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KURUKSHETRA UNIVERSITY, KURUKSHETRA Curriculum for M.Sc. Statistics (CBCS)

Scheme of Examination (Effective from the Academic Session 2016-

2017)

The duration of the course leading to the degree of Master of Science (M.Sc.) Statistics shall be of two academic years. The course will be run under semester system. The examination of Semester –I and Semester- II will ordinarily be held in the month of December and that of Semester-II and Semester-IV in the month of May on the dates to be notified by the controller of examinations.There will be five papers in each semester, four theory and one practical. In Semester-I & II all the papers will be core papers. In Semester –III and IV out of four theory papers, two papers will be core and the other two will be elective ones. The elective papers will be offered from the list given in the scheme provided the staff is available to teach them.

Every student will be required to go for training in reputed institute and after the completion of the training the student will submit the training report. The report will be evaluated by practical examiner.

Each student will be required to give one test /seminar for each paper. and one class test for the purpose of internal assessment.

The details of the scheme of examination

Semester-1Paper No.

Nomenclature Paper type

Credits Contact hours per

week

Internal Marks

External Marks

Total Marks

Duration of Exam

(Hours)

ST-101 Measure and Probability

core 4 4 25 75 100 Three

ST-102 Statistical Methods and Distribution Theory

core 4 4 25 75 100 Three

ST-103 Inference-I core 4 4 25 75 100 Three

ST-104 Applied Statistics core 4 4 25 75 100 Three

ST-105 Practical (Calculator and SPSS/SYSTAT based)

core 4 8 25 75 100 Four

Total Credits-20 Total Marks -500

KURUKSHETRA UNIVERSITY, KURUKSHETRACurriculum for M.Sc. Statistics (CBCS)

Scheme of Examination (Effective from the Academic Session 2016-2017)

Semester – ll

Paper No.

Nomenclature Paper

type

Credits Contact

Hours

Per week

Internal marks

External

marks

Total marks

Duration of Exam (Hours)

ST-201 Demography core 4 4 25 75 100 Three

ST-202 Operations Research core 4 4 25 75 100 Three

ST-203 Inference-II core 4 4 25 75 100 Three

ST-204 Computer Fundamentals and Problem Solving Using C

core 4 4 25 75 100 Three

ST-205 Practical

(Computer based)

core 4 8 25 75 100 Four

*OE- *Open elective *open

elective

2 2 15 35 50 Three

Total Credits-22 Total Marks- 550

*To be opted from the list of the papers of the other departments of faculty of Science.

KURUKSHETRA UNIVERSITY, KURUKSHETRA Curriculum for M.Sc. Statistics (CBCS)

Scheme of Examination

(Effective from the Academic Session 2016-2017)

Semester –llI

Paper No.

Nomenclature Paper

type

Credits ContactHours Per week

Internal marks

External

marks

Total marks

Duration of Exam.

(Hours )

ST-301 Sampling Theory core 4 4 25 75 100 Three

ST-302 Object-Oriented Programming with C++

core 4 4 25 75 100 Three

ST-303

&

ST-304

Opt.(i) Theory of Queues

Opt. (ii) Linear Programming

Opt.(iii) Stochastic Processes

Opt.(iv) Bio-Statistics

Opt. (v) Statistical Methods in Epidemiology

Opt. (vi) Statistical Ecology

Any Two

elective 4 4 25 75 100 Three


ST-305 Practical (Computer based)

core 4 8 25 75 100 Four

*OE- *Open elective *Open

elective

2 2 15 35 50 Three


* To be opted from the list of the papers of the other departments of faculty of Science.

KURUKSHETRA UNIVERSITY, KURUKSHETRACurriculum for M.Sc. Statistics (CBCS)

Scheme of Examination (Effective from the Academic Session 2016-2017)

Semester –IV

Paper No.

Nomenclature Paper

type

Credits ContactHours Per week

Internal marks

External

marks

Total marks

Duration of

Exam.

(Hours )

ST-401 Multivariate Analysis core 4 4 25 75 100 Three

ST-402 Linear Estimation & Design of Experiments

core 4 4 25 75 100 Three

ST-403

&

ST-404

Opt. (i) Reliability and Renewal Theory

Opt.(ii) Non-Linear and Dynamic Programming

Opt. (iii) Information Theory

Opt. (iv) Game Theory

Opt. (v) Econometrics

Opt. (vi) Acturial Statistics

Any Two



ST-405 Practical (Calculator and SPSS/SYSTAT based)

core 4 8 25 75 100 Four


Total Credits for two academic years: 84 (42+42).

M.Sc. Statistics Semester – I

Paper – I Measure and Probability(ST-101) Course Objectives:

The objective of this course is to provide an introduction to the basic notations and results of

measure theory and how these are used in probability theory. The aim of the course is to pay a

special attention to applications of Measure Theory in the Probability Theory. We will develop a

proper understanding of probability spaces for random variables and their finite and infinite

sequences. Using these concepts we will discuss Strong Laws of Large Numbers and their

applications. We will derive the Central Limit Theorem and we will discuss some of its

applications.

Learning Outcomes: On completion of this course students will be able to:

Understand the concepts of random variables, sigma-fields generated by random

variables, probability distributions and independence of random variables related to

measurable functions.

Knowledge about integration with respect to probability distributions, absolutely

continuous measures, expectation of a random variable and characteristic functions as

applications of Lebesgue integral.

Understand the concept of integrable functions, moments, independence and first

construction of conditional expectation.

Analyze modes of convergence,have knowledge about convergence in probability.

Understand weak and strong laws of large numbers, prove Borel-Cantelli lemmas and

central limit theorem.


Paper – I Measure and Probability(ST-101) (4 Credits)

Max Marks: 75+25* *Internal Assessment

Time: 3 hrs.

Note:There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all selecting one from each unit and the compulsory one. The weightage of all the questions will be the same.

Unit –IFields; sigma field, sigma-field generated by a class of subsets, Borel fields. Sequence of sets, limsup and liminf of sequence of sets, random variables, distribution function. Measure, probability measure, properties of a measure, Concept of outer measures, inner measures, lebesgue measures, concept of Lebesgue-Stieltjes measure.

Unit –IIMeasurable functions, sequence of measurable functions; their convergence of various types. Integration of measurable function. Monotone convergence theorem. Fatou's Lemma. Dominated convergence theorem, Product measure, Fubin’s Theorem.

Unit –IIIBorel-Contelli Lemma, Tchebycheff's and Kolmogorov's inequalities, various modes of convergence: in probability, almost sure, in distribution and in mean square and their interrelationship.

Unit –IVLaws of large numbers for i.i.d. Sequences. Characteristic function its uniqueness, continuity and inversion formula. Applications of characteristic functions. Central limit theorems: De Moivre’s-Laplace, Liapounov, Lindeberg-Levy and their applications

References:1. Kingman, J. F. C. & Taylor, : Introduction to Measure and Probability, Cambridge S.J. (1966). University Press.2. Bhat, B.R. : Modern Probability Theory, Wiley Eastern Limited3. Taylor, J. C. : An Introduction to Measure and Probability, Springer.4. Royden, H.L. : Real Analysis, Pearson Prentice Hall.5. Billingsley, P. (1986). : Probability and Measure, Wiley.6. Halmos, P.R. : Measure Theory, Springer7. Basu,A.K. : Measure Theory and Probability,PHI Learning(Pt.Lim.)


Paper-II Statistical Methods and Distribution Theory(ST-102) Course Objectives:The course aims to shape the attitudes of learners regarding the field of statistics. This course will lay the foundation to probability theory and Statistical modeling of outcomes of real life random experiments through various Statistical distributions. Specifically, the course aim to Motivate in students an intrinsic interest in statistical thinking.


Explain the concepts of probability.

Understand the Mathematical, Tchebycheff's, Markov, Jensen and Holder and

Minkowski inequalities.

Understand the concepts of distribution theory.

Test the hypothesis using suitable statistical test.

Understand the discrete and continuous distributions like Binomial, Poisson,

Geometric, Negative binomial, Hypergeometric and Multinomial, Normal, log

normal distributions, Uniform, Exponential, Cauchy, Beta, Gamma distribution.


Paper-II Statistical Methods and Distribution Theory(ST-102) (4 Credits)

` Max Marks: 75+25* *Internal Assessment Time: 3 hrs.

Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same.

Unit-IBasic concepts of probability: Random variable, sample space, events, different definitions of probability, notations, distribution functions. Additive law of probability, theorem of total probability, theorem of compound probability and Baye’s theorem. Concept of bivariate, marginal and conditional distributions.

Unit-IIMathematical Expectation : Expectation and moments, expectation of sum of variates, expectation of product of independent variates, moment generating function. Tchebycheff's, Markov , Jensen and Holder and Minkowski inequalities, Covariance, correlation coefficient , rank correlation, regression lines partial correlation coefficient, multiple correlation coefficient . Relation between characteristic function and moments.

Unit – IIIBinomial, Poisson, Geometric, Negative binomial, Hypergeometric and Multinomial, Normal and log normal distributions.

Unit –IVUniform, Exponential, Cauchy, Beta, Gamma distribution, Sampling distributions: Student – t distributions, F- distribution, Fisher’s z – distribution and Chi-square distribution. Inter relations, asymptotic derivations. Simple tests based on t, F, chi square and normal variate z.

References:1. Feller, W. : Introduction to probability and its applications, Vol.I, Wiley2. Parzen, E. : Modern Probability Theory and its Applications, Wiley Interscience3. Meyer, P.L. : Introductory Probability and Statistical Applications, Addison wesely.4. Cramer, H. : Random variable and Probability Distribution, Cambridge University Press.5. Kapur, J.N. Sexena, H.C.: Mathematical Statistics & S.Chand & Co.


Paper – III Inference –I(ST-103)Course Objectives:

The main objective of the course is to draw statistically valid conclusions about a population on

the basis of a sample in a scientific manner. This course deals with fundamental concepts and

techniques of statistical inference including point and interval estimation. A brief revision will

also be given of some basic topics in probability theory as well as single and multiple random

variables. Alternative philosophical approaches to inference such as likelihood methods, and

fiducial methods will be described. The impact that statistics has made and will continue to make

in virtually all fields of scientific and other human endeavors is considered.

During this course students will develop a deeper understanding of the basis underlying modern

statistical inference and equip themselves with a statistical tool kit which will enable them to

apply their knowledge and skills to real world tasks.

Learning Outcomes: On completion of the course, students will be able to:

Apply various discrete and continuous univariate and multivariate probability

distributions in modeling statistical processes.

Understand how sampling distributions are used in making statistical inferences by

defining sampling distribution.

Estimate unknown parameters of a given probability distribution using standard and non-

standard estimation techniques.

Understand (i) how probability is used to make statistical inferences, (ii) what inferential

statistics are used for and (iii) know how to perform point and interval estimation.

Familiar with the fundamental concepts of random variables as they apply to statistical

inferences.

Familiar with the fundamental concepts of statistical inference as they apply to problems

found in other disciplines.


Paper – III Inference –I(ST-103) (4 Credits)

Max Marks: 75+25* *Internal Assessment Time: 3 hrs.

Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same. Unit – I

Elements of Statistical Inference. Concept of likelihood function. Point estimation. Concept of consistency, unbiased estimators, correction for bias, minimum variance estimator, Cramer – Rao inequality, Minimum Variance-Bound (M.V.B.) estimator, Bhattacharya Bounds, Uniqueness of minimum variance estimators, efficiency, Minimum mean- square estimation.

Unit – IISufficient statistic , sufficiency and minimum variance. Rao- Blackwell theorem. Distributions possessing sufficient statistics. Sufficiency when range depends on the parameter. Least squares method of estimation and its properties.

Unit – IIIMethods of estimation : Method of moments, Method of minimum chi-square, Method of maximum likelihood estimators and their properties, sufficiency, consistency of ML estimators. Hazurbazar’s theorem, unique consistent ML estimators, efficiency and asymptotic normality of ML estimators.

Unit – IVInterval estimation : Confidence intervals, confidence statements , central and non-central intervals , confidence intervals, Most selective intervals , Fiducial intervals : Fiducial inference in student’s distribution , Problem of two means and its fiducial solution . Exact confidence intervals based on student’s distribution, Approximate confidence- intervals solutions. Ideas of subjective probability, prior and posterior distribution, Bayesian intervals, Discussion of the methods of interval estimation.

References:1. Kendall and Stuart : Advanced Theory of Statistics Vol.-II, Charles Griffin Co .Ltd

London.2. Rohtagi,V.K. : Introduction to probability Theory and Mathematical Statistics (for Numerical and Theoretical Applications), John Wiley and Sons.3. Wald, A: : Sequential Analysis, Dover publications, INC, New York.4. Rao, C.R. : Advanced Statistical Methods in Biometric Research. John Wiley &Sons, INC, New York.


Paper IV Applied Statistics

(ST-104)

Course Objectives:

The course aims to provide the theoretical knowledge about time series and S.Q.C.’s skills for

the applied scientist who needs to monitor and improve the quality of service or industrial

processes. It focuses on concepts and various techniques used in sampling and design in the

context of quality control. It provides the knowledge to the students with the qualitative and

analytical skills necessary to assist in planning, decision making and research within various

institutions. It also help to apply statistical techniques to model relationships between variables

and make predictions.


Explain the concepts of Statistical Quality Control and associated techniques.

Construct appropriate Quality Control Charts and Forecasting models useful in

monitoring a process.

Apply various sampling inspection plans to real world problems for both theoretical and

applied research

Assess the ability of a particular process to meet customer expectations.

Develop an appropriate quality assurance plan to assess the ability of the service to meet

requisite national and international quality standards.

Understand to identify whether a process in statistical control or not.

Understand to estimate Trend, Seasonal and Cyclic components of time series.

Understand past and future behavior of phenomena under study.

Understand how a product quality can be improved and elimination of assignable causes

of variations.


Paper IV Applied Statistics(ST-104) (4 Credits)


Time: 3 hrs.Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same.

Unit- IAnalysis of time Series, Trend measurement ; use of polynomial, logistic, Gompertz and lognormal functions . Moving average method, Spencer's formulae; variate difference method, its use for estimation of variance of the random component. Measurement of seasonal fluctuations, measurement of cyclical movement.

Unit- IIPeriodogram analysis. Concept of stationary time series, correlogram analysis, correlogram of an autoregressive scheme, a moving average scheme and a Harmonic series. Statistical quality control and its purposes; 3 sigma control limits, control charts for variables (mean and range, mean and standard deviation) Control chart for fraction defective, control chart for the number of defects per unit.

Unit- IIINatural tolerance limits and specification limits; Modified control limits. Sampling Inspection Plan : Concepts of Acceptance quality level (A.Q.L) ,Lot tolerance proportion defective ( LTPD)and indifference quality. The single and double sampling plans, and their four curves viz, AOQ, Operating characteristic (OC), Average Sample Numper (ASN) and Average Total Inspection (ATI) curves.

Unit- IVSequential sampling plan and its AOQ, OC, ASN and ATI. The choice of sampling plans by attributes and by variables. Acceptance Sampling plan by variables, single and sequential Sampling Plans, acceptance sampling by variables (known and unknown sigma cases.

References:1. Kendall, M.G. : Time Series,Griffin London2. Gupta, S.C. & Kapoor, V.K. : Fundamentals of Applied Statistics, Sultan Chand and Sons.3. Ekambaram, S.K. : The Statistical Basis of Acceptance Sampling, Asia Publishing House.4. Goon, A.M., Gupta, : Fundamentals of Statistics, Vol. II, ed. VI, M.K. & Dasgupta, B. Word Press Calcutta 1988

5. Cooray, T.M.J.A. : Applied Time Series –Analysis and

Forecasting, Narosa Publishing House6. Hansen, B.L. & Ghare, P.M. : Quality Control and Application, PHI. 1987.7. Montgomery, D.C. : Introduction to Statistical Quality Control, J. Wiley. 1985

11. Gowden, D.J. : Statistical Methods in Quality Control, Prentice Hall.

12. Grant, E.L. : Statistical Quality Control, Wiley Eastern. 13. Duncan, A.C : Quality Control and Industrial Statistics,

Richard O.Irwin, Homewood.IL


Paper-V Practical (Calculator and SPSS/SYSTAT based)(ST-105)

Course Objectives: The main objective of the course is to introduce the statistical concepts and Software of quantitative data analysis used in statistics to students and to provide an understanding of basic statistical methods and ability to use them. During the course the students will train how to use Statistical Package for Social Studies (SPSS) and SYSTAT. Interpret statistical analysis and draw conclusions in context and in the presence of uncertainty. To acknowledge students the use of testing hypotheses for different parameter(s). It also provides practical knowledge about the concepts of Statistical Quality Control and Time Series.


Interpret the results of Statistical Analysis.

Obtain experience in using Statistical Package for Social Studies (SPSS) and SYSTAT.

Test the hypothesis using suitable statistical test.

Understand to identify whether a process in statistical control or not.

Understand to estimate Trend, Seasonal and Cyclic components of time series.

Ability to convert various problems of practical interest into statistical models and make

inferences on it.


Paper-V Practical (Calculator and SPSS/SYSTAT based) (ST-105) (4 Credits)

Max Marks : 75**+25* ** Practical : 60 Class Record : 10 Viva-Vice : 10 * Internal Assessment Time: 4 hrs.

Note: There will be 4 questions, candidates will be required to attempt any 3 questions. List of Practicals:

1. Testes of significance based on t distribution.

(i) Testing the significance of the mean of a random sample from a normal population.(ii) Testing the significance of difference between two sample means,(iii) Testing the significance of an observed correlation coefficient.(iv) Testing the significance of an observed partial correlation coefficient.(v) Testing the significance of an observed regression coefficient.

2. Tests based on F distribution.

(i) Testing the significance of the ratio of two independent estimates of the population variance.

(ii) Testing the homogeneity of means (Analysis of variance).

3. Testing the significance of the difference between two independent correlation coefficients.

4. Testing the significance for

(i) A single proportion(ii) Difference of proportions for large samples.

5. Obtaining confidence or fiducial limits for unknown mean.

6. Testing the significance of the difference between means of two large samples.

7. Testing the significance of difference between standard deviations of two large samples.

8. Estimