models simplify the problematic part UNIT V: Stochastic ProcessesAfter the completion of this unit, the students should be able to:Explain random processExplain Markov chains and Markov process and he will know applications to real life problem11.Class Time Tables II A,B,C & D secGeethanjali College of Engineering & TechnologyDepartment of Computer Science & EngineeringYear/Sem/Sec: II-B. Tech I-Semester A-SectionRoom No: A.Y : 2016 -17WEF:Class Teacher: Time09.30-10.2010.20-11.1011.10-12.0012.00-12.5012.50-1.301.30-2.202.20-3.103.10-4.00Period1234LUNCH567MondayTuesdayWednesdayThursdayFridaySaturdayS.NoSubject(T/P)Faculty NameContact No1PROBABILITY AND STATISTICS2MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE3DIGITAL LOGIC DESIGN4ELECTRONIC DEVICES AND CIRCUITS5BASIC ELECTRICAL ENGINEERING6DATA STRUCTURES7ELECTRICAL AND ELECTRONICS LAB8DATA STRUCTURES LAB9SEMINAR10*-Tutorial Hour/Discussion HourTT. Cord:___________ HOD:________ Dean Academics:-_______ Principal:___________________Geethanjali College of Engineering & TechnologyDepartment of Computer Science & EngineeringYear/Sem/Sec: II-B.Tech I-Semester B-Section Room No:A. Y : 2016--167 WEF: Class Teacher: Time09.30-10.2010.20-11.1011.10-12.0012.00-12.5012.50-1.301.30-2.202.20-3.103.10-4.00Period1234LUNCH567MondayTuesdayWednesdayThursdayFridaySaturdayS.NoSubject(T/P)Faculty NameContact No12345678910TT. Coord:___________ HOD:________ Dean Academics:-_______ Principal:___________________Geethanjali College of Engineering & TechnologyDepartment of Computer Science & EngineeringYear/Sem/Sec: II-B. Tech I-Semester C-SectionRoom No: A.Y : 2016 -17WEF:Class Teacher: Time09.30-10.2010.20-11.1011.10-12.0012.00-12.5012.50-1.301.30-2.202.20-3.103.10-4.00Period1234LUNCH567MondayTuesdayWednesdayThursdayFridaySaturdayS.NoSubject(T/P)Faculty NameContact No1PROBABILITY AND STATISTICS2MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE3DIGITAL LOGIC DESIGN4ELECTRONIC DEVICES AND CIRCUITS5BASIC ELECTRICAL ENGINEERING6DATA STRUCTURES7ELECTRICAL AND ELECTRONICS LAB8DATA STRUCTURES LAB9SEMINAR10*-Tutorial Hour/Discussion HourTT. Cord:___________ HOD:________ Dean Academics:-_______ Principal:___________________Geethanjali College of Engineering & TechnologyDepartment of Computer Science & EngineeringYear/Sem/Sec: II-B.Tech I-Semester D-Section Room No:A. Y : 2016--167 WEF: Class Teacher: Time09.30-10.2010.20-11.1011.10-12.0012.00-12.5012.50-1.301.30-2.202.20-3.103.10-4.00Period1234LUNCH567MondayTuesdayWednesdayThursdayFridaySaturdayS.NoSubject(T/P)Faculty NameContact No12345678910TT. Cord:___________ HOD:________ Dean Academics:-_______ Principal:___________________ 12.Individual Time Table:Time table CSE-A Name of the faculty:Time9.30-10.2010.20-11.1011.10-12.0012.00-12.50Lunch Break1.30-2.202.20-3.103.10-4.00Period1234567MonTuesWedThursFriSatTime table CSE-B Name of the faculty:Time9.30-10.2010.20-11.1011.10-12.0012.00-12.50Lunch Break1.30-2.202.20-3.103.10-4.00Period1234567MonTuesWedThursFriSat Time table CSE-C Name of the faculty:Time9.30-10.2010.20-11.1011.10-12.0012.00-12.50Lunch Break1.30-2.202.20-3.103.10-4.00Period1234567MonTuesWedThursFriSatTime table CSE-D Name of the faculty:Time9.30-10.2010.20-11.1011.10-12.0012.00-12.50Lunch Break1.30-2.202.20-3.103.10-4.00Period1234567MonTuesWedThursFriSat13. Lecture Schedule with methodology being used/adoptedLecture Schedule for CSE-A Section:S. NoUnit NoTotal no. of PeriodsTopics to be coveredRegular / AdditionalTeaching aids usedLCD/OHP/BBDate1I1Introduction to ProbabilityregularBB21Introduction to Random variablesregularBB31Discrete Random variables- probability distributions, mass functionregularBB41Problems on D.R.VregularBB51continuous Random variables- probability distributions, density function, mathematical expectationregularBB61Problems on C.R. VregularBB71Moment about origin, central momentsregularBB81Moment generating function of probability distributionregularBB91Binomial distribution- Generating function, mean and varianceregularBB101Problems on Binomial distributionregularBB111Poisson distribution- Generating function, mean and varianceregularBB121Problems on Poisson distributionregularBB131Normal distribution- Generating function, mean and varianceregularBB141Normal distribution- median, moderegularBB151Standard normal variate- mean, varianceregularBB161Areas under normal curvesregularBB17II1Introduction to joint probability distributionsregularBB181Joint probability mass / density functionregularBB191Marginal probability mass / density functionregularBB201Covariance of two random variablesregularBB21II1Correlation – coefficient of correlationregularBB221Problems on correlationregularBB231Problems on correlation for rounded meanregularBB241Rank correlation- non repeated ranksadditionalBB251Rank correlation- repeated ranksregularBB261Multiple correlationregularBB271Lines of regressionregularBB281Finding out correlation coefficient using regression linesregularBB291Multiple regressionregularBB30III1Basic definitions on samplingregularBB311Expected values of sample mean and varianceregularBB321Sampling distribution- Sampling distribution of mean and variancesregularBB33III1Introduction to estimation – likelihood estimationregularBB341Interval estimation- confidence intervalregularBB351ProblemsregularBB361Testing of hypothesis- procedureregularBB371Large samples – Z- test for single meanregularBB381Z- test for difference between two meansregularBB391Test for standard deviationregularBB401Test for single proportionregularBB411Test for difference between two proportionsregularBB421Small samples- t- distribution and its propertiesregularBB431t- test for single meanregularBB441Paired t- testregularBB451t- test for difference between two meansregularBB461F- distribution and its propertiesregularBB47III1F- test for variancesregularBB481Chi square distribution and its propertiesregularBB491Chi square test for goodness of fitregularBB50IV1Introduction to queuing theory- terminologyregularBB511States of queuing theory, arrival and service processesregularBB521Pure birth-death processregularBB531Problems on M/M/1/∞regularBB541Problems on M/M/1/NregularBB55V1Introduction to Stochastic processes- Classification and methods of random processesregularBB561Stationary and non stationary random processesregularBB571Average values of single and two or more random processesregularBB581Markov processregularBB591Markov chainregularBB601Classification of statesregularBB611Stochastic matrixregularBB621Summary And RevisionregularBB Lecture Schedule for CSE-B Section:S. NoUnit NoTotal no. of PeriodsTopics to be coveredRegular / AdditionalTeaching aids usedLCD/OHP/BBDate1I1Introduction to ProbabilityregularBB21Introduction to Random variablesregularBB31Discrete Random variables- probability distributions, mass functionregularBB41Problems on D.R.VregularBB51continuous Random variables- probability distributions, density function, mathematical expectationregularBB61Problems on C.R. VregularBB71Moment about origin, central momentsregularBB81Moment generating function of probability distributionregularBB91Binomial distribution- Generating function, mean and varianceregularBB101Problems on Binomial distributionregularBB111Poisson distribution- Generating function, mean and varianceregularBB121Problems on Poisson distributionregularBB131Normal distribution- Generating function, mean and varianceregularBB141Normal distribution- median, moderegularBB151Standard normal variate- mean, varianceregularBB161Areas under normal curvesregularBB17II1Introduction to joint probability distributionsregularBB181Joint probability mass / density functionregularBB191Marginal probability mass / density functionregularBB201Covariance of two random variablesregularBB21II1Correlation – coefficient of correlationregularBB221Problems on correlationregularBB231Problems on correlation for rounded meanregularBB241Rank correlation- non repeated ranksadditionalBB251Rank correlation- repeated ranksregularBB261Multiple correlationregularBB271Lines of regressionregularBB281Finding out correlation coefficient using regression linesregularBB291Multiple regressionregularBB30III1Basic definitions on samplingregularBB311Expected values of sample mean and varianceregularBB321Sampling distribution- Sampling distribution of mean and variancesregularBB33III1Introduction to estimation – likelihood estimationregularBB341Interval estimation- confidence intervalregularBB351ProblemsregularBB361Testing of hypothesis- procedureregularBB371Large samples – Z- test for single meanregularBB381Z- test for difference between two meansregularBB391Test for standard deviationregularBB401Test for single proportionregularBB411Test for difference between two proportionsregularBB421Small samples- t- distribution and its propertiesregularBB431t- test for single meanregularBB441Paired t- testregularBB451t- test for difference between two meansregularBB461F- distribution and its propertiesregularBB47III1F- test for variancesregularBB481Chi square distribution and its propertiesregularBB491Chi square test for goodness of fitregularBB50IV1Introduction to queuing theory- terminologyregularBB511States of queuing theory, arrival and service processesregularBB521Pure birth-death processregularBB531Problems on M/M/1/∞regularBB541Problems on M/M/1/NregularBB55V1Introduction to Stochastic processes- Classification and methods of random processesregularBB561Stationary and non stationary random processesregularBB571Average values of single and two or more random processesregularBB581Markov processregularBB591Markov chainregularBB601Classification of statesregularBB611Stochastic matrixregularBB621Summary And RevisionregularBB Lecture Schedule for CSE-C Section:S. NoUnit NoTotal no. of PeriodsTopics to be coveredRegular / AdditionalTeaching aids usedLCD/OHP/BBDate1I1Introduction to ProbabilityregularBB21Introduction to Random variablesregularBB31Discrete Random variables- probability distributions, mass functionregularBB41Problems on D.R.VregularBB51continuous Random variables- probability distributions, density function, mathematical expectationregularBB61Problems on C.R. VregularBB71Moment about origin, central momentsregularBB81Moment generating function of probability distributionregularBB91Binomial distribution- Generating function, mean and varianceregularBB101Problems on Binomial distributionregularBB111Poisson distribution- Generating function, mean and varianceregularBB121Problems on Poisson distributionregularBB131Normal distribution- Generating function, mean and varianceregularBB141Normal distribution- median, moderegularBB151Standard normal variate- mean, varianceregularBB161Areas under normal curvesregularBB17II1Introduction to joint probability distributionsregularBB181Joint probability mass / density functionregularBB191Marginal probability mass / density functionregularBB201Covariance of two random variablesregularBB21II1Correlation – coefficient of correlationregularBB221Problems on correlationregularBB231Problems on correlation for rounded meanregularBB241Rank correlation- non repeated ranksadditionalBB251Rank correlation- repeated ranksregularBB261Multiple correlationregularBB271Lines of regressionregularBB281Finding out correlation coefficient using regression linesregularBB291Multiple regressionregularBB30III1Basic definitions on samplingregularBB311Expected values of sample mean and varianceregularBB321Sampling distribution- Sampling distribution of mean and variancesregularBB33III1Introduction to estimation – likelihood estimationregularBB341Interval estimation- confidence intervalregularBB351ProblemsregularBB361Testing of hypothesis- procedureregularBB371Large samples – Z- test for single meanregularBB381Z- test for difference between two meansregularBB391Test for standard deviationregularBB401Test for single proportionregularBB411Test for difference between two proportionsregularBB421Small samples- t- distribution and its propertiesregularBB431t- test for single meanregularBB441Paired t- testregularBB451t- test for difference between two meansregularBB461F- distribution and its propertiesregularBB47III1F- test for variancesregularBB481Chi square distribution and its propertiesregularBB491Chi square test for goodness of fitregularBB50IV1Introduction to queuing theory- terminologyregularBB511States of queuing theory, arrival and service processesregularBB521Pure birth-death processregularBB531Problems on M/M/1/∞regularBB541Problems on M/M/1/NregularBB55V1Introduction to Stochastic processes- Classification and methods of random processesregularBB561Stationary and non stationary random processesregularBB571Average values of single and two or more random processesregularBB581Markov processregularBB591Markov chainregularBB601Classification of statesregularBB611Stochastic matrixregularBB621Summary And RevisionregularBB Lecture Schedule for CSE-D Section:S. NoUnit NoTotal no. of PeriodsTopics to be coveredRegular / AdditionalTeaching aids usedLCD/OHP/BBDate1I1Introduction to ProbabilityregularBB21Introduction to Random variablesregularBB31Discrete Random variables- probability distributions, mass functionregularBB41Problems on D.R.VregularBB51continuous Random variables- probability distributions, density function, mathematical expectationregularBB61Problems on C.R. VregularBB71Moment about origin, central momentsregularBB81Moment generating function of probability distributionregularBB91Binomial distribution- Generating function, mean and varianceregularBB101Problems on Binomial distributionregularBB111Poisson distribution- Generating function, mean and varianceregularBB121Problems on Poisson distributionregularBB131Normal distribution- Generating function, mean and varianceregularBB141Normal distribution- median, moderegularBB151Standard normal variate- mean, varianceregularBB161Areas under normal curvesregularBB17II1Introduction to joint probability distributionsregularBB181Joint probability mass / density functionregularBB191Marginal probability mass / density functionregularBB201Covariance of two random variablesregularBB21II1Correlation – coefficient of correlationregularBB221Problems on correlationregularBB231Problems on correlation for rounded meanregularBB241Rank correlation- non repeated ranksadditionalBB251Rank correlation- repeated ranksregularBB261Multiple correlationregularBB271Lines of regressionregularBB281Finding out correlation coefficient using regression linesregularBB291Multiple regressionregularBB30III1Basic definitions on samplingregularBB311Expected values of sample mean and varianceregularBB321Sampling distribution- Sampling distribution of mean and variancesregularBBI-MID EXAMRegression- Regression coefficient33III1Introduction to estimation – likelihood estimationregularBB31/08/2015341Interval estimation- confidence intervalregularBB01/09/2015351ProblemsregularBB02/09/2015361Testing of hypothesis- procedureregularBB04/09/2015371Large samples – Z- test for single meanregularBB07/09/2015381Z- test for difference between two meansregularBB08/09/2015391Test for standard deviationregularBB09/09/2015401Test for single proportionregularBB11/09/2015411Test for difference between two proportionsregularBB14/09/2015421Small samples- t- distribution and its propertiesregularBB15/09/2015431t- test for single meanregularBB16/09/2015441Paired t- testregularBB18/09/2015451t- test for difference between two meansregularBB19/09/2015461F- distribution and its propertiesregularBB21/09/201547III1F- test for variancesregularBB22/09/2015481Chi square distribution and its propertiesregularBB23/09/2015491Chi square test for goodness of fitregularBB26/09/201550IV1Introduction to queuing theory- terminologyregularBB28/09/2015511States of queuing theory, arrival and service processesregularBB29/09/2015521Pure birth-death processregularBB03/10/2015531Problems on M/M/1/∞regularBB05/10/2015541Problems on M/M/1/NregularBB06/10/201555V1Introduction to Stochastic processes- Classification and methods of random processesregularBB07/10/2015561Stationary and non stationary random processesregularBB09/10/2015571Average values of single and two or more random processesregularBB12/10/2015581Markov processregularBB14/10/2015591Markov chainregularBB16/10/2015601Classification of statesregularBB17/10/2015611Stochastic matrixregularPPT19/10/2015621Summary And RevisionregularBB23/10/2015 14.Detailed Notes : Hardcopy available 15.Additional topics :Fundamentals on ProbabilityMutually Exclusive EventsMutually Exhaustive EventsConditional ProbabilityAddition theorem on two and three eventsBaye’s theorem, Total probability Theorem16.University Question papers of previous years:GEETHANJALI COLLEGE OF ENGINEERING & TECHNOLOGYCheeryal (v), Keesara (M), R.R.Dist.-501301.R13 Model Question Paper-IB.Tech. II-year I-Semester, Subject: Probability and StatisticsTime: 3 hours Max.Marks:75 _______________________________________________________________Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 units. Answer any one full question from each unit. Each Question carries 10 marks and may have a, b, c as sub questions.Part-A (25 Marks) a) Define continuous random variable? (2M) b) Derive mean of a Poisson distribution? (3M) Define joint probability mass function? (2M) If x=28, y=28 and xy=112. Then find cov (x,y)? (3M)Define confidence interval? (2M)How many different samples of size 2 can be chosen from a finite population of size 25? (3M) What is pure birth and death process? (2M) A machine repairing shop gets on average 16 machines per day ( of eight hours) for repair and the arrival pattern is Poisson. Find arrival rate per hour? (3M) Explain about Stationary Processes. (2M)Explain the process of finding expected duration of the game? (3M) Part – B (50 Marks)A random variable x has the following probability distribution. x12345678P(x=x)k2k3k4k5k6k7k8k Find the value of (i)K (ii) p(x QUOTE 2) (iii) p(2 x ≤ 5). (10M) (OR)(i) The mean and variance of binomial distribution are 4 and 4/3 respectively. Find p(x>1).(ii)Let x denote the number of heads is a single toss of 4 fair coins. determine (a) p(x2) (b) p(1≤x 3)(10M)Ten participants in a contest are ranked by two judges as follows X16510324978y64981231057 Calculate the rank correlation coefficient?(10M)(OR)If θ is an angle between two regression lines show that tanθ= 1-r2r.σxσyσx2+σy2 . Explain the significance when r = 0 and r = ±1. (10M)An unbiased coin is thrown n times. It is desired that the relative frequency of the appearance of heads should lie between 0.49 and 0.51. Find the smallest value of n that will ensure this result with 90% confidence? (10M)(OR)Two horses A and B were tested according to the time in seconds to run a particular race with the following results. Test whether you can discriminate between two horses (10M)Horse A28303233332934Horse B293030242729--At the election commission office, for the Voter’s identity Card, a Photographer takes passport size photo at an average rate of 24 photos per hour. The photographer must wait until the voter blinks or scowls, so the time to take a photo is exponentially distributed. Customers arrive at Poisson distributed average rate of 20 voters per hour. Find i) what the utilization of Photographer is. (ii) How much time, the voter has to spend at the election commission office on an average to get the service. (10M)(OR)a) Explain about queuing theory characteristics? (10M) b) Define preemptive discipline and non-preemptive priority? (10M)The transition probability matrix of a markov chain is given by0.30.700.10.40.500.20.8 verify Whether the matrix is irreducible or not?(10M) (OR)Find the nature of states of the markov chain and explain with transition probability matrix (10M)GEETHANJALI COLLEGE OF ENGINEERING & TECHNOLOGYCheeryal (v), Keesara (M), R.R.Dist.-501301.R13 Model Question Paper-IIB.Tech. II-year I-Semester, Subject: Probability and StatisticsTime: 3 hours Max.Marks:75 ________________________________________________________________Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 units. Answer any one full question from each unit. Each Question carries 10 marks and may have a, b, c as sub questions.Part-A (25 Marks) a) Define continuous random variable? (2M) b) Derive mean of a Poisson distribution? (3M)Define joint probability mass function? (2M)If x=28, y=28 and xy=112. Then find cov (x,y)? (3M)Define confidence interval? (2M)How many different samples of size 2 can be chosen from a finite population of size 25? (3M) What is pure birth and death process? (2M)A machine repairing shop gets on average 16 machines per day ( of eight hours) for repair and the arrival pattern is Poisson. Find arrival rate per hour? (3M) Explain about Stationary Processes. (2M)Explain the process of finding expected duration of the game? (3M) Part – B (50 Marks)(i) Using recurrence formula find the probabilities when x= 0,1,2,3,4 and 5: if the mean of Poisson distribution is 3? (ii) Out of 800 families with 5 children each, how many would you expect to have (i) 3 boys (ii) either 2 or 3 boys (10M) (OR)Fit a normal curve to the following distribution(10M)x246810f14641The correlation table given below shows that the ages of husband and wife of 53 married couples living together on the census night of 1991.Calculate the coefficient of correlation between the age of the husband and that of the wife. (10M) Age of husbandAge of wifeTotal15-2525-3535-4545-5555-6565-7515-2511----225-352121---1535-45-4101--1545-55--361-1055-65---242865-75----123Total3171496453(OR)Find the regression line of x on y and y on x for the following data(10M)X10121316172025Y10222427293337A certain stimulus administered to each of 12 patients resulted in the following increases of blood pressure.5,2,8,-1,3,0,-2,1,5,0,4,6. Can it be concluded that the stimulus will in general be accompanied by an increase in blood pressure?(10M)(OR)The measurements of the output of two units have given the following results. Assuming that both samples have been level whether the two populations have the same variance. (10M)Unit-A14.110.114.713.714.0Unit-B14.014.513.712.714.1(OR)A fast food restaurant has one drive is window. It is estimated that cars arrive according to a Poisson distribution at the rate of 2 every 5 minutes and that there is enough space to accommodate a line of 10 cars. Other arriving cars cannot wait outside this space, if necessary. If takes 1.5 minutes on the average to fill an order, but the service time actually varies according to an exponential distribution. Determine the following.a)The probability that the facility is idle.b)The expected number of customers waiting to be served.(10M) (OR)A car park contains 5 cars. The arrival of cars is Poisson with a mean rate of 10 per hour. The length of time each car spends in the car park has negative exponential distribution with mean 2 hours. How many cars are in the car park on average and what is the probability of a newly arriving customer finding the car park full and having to park his car elsewhere?(10M) 13234312112 is this matrix stochastic?(10M) (OR)Find the nature of states of the Markova chain with transition probability matrixP=01012012010 (10M) GEETHANJALI COLLEGE OF ENGINEERING & TECHNOLOGYCheeryal (v), Keesara (M), R.R.Dist.-501301.R13 Model Question Paper-IIIB.Tech. II-year I-Semester, Subject: Probability and StatisticsTime: 3 hours Max.Marks:75 ________________________________________________________________Note: This question paper contains two parts A and B. Part A is compulsory which carries 25 marks. Answer all questions in Part A. Part B consists of 5 units. Answer any one full question from each unit. Each Question carries 10 marks and may have a, b, c as sub questions.Part-A (25 Marks)1Define discrete random variable? (2M) Derive mean of aNormal distribution? (3M) Define marginal probability mass function? (2M)The two regression equations of the variables x and y are (3M)x=13.42-0.87y and y= 16.94-0.50x. Then find mean of y? Define population, sample and sampling with examples? (3M) For an F- distribution find F0.05 with ?1=7 and ?2=15 ? (2M) Explain the terms of queuing theory and give two examples. (2M) Show that Average queue length is σ21-σ (3M) Explain about Markov Processes. (2M) Let p=12, q=12, z=500,a=1000. then find the expected duration of the game. (3M) Part – B (50 Marks)If f(x) is the distribution function x given by Fx=0 if x≤1,kx-14 if 1<x≤3,1 if x>1 Determine (i) f(x) (ii) k (iii) mean (10M)(OR)If the masses of 300 students are normally distributed with mean 68 kgs and standard deviation 3 kgs how many students have masses (i)Greater than 72 kgs (ii) Less than or equal to 64kgs (iii) Between 65 and 71 kgs inclusive ?(10M)Psychological tests of intelligence and of engineering ability were applied to 10 students. Hence is a record of ungrouped data showing intelligence ratio (I.R.) and engineering ratio (E.R.) Calculate the coefficient of correlation. (10M)StudentABCDEFGHIJI.R.1051041021011009998969392E.R.101103100989596104929794(OR)In the following table are recorded data showing the test scores made by salesmen on an intelligence test and their weekly sales.Sales men12345678910Test Scores40705060805090406060Sales(‘000)2.56.04.55.04.52.05.53.04.53.0 Calculate the regression line of sales on test scores and estimate the most probable weekly sales volume if a sales man makes a score of 70? (10M) A 11 students were given a test in statistics they were given a month’s further tuition and a second test of equal difficulty was held at the end of it. Do the marks give evidence that the students have benefited by extra coaching? (10M)Boys1234567891011Marks I test2320192118201817231619Marks II test2419221820222020232017(OR)A sample of 900 members is found to have a mean of 3.4 cm. Can it be reasonably regarded as a truly random sample from a large population with mean 3.25 cm and S.D. 1.61cm? (10M)A manager of a Local hamburger restaurant is preparing to open a new fast food restaurant called Hasty Burgers. Based on the arrival rates at existing outlets. Manager expects customers to arrive at the drive in window according to a Poisson distribution with a mean of 20 customers per hour. The service rate is flexible however, the service times are expected to follow an exponential distribution the drive in window in single ever operation.What service rate is needed to keep the average number of customers is the service system to 4.For the service rate in part (a). What is the probability that more than 4 customers are in the line and being served? (10M) (OR)A fast food restaurant has one drive is window. It is estimated that cars arrive according to a Poisson distribution at the rate of 2 every 5 minutes and that there is enough space to accommodate a line of 10 cars. Other arriving cars cannot wait outside this space, if necessary. If takes 1.5 minutes on the average to fill an order, but the service time actually varies according to an exponential distribution. Determine the following.The probability that the facility is idle.The expected number of customers waiting to be served. (10M)Explain classification of states and chains of markov process. (10M) (OR)If the transition probability matrix of market shares of three brands A,B and C is 0.40.30.30.80.10.10.350.250.4 Find (a) The market shares in second and third periods (b) The limiting probabilities (10M)15554715240030794730480045720015049529683-59843617853848608530480030289517.Question Bank: Unit – IShort answer QuestionsDefine probability.What are the axioms of probability?Define random experiment.Define random variable and types of random variable.Define an Event.Define Sample space.Define discrete random variable.Define continuous random variable.Derive probability distribution for D.R.V,?Derive probability distribution for C.R.V,?Define Binomial Distribution. Derive probability function of a Binomial distribution?Derive mean of a Binomial distribution?Derive variance of a Binomial distribution?Define Moment generating function.Find the moment generating function for Binomial distribution.Define Poisson distribution.Find the moment generating function for Poisson distribution.Derive probability function of a Poisson distribution?Derive mean of a Poisson distribution?Derive variance of a Poisson distribution?Define Normal distribution.Derive probability function of a Normal distribution?Derive mean of a Normal distribution?Derive variance of a Normal distribution?Derive median of a Normal distribution?Derive mode of a Normal distribution?Find the ratio of Mean Deviation, Standard deviation, Standard deviation and quartile deviation.Long Answer Questions: If f(x) is the distribution function x given by Fx=0 if x≤1,kx-14 if 1<x≤3,1 if x>1 Determine (i) f(x) (ii) k (iii) meanA random variable x has the following probability distribution. x12345678P(x=x)k2k3k4k5k6k7k8k Find the value of K (ii) p(x QUOTE 2) (iii) p(2 x ≤ 5).If X and Y are discrete random variables and k is constant then prove that(i) E(X+K) = E(X) +K (ii) E(X+Y) = E(X) + E(Y)Let F(X) be the distribution function of random variable X given by fx=cx3, when 0≤ &x≤31, when &x>30, when x≤0 Determine (i) c (ii) mean (iii) p(x>1)(i) The mean and variance of binomial distribution are 4 and 4/3 respectively. Find p(x>1).(ii)Let x denote the number of heads is a single toss of 4 fair coins. determine (a) p(x2) (b) p(1≤x 3)Average number of accidents on any duty on a national highway is 1.6. Determine the probability that the number of accidents are (i)at least one (ii) at most one. (i) Using recurrence formula find the probabilities when x= 0,1,2,3,4 and 5: if the mean of Poisson distribution is 3? (ii) Out of 800 families with 5 children each, how many would you expect to have (i) 3 boys (ii) either 2 or 3 boysIf the masses of 300 students are normally distributed with mean 68kgs and standard deviation 3kgs how many students have masses (i)Greater than 72kgs (ii) Less than or equal to 64kgs (iii) Between 65 and 71kgs inclusive ? In a normal distribution, 31% of the items are under 45 and 8% are over 64. Find the mean and S.D. of the distribution?Fit a normal curve to the following distributionx246810f14641 Unit – IIShort Answer Questions:Define Joint probability.Define joint probability mass function.Define joint probability density function.Define marginal probability mass function.Define marginal probability density function.Define correlation.Define Covariance.Explain the rank of correlation.What is meant by regression?Define regression coefficient.Write the relation between coefficient of correlation and regression coefficient?The two regression equations of the variables x and y are x= 19.13-0.87y and y = 11.64-0.50x. Then find correlation coefficient between x &y?The two regression equations of the variables x and y are x= 20.12-0.87y and y = 13.64-0.50x. Then find mean of x?The two regression equations of the variables x and y are x= 13.42-0.87y and y = 16.94-0.50x. Then find mean of y?If x=28, y=28 and xy=112. Then find cov (x,y)?Long answer Questions:Ten participants in a contest are ranked by two judges as followsX16510324978y64981231057 Calculate the rank correlation coefficient?Psychological tests of intelligence and of engineering ability were applied to 10 students. Hence is a record of ungrouped data showing intelligence ratio (I.R.) and engineering ratio (E.R.) Calculate the coefficient of correlation.StudentABCDEFGHIJI.R.1051041021011009998969392E.R.101103100989596104929794The correlation table given below shows that the ages of husband and wife of 53 married couples living together on the census night of 1991. Calculate the coefficient of correlation between the age of the husband and that of the wife.Age of husbandAge of wifeTotal15-2525-3535-4545-5555-6565-7515-2511----225-352121---1535-45-4101--1545-55--361-1055-65---242865-75----123Total3171496453In the following table are recorded data showing the test scores made by salesmen on an intelligence test and their weekly sales.Sales men12345678910Test Scores40705060805090406060Sales(‘000)2.56.04.55.04.52.05.53.04.53.0Calculate the regression line of sales on test scores and estimate the most probable weekly sales volume if a sales man makes a score of 70?If θ is an angle between two regression lines show that tanθ= 1-r2r.σxσyσx2+σy2 . Explain the significance when r = 0 and r = ±1.Find if there is any significance correlation between the heights and weights given below Heights in inches575962636465555857Weights in lbs113117126126130129111116112Find Karl Pearson’s coefficient of correlation from the following dataWages1001011021021009997989695Cost of living98999997959295949091A random sample of 5 college students is selected and their grades in Mathematics and Statistics are found to be the following. Find the coefficient of correlation between them12345Mathematics8560734090Statistics9375655080Find the regression line of x on y and y on x for the following dataX10121316172025Y10222427293337Calculate coefficient of correlation from the following dataX12981011137Y1486911123Unit – III: Short Answer QuestionsDefine population, sample and sampling with examples?Explain types of sampling?If ‘N’ refers population and ‘n’ refers sample size then mention i) The number of samples with replacement. ii) The number of samples without replacement.Define large sample and small sample.Define Parameters and Statistic.Define sample mean , sample variation, Sample Standard deviation.Explain sampling distribution and sampling distribution of statistic.Write the central limit theorem.Define Standard error and probable error of sample mean. Define confidence interval?Define estimate ,estimator and estimation .Define Bayesian estimation.Define statistical inference. and types of problems under statistical inference.Explain briefly types of estimations.Define an unbiased estimator and show that x is an unbiased estimator of the population mean μ.Define statistical hypothesis.Define hypothesis testing, null hypothesis and alternate hypothesis.Explain briefly type I error and type II error .Explain i) critical region ii)left tailed test iii)Right tailed test iv) Two tailed est.A sample of size 300 was taken whose variance is 225 and mean is 54. Construct 95% confidence interval for the mean?What is the value of correction factor if n = 5 and N = 200?How many different samples of size 2 can be chosen from a finite population of size 25?What is meant by degree of freedom?Define the statistics of ‘t’ ,‘F’ and χ2 distributions and their major applications.For an F- distribution find F0.05 with ?1=7 and ?2=15 ? Long Answer Questions: Prove that for a random sample of size n,X1,X2,….Xn taken from an infinite population s2 =1ni=1n(Xi-X)2 is not unbiased estimator of the parameter σ2 but 1n-1i=1n(Xi-X)2 is unbiased.(i)A random sample of 100 teachers in a large metropolitan area revealed a mean weekly salary of Rs.487 with a standard deviation Rs.48. with what degree of confidence can we assert that the average weekly salary of all teachers in the metropolitan area is between 478.6 to 495.4. (ii) Among 900 people is a state 90 are found to be chapatti eaters. Construct 99% confidence interval for the true proportionSample of size 2 are taken from the population 4,8,12,16,20,24 without replacement. Find (a) Mean of the population (b) Standard deviation of the population(c) The mean of the sampling distribution of the means (d) the standard deviation of the sampling distributions of means.A sample of 900 members is found to have a mean of 3.4 cm. Can it be reasonably regarded as a truly random sample from a large population with mean 3.25 cm and S.D. 1.61cm?The means of simple samples of sizes 1000 and 2000 are 67.5 and 68.0 cm respectively. Can the samples be regarded as drawn from the same population of S.D. 2.5cm?An unbiased coin is thrown n times. It is desired that the relative frequency of the appearance of heads should lie between 0.49 and 0.51. Find the smallest value of n that will ensure this result with 90% confidence?A certain stimulus administered to each of 12 patients resulted in the following increases of blood pressure.5,2,8,-1,3,0,-2,1,5,0,4,6. Can it be concluded that the stimulus will in general be accompanied by an increase in blood pressure?A 11 students were given a test in statistics they were given a month’s further tuition and a second test of equal difficulty was held at the end of it. Do the marks give evidence that the students have benefited by extra coaching?Boys1234567891011Marks I test2320192118201817231619Marks II test2419221820222020232017Two horses A and B were tested according to the time in seconds to run a particular race with the following results. Test whether you can discriminate between two horsesHorse A28303233332934Horse B293030242729--The results of polls conducted 2 weeks and 4 weeks before a election are shown in the following table:Two weeks before election4 weeks before electionFor Democratic candidate8466Undecided3743For Republican candidate7991Use the 0.05 level of significance to test whether there has been a change in opinion during the 2 weeks between the rolls.The measurements of the output of two units have given the following results. Assuming that both samples have been level whether the two populations have the same variance.Unit-A14.110.114.713.714.0Unit-B14.014.513.712.714.1 Unit – IV Short Answer QuestionsExplain the terms of queuing theory and give two examples.Explain about arrival pattern and service pattern.Explain the characteristics of queuing theory.Explain about The queue discipline and queue behavior.What is pure birth and death process?Probability that there are n customers in the system.Probability that there are n or more customers in the system.Show that Average number of customers in the system is σ1-σShow that Average queue length is σ21-σA TV repair man finds that the time spent on his jobs has an exponential distribution with mean 30 minutes. He repairs sets in the order in which they arrive. The arrival of sets is approximately Poisson with an average rate of 10 per eight hour day. Then find service rate per hour and arrival rate per hour.A machine repairing shop gets on average 16 machines per day (of eight hours) for repair and the arrival pattern is Poisson. Find arrival rate per hour? Long Answer Questions:a) Write the operational characteristics of Queuing theory.b) Assume the goods trains are coming is a yard at the root of 30 trains per day and suppose that inter arrival times follow an exponential distribution the service time for each train is assumed to be exponential with an average of 36 minutes. If the yard can admit a trains at time (there being 10 lines), one of which is reserved for shunting purpose), calculate the probability that the yard is empty and find the average queue length.a) A T.V repair man finds that the time spent on has jobs has an exponential distribution with mean 30 minutes. If he repairs sets in the order in which they came in, and if the arrival of sets in approximately Poisson with an average rate of 10 per-8-hour day. What is repairman’s expected idle time each day? How many jobs are a head of the average set just brought us?b) Patients arrive at a clinic according to a poisson distribution at a rate of 30 patients per hours, the waiting room does not accommodate more than 14 patients. Examination time per patients is exponential with mean rate 20 per hour.i) Find effective arrival rate at the clinicii) What is the probability that an arriving patients will not wait?A manager of a Local hamburger restaurant is preparing to open a new fast food restaurant called Hasty Burgers. Based on the arrival rates at existing outlets. Manager expects customers to arrive at the drive in window according to a Poisson distribution with a mean of 20 customers per hour. The service rate is flexible however, the service times are expected to follow an exponential distribution the drive in window in single ever operation.What service rate is needed to keep the average number of customers is the service system to 4.For the service rate in part (a). What is the probability that more than 4 customers are in the line and being served?At a certain petrol pump, customers arrive in a poission process with an average time of five minutes between arrivals, the time intervals between serves at the petrol pump follows exponential distribution and the mean time taken to service a unit is two minutes. Find the following:Average time a customer has to wait is the queue.By how much time the flow of the customers be increases to justify the opening of another service point, where the customer has to wait for five minutes for the service. A fast food restaurant has one drive is window. It is estimated that cars arrive according to a Poisson distribution at the rate of 2 every 5 minutes and that there is enough space to accommodate a line of 10 cars. Other arriving cars cannot wait outside this space, if necessary. If takes 1.5 minutes on the average to fill an order, but the service time actually varies according to an exponential distribution. Determine the following.The probability that the facility is idle.The expected number of customers waiting to be served.a) Explain about queuing theory characteristics?Define preemptive discipline and non-preemptive priority?Consider a self service store with one cashier. Assume Poisson arrivals and exponential service time. Suppose that 9 customers arrive on the average of every 5 minutes and the cashier can serve 19 in 5 minutes. Find (i) the average number of customers queuing for service.(ii) The probability of having more than 10 customers in the system. (iii) The probability that the customer has to queue for more than 2 minutes.A car park contains 5 cars. The arrival of cars is Poisson with a mean rate of 10 per hour. The length of time each car spends in the car park has negative exponential distribution with mean 2 hours. How many cars are in the car park on average and what is the probability of a newly arriving customer finding the car park full and having to park his car elsewhere?At the election commission office, for the Voter’s identity Card, a Photographer takes passport size photo at an average rate of 24 photos per hour. The photographer must wait until the voter blinks or scowls, so the time to take a photo is exponentially distributed. Customers arrive at Poisson distributed average rate of 20 voters per hour. Find i) what the utilization of Photographer is. (ii) How much time, the voter has to spend at the election commission office on an average to get the service. Barber A takes 15 minutes to complete one hair cut. Customers arrive in his shop at an average rate of one every 30 minutes, Barber B takes 25 minutes to complete one hair cut and customers arrive at his shop at an average rate of one every 50 minutes. The arrival processes are Poisson and the service times follow an exponential distribution.Unit – V Short Answer Questions:What do you mean by stochastic processes and what are the types of stochastic process? Define them.Explain about Markov Processes.Explain about Stationary Processes.Explain about dependent and independent Stochastic Processes.Explain about gamblers Ruin Problem.Show that the probability that the game never ends is zero.Explain the process of finding expected duration of the game?Let p=12, q=12, z=500,a=1000. then find the expected duration of the game.Calculate the probability of ruin and expected duration of the game, when (i)a=100,z=5,p=0.6 iia=50,z=40, p=0.5 Explain about Transition matrix. Which of the following matrices are stochastici100010ii1001iii011314 (iv)12121212 Define the Stochastic matrix Which of the stochastic matrices are regular.i12141401012012 ii 1212012120141412 iii00112012010Long Answer Questions:The transition probability matrix of a morkov chain is given by0.30.700.10.40.500.20.8 verifyWhether the matrix is irreducible or not?13234312112 is this matrix stochastic?0010.50.250.25010 is this matrix regular?If the transition probability matrix of market shares of three brands A,B and C is0.40.30.30.80.10.10.350.250.4 Find (a) The market shares in second and third periods (b)The limiting probabilities. Explain classification of states and chains of markov process.Three boys A,B and C are throwing a ball to each other. A always throws the ball to B and B always throws the ball to C, but C is just as likely to throw the ball to B as to A. Show that the process is Markovian. Find the transition matrix and classify the states. Do all the states ate ergodic?Find the nature of states of the markov chain with transition probability matrixP=01012012010A fair die is tossed repeatedly. If Xn denotes the maximum of the numbers occurring in the first n tosses, find the transition probability matrix P of the markov chain Xn. Find also P2 and PX2=6.Define Markov chain,regular, ergodic and Stochastic matrices?Check whether the following markov chain is regular and ergodic?10.50.50 0.5 000.50.5 000.500.50.50.5A gambler has Rs.2. He bets Rs.1 at a time and wins Rs.1 with the probability 0.5. He stops playing if he looses Rs.2 or wins Rs.4.(a) What is the transition probability matrix of the related Markov chain?(b) What is the probability that he has lost his money at the end of 5 plays?18.Assignment questions:Unit-I a) Explain, with suitable examples, discrete and continuous random variable. b) Find the first 3 moments about origin from Moment generating function of the Binomial distribution. a) If ‘X’ is a continuous random variable whose probability density function is given by fx=13, 0, -1<x<2elsewhere b) A sample of 3 items is selected from a box having 6 items of which 3 are defective then find the mean of the distribution of defective items. a) If X is the continuous random variable whose probability density function is fx=ax+bx2,0, 0<x<1else where and EX=0.6. Find the values a and b. b) The mean variance of a binomial distribution are 2 and 85. Find : in and Mode iiMaximum probability cPX>2.a) If the weight of 1000 students are normally distributed with mean 75 kgs. And standard deviation 10kgs. How many students have weight greater than 90 kgs.b) If X and Y are two random variables with joint probability density function fx,y=Ke-X-|Y|. Find the value of K.a) Two dice are thrown 4 times. If getting a sum of 7 is a success. Find the probability that getting the success i) Twice ii) only once.b) Students of a class were given an examination. Their marks were found to be normally distributed with mean 55 marks and standard deviation 5. Find the number of students who got the marks more than 60 if 500 students wrote the examination.a) Poisson variable has double mode at x=2 and x=3, find the maximum probability and also find p(x≥2)b) If the masses of 300 students are normally distributed with mean 68 kgs and standard deviation 3 kgs how many students have masses greater than 72 kgs.Unit-2Calculate the coefficient of rank correlationX68647550648075405564Y62586845816068485070 Write the relation between correlation and regression coefficients. Is it possible to have two variables X and Y with regression coefficient as 2.8 and -0.5? Explain.If X and Y are two random variables having joint density function fx,y=186-x-y,0, 0≤x≤2, 2≤y<4otherwise Find: i PX<1Y<3 ii fXx & fY(y) For the following data, find equations of the two regression lines.X12345Y1525354555 The joint probability density function is given by fx,y=10xy2,0, 0<x<y<1elsewhereFind: a) Marginal probability density function for X b) Marginal probability density function for Y c) Conditional P.D.F. of X given Y d) Conditional P.D.F. of Y given X e) PY>0.5X=0.25. Calculate the coefficient of correlation between the two variable x and y. Also find the regression coefficients.X6566676768697072Y6768656872726971 The equations of two regression lines obtained in a correlation analysis are 3x+12y=19, 3y+9x=46. Find the means of x and y.Unit-3 In a city A 20% of a ramdom sample of 900 school boys had a certain physical defect. In another city B 18.5% of a random sample of 1600 school boys had the same defect. Is the difference between the proportions significant?Write the standard error of (i) sample mean (ii) difference of two sample means.Mean of population =0.700, mean of the sample = 0.742, standard deviation of the sample =0.040, sample size=10. Test the null hypothesis for population mean=0.700. A die is thrown 60 times with the following results.Face123456Frequency871281411 Test 5% level of significance if the die is honest. Two horses A and B were tested according to the time (in seconds) to run a particular track with the following results.Horse A28303233352934Horse B 293030242729Test whether the two horses have the same running capacity.Define type-I and type-II error.A sample of 150 items is taken from a population whose standard deviation is 12. Find the standard error of means.A researcher wants to know the intelligence of students in a school. He selected two groups there 150 students having mean IQ of 75 with a S.D. of 15 in the second group there are 250 students having mean IQ of 70 with S.D. of 20. Is there a significant difference between the means of two groups?Fit a Poisson distribution to the following data and test the goodness of fit.X0123456Observed frequency27572307521In a city A 20% of a random sample of 900 school boys had a certain physical defect. In another city B 18.5% of a random sample of 1600 school boys had the same defect. Is the difference between the proportion significant?Two independent samples of 8 and 7 items respectively have the following values.Sample I111113111591214Sample II91110139810-Is the difference between the means of sample significant? Given below is the number of male births in 1000 families having five children in each family.Male Children012345No. of families4030025020030180Test whether the given data is consistent with the hypothesis that the binomial law holds if the chance of a male birth is equal to that of female births.Unit-4Patients arrive at a hospital at random with a mean arrival rate of 3 per hour. The department is served by one doctor, who spends on average 15 minutes with each patient. Actual consulting times being exponentially distributed. Find The portion of time that the doctor is idle.The mean number of patients waiting to see the doctor.The probability of there being more than 3 patients waiting.The mean waiting time for patients.Derive formulaeTo find the expected number of units in the system.To find the expected number of units is the queue.Define Balking, reneging and jockeying.Define mean arrival rate and Transient state in queuing system.In a telephone exchange the arrival of calls follow Poisson distribution with an average of 8 minutes between two consecutive calls. The length of a call is 4 minutes. Determine The probability that the call arriving at the booth will have to wait.The average queue length that forms from time to time.The probability that an arrival will have to wait for more than 10 minutes before the phone is free.A computer shop has a laser printer. The jobs for laser printing are randomly distributed approximately poisson distribution with mean service rate of 10 jobs per hour, since job pages vary in length ( pages to be printed). The jobs arrive at a rate of 6 perhour during the entire 8 hours working day. If the laser printer is valued Rs. 30 per hour, determine:The percent time an arriving job has to wait.Average system time.Average idle time cost of the printer per day.Prove that the probability of having ‘n’ customers in the queuing system M /M /I:∞,FCFS is Pn=ρn1-ρ, Where is ρ traffic intensity of the system?In a public Telephone both the arrivals are on the average 15 per hour. A call on the average takes 3 minutes. If there is just one phone, find (i) expected number of callers in the booth at any time (ii) The proportion of the time the booth is expected to be idle.Unit-5 a) Define a Regular transition matrix. If 0.2x0.20.10x+yz0.20.1 is a transmission probability matrix, then find the values of x,y and z. Define a regular Marcov Chain.Find the values of of x,y and z inorder for 0x1300y1314z to be transition matrix.Write about the different states of the stochastic process.The three state markov chain is given by the transition probability matrix P=023131201212120. prove that the chain is irreduciable.The transition probability matrix of a Markov chain is given by 0.30.700.10.40.500.20.8. Is this Matrix irreducible. The transition probability matrix is given by 00.40.60.10.20.70.30.30.4 and P0=0.2,0.3,0.5. Find a) The distribution after three transitions and the limiting probabilities. A country is divided into three demographic regions. It is found that each year 5% of the residents of region1 move to region2 and 5% move to region3, from residents of region2, 15% move to region1 and 10% move to region3 and from the residents of region3, 10% move to region1 and 5% move to region2. What percentage of the population resides in each of the three regions after a long period of time?Study of passage of English text to find a vowel followed by a vowel or a consonant followed by a consonant or a reveal the following transition probability matrix 0.120.880.540.46. Find the percentage of letters in the text book which are expected to be vowels. 19.Unitwise Quizz Questions and long answer questionsObjective Type Questions:Unit- I1. If f(x) =A x2 in 0≤x≤1 is a probability function then A= (a) 1 (b)2 (c)3 (d) 42.The area under the whole normal curve is---- (a) 0 (b)2 (c)3 (d) 13.The mean, median and mode of Normal distribution are... (a) equal (b)2 (c)3 (d) 14.In the standard normal curve the area between z=-1 and z=1 is nearly..... (a) 68% (b)67% (c)66% (d) 65%5.In a normal distribution, mean deviation : standard deviation =... (a) 4 : 5 (b)5 : 4 (c)3 :5 (d) 5 : 36.The mean of the binomial distribution is.......... (a) np (b)npq (c)p (d) p+q7.If the mean and variance of the binomial variate are 4 and 2 then p = (a) 0.5 (b)0.3 (c)0.4 (d) 0.18.If the mean the binomial is 6 and variance is 2,then the mode of this distribution is= (a) 5 (b) 3 (c) 4 (d) 69.The probability of getting one boy in a family of 4 children is = (a)0.25 (b) 0.24 (c) 0.22 (d) 0.26510.If the variance of a poisson variate is 3 thenP(x=0)=..... (a) e-3 (b) e-4 (c) e-5 (d) e-211. If the mean and variance of the poisson distribution are........ (a) equal (b)2 (c)3 (d) 112.In a poisson distribution if 2P(X=1)=P(X=2)........... (a) 5 (b) 3 (c) 4 (d) 613.A die is thrown 8 times . the probability that 3 will show exactly 2 times is...... (a) 28×5668 (b) 28×5768 (c) 28×5667 (d) 28×566 14.The binomial distribution whose mean is 5 and variance is 103 is.........(a) 15cr 13r 2315-r (b) 14cr 13r 2314-r (c) 13cr 13r 2313-r (d) 11cr 13r 2311-r 15.If P(1)=P(2), then the mean of the Poisson distribution is (a) 4 (b)2 (c)3 (d) 1Unit-IIIf the regression coefficients are -0.4 and -0.9, then the correlation coefficient is------The value of coefficient of correlation lies between---and ----The rank coefficient is given by-----------------If two regression lines are coincide then the coefficient correlation is ---------Arithmetic mean of the coefficients of regression is -------than the coefficient of correlation.When variables are independent, the two lines of regression are-------Correlation coefficient is the geometrical mean between----------The angle between two regression lines is given by_________The equation of line of regression of y on x is---------------The point of intersection of two regression lines is-----------If X and Y are independent, then the correlation coefficient between X and Y is ---------If two regression lines are perpendicular to each other, then their coefficient of correlation is—If y=x+1 and x= 3y-7 are the two lines of regression, then x=--,y=--and r=----Regression coefficient of y on x is 0.7 and that of x on y is 3.2. Is the correlation coefficient r is consistent? If r1and r2 are two regression coefficients,then signs of r1and r2 depends on----- The correlation coefficient is 0, the regression lines are------The equations of regression lines are y=0.5x+a and x=0.4y+b.The correlation Coefficient is -----The coefficient of correlationcannot be positive (b) cannot be negative (c) either positive or negative (d) noneWhich of the following is the highest range of r ?0 an 1 (b) - 1 and 0 (c) -1 and 1 (d) 1 and 1The coefficient of correlation is independent of change of scale property (b) change of origin only (c) both (d) noneThe value of r2 for a particular situation is 0.81. the coefficient of correlation is ----------------The coefficient of correlationHas no limits (b) can be less than 1 (c) can be more than 1 (d) varies between ± 1Coefficient of correlation = bxy*byx (b) √ bxy*byx (c) √ bxy (d) √ byxOne regreesion coefficient is positive then the other regression coefficinet isPositive (b) negative (c) equal to zero (d) cannot sayRegression coefficient is independent of Origin (b) scale (c) origin and scale (d) noneWhen two regression lines coincide then r is0 (b) – 1 (c) 1(d) 0.5The regression lines cut each other at the point of Average of X and Y (b) average of X only (c) average of Y only (d) noneUnit – IIIThe totality of the observation is called.... (a) population (b)sample (c)parameter (d)noneThe statistical constants of the population are called................ (a) statistic (b)sample (c)parameter (d)noneThe probability distribution of a statistic is called......... (a) Normal distribution (b)sampling distribution (c)binomial distribution (d)noneThe finite population correction factor is (a)n-Nn-1 (b) N-nN-1 (c) N-1N-n (d)noneThe standard error of the statistic sample meanx is (a) σn (b) 2σn (c) 3σn (d)noneIf x1,x2,x3,x4,x5,x6 ,......,xn constitute arandom sample from an infinite population with the mean μ and the variance σ2 then μx =........and σ2x =........... (a) μ , σ2 n (b) 3μ , 2 n (c) μ , σ2 3n (d)nonet1-∝ = (a)t∝ (b)-t∝ (c) t∝-1 (d)noneIf x =157 , μ=155 , σ =15 and n=36 then z is (a) 0.8 (b) 0.08 (c) 0.6 (d)noneThe shape of t-distribution is similar to that of (a) chi-square distribution (b)Normal distribution (c) F-distribution (d)noneChi-square distribution curve varies from (a) -∞ to ∞ (b) 0 to ∞ ) (c) -∞ to 0 (d)noneP(z>1.76) = (a) 0.5+P(0<z<1.76) (b) ) 0.5-P(0<z<1.76) (c) ) P(0<z<1.76) (d)noneThe sample of size 4 has values 25,28,26,25 then the variance of the sample is (a) 2 (b) ) 2.5 (c) ) 4.2 (d)noneThe marks of 5 students in one subject are 45,47,49,61,48 and the mean of the population is 52 then t =..... (a) 0.7 (b) ) 0.5 (c) ) 0.6 (d)noneIf the size of the sample is 5 and size of the population is 2000 .then the correction factor is (a) 0.999 (b) ) 9.99 (c) ) 99.9 (d)noneIf x =47.5 , μ =42.1, S = 8.4 and n =25 then t is...... (a) 3.21 (b) ) 4.5 (c) ) 3.12 (d)noneUnit – IV1.For the system (M/M/S):(∞/FCFS), Average number of customers is the system =Ls =(a) 1μ-λ (b) μμ-λ (c) λμ-λ (d) 1- σ2.For the system (M/M/1):(∞/FCFS), trafic intensity=-----------(a) μλ (b) μ λ (c) λμ (d) μ +λ 3.Suppose the inter arrival time is 15 minutes and inter service time is 10 minutes. The traffic intensity is------------(a) 45 (b) 1510 (c) 54 (d) 1015Average number of customers is the system=(a) 1μ-λ (b) μμ-λ (c) λμ-λ (d) 1- σAverage number of customers is the queue=(a) 1μ-λ (b) μμ-λ (c) λμ-λ (d) σ21- σAverage waiting time of a customer in the queue =(a) σμ-λ (b) μμ-λ (c) λμ-λ (d) σ21- σAverage waiting time of a customer who has to wait =(a) 1μ-λ (b) μμ-λ (c) λμ-λ (d) σ21- σ8.If average number of arrivals is 4 per hour and average number of services is 6 per hour, probability that a new arrival need not wait for the service is--------(a) 23 (b) 29 (c) 13 (d) 49Arrival rate is 10 per day; service rate is 16 per day. The day consists of 8 working hours. Expected idle time per day is--------(a) 2 hours (b) 3 hours(c) 3.5 hours (d) 4 hoursIn the above problem, average number of customers in the system is=-----------(a) 13 (b) 43 (c) 53 (d) 2If λ =8 and μ = 12 per hour,the average time spent by a customer is the system is----------(a) 10 minutes (b) 12 minutes (c)15 minutes (d) 20 minutesIn the above problem, the expected time a customer spends in the queue is(a) 10 minutes (b) 7 minutes (c)15 minutes (d) 5 minutesArrival rate is 3 per hour; service rate is 5 per hour. Then the traffic intensity is—(a) 35 (b) 25 (c) 53 (d) 52 Probability that there are more than or equal to 10 customers is the system =----(a) 3510(b) 3511 (c) 3512 (d) 351315.Probability that there are n customers in the system =-----(a) σn (1- σ) (b) μ λ (c) λμ (d) μ +λ Unit – VIf the probability transition, matrix is .6.40.20.8then P (2) = ………………. If the probability transition, matrix is 0.1.6.3.5.1.4.1.2.7 then P (2) =______A random process X(t) is called a markov process if _______________If the tpm is .4.60.20.8 , then the limiting probabilities are---------------- A state forming a closed set by itself is called ___________________ state.A state i is said to be ________________________ if it is a periodic.The transition probability matrix of an irreducible chain is a ___ [ ](a) Reducible matrix (b) irreducible matrix (c) identity matrix (d) all of the above______is said to be regular if its transition probability matrix is regular.A non-null persistent and aperiodic state is called ___________________If the possible values of Xn are …….-2,-1,0,1,2,….Then the Markov chain is said to be…………..20.Tutorial Problems:Unit-IIf X is a continuous random variable whose probability density function is given byfx=13, -1<x<20, else where . Find the moment generating function. If X is a Poisson vitiate such that px=1=24px=3, find the mean. Two dice are thrown 4 times. If getting a sum of 7 is a success. Find the probability that getting the success i Twice ii only once If the masses of 300 students are normally distributed with mean 68 kgs and standard deviation 3 kgs how many students have masses (i)Greater than 72 kgs (ii) Less than or equal to 64kgs (iii) Between 65 and 71 kgs inclusive ? Fit a normal curve to the following distributionx246810f14641 Unit-2Psychological tests of intelligence and of engineering ability were applied to 10 students. Hence is a record of ungrouped data showing intelligence ratio (I.R.) and engineering ratio (E.R.) Calculate the coefficient of correlation. StudentABCDEFGHIJI.R.1051041021011009998969392E.R.101103100989596104929794If θ is an angle between two regression lines show that tanθ= 1-r2r.σxσyσx2+σy2 . Explain the significance when r = 0 and r = ±1.Find the regression line of x on y and y on x for the following dataX10121316172025Y10222427293337A random sample of 5 college students is selected and their grades in Mathematics and Statistics are found to be the following. Find the coefficient of correlation between themS.No.12345Mathematics8560734090Statistics9375655080unit-3Sample of size 2 are taken from the population 4,8,12,16,20,24 without replacement. Find (a) Mean of the population (b) Standard deviation of the population(c) The mean of the sampling distribution of the means (d) the standard deviation of the sampling distributions of means.An unbiased coin is thrown n times. It is desired that the relative frequency of the appearance of heads should lie between 0.49 and 0.51. Find the smallest value of n that will ensure this result with 90% confidence?A random sample of 100 teachers in a large metropolitan area revealed a mean weekly salary of Rs.487 with a standard deviation Rs.48. with what degree of confidence can we assert that the average weekly salary of all teachers in the metropolitan area is between 478.6 to 495.4.A 11 students were given a test in statistics they were given a month’s further tuition and a second test of equal difficulty was held at the end of it. Do the marks give evidence that the students have benefited by extra coaching?Boys1234567891011Marks I test2320192118201817231619Marks II test2419221820222020232017Unit-4A manager of a Local hamburger restaurant is preparing to open a new fast food restaurant called Hasty Burgers. Based on the arrival rates at existing outlets. Manager expects customers to arrive at the drive in window according to a Poisson distribution with a mean of 20 customers per hour. The service rate is flexible however, the service times are expected to follow an exponential distribution the drive in window in single ever operation. a) What service rate is needed to keep the average number of customers is the service system to 4. b) For the service rate in part (a). What is the probability that more than 4 customers are in the line and being served? At the election commission office, for the Voter’s identity Card, a Photographer takes passport size photo at an average rate of 24 photos per hour. The photographer must wait until the voter blinks or scowls, so the time to take a photo is exponentially distributed. Customers arrive at Poisson distributed average rate of 20 voters per hour. Find i) what the utilization of Photographer is (ii) How much time, the voter has to spend at the election commission office on an average to get the serviceConsider a self service store with one cashier. Assume Poisson arrivals and exponential service time. Suppose that 9 customers arrive on the average of every 5 minutes and the cashier can serve 19 in 5 minutes. Find (i) the average number of customers queuing for service.(ii) The probability of having more than 10 customers in the system. (iii) The probability that the customer has to queue for more than 2 minutes.Unit-5If the transition probability matrix of market shares of three brands A,B and C is 0.40.30.30.80.10.10.350.250.4 Find (a) The market shares in second and third periods (b)The limiting probabilities. Check whether the following Markov chain is regular and ergodic? 10.50.50 0.5 000.50.5 000.500.50.50.5A gambler has Rs.2. He bets Rs.1 at a time and wins Rs.1 with the probability 0.5. He stops playing if he looses Rs.2 or wins Rs.4.(a) What is the transition probability matrix of the related Markov chain?(b) What is the probability that he has lost his money at the end of 5 plays?The transition probability matrix of a morkov chain is given by0.30.700.10.40.500.20.8 Verify? Whether the matrix is irreducible or not?21.Known gaps, if any and inclusion of the same in lecture schedulePermutationsCombinationsCentral deviationsFrequency distributionProbabilityConditional probabilityCurve fitting(regression lines)Queue concept22.Discussion topicsConcept of ProbabilityConditional ProbabilityRandom VariableBionomial DistributionPoison Distribution23.References,Journals,Websites and E-links:23.1Text BooksHigher Engineering Mathematics by Dr B S Grewal, Khanna PublicationsProbability and Statistics for Engineers and Scientists by Sheldon M Ross, Academic PressOperations Research by S D Sarma23.2. Reference Books Mathematics for engineers by K B Datta and M A S Srinivas, Cengage Publications Probability and Statistics by T K V Iyengar & B Krishna Gandhi Et Fundamental of mathematical statistics by S C Gupta and V K Kapoor Probability and Statistics for Engineers and Scientists by Jay I Devore.23.3. Websites1. 2. mathworld.3. 4. 5. 23.4. Journals1. Numerical Linear Algebra with Applications2. International Journal for Numerical Methods in Engineering3 Journal of Inequalities in Pure and Applied Mathematics4. SIAM Journal of Applied Mathematics5. Journal of Partial Differential Equations24.Quality Measurement Sheets:A. Course End SurveyB. Teaching EvaluationHard copy available25.Student List: 26.Group-Wise students list for discussion topics: ................
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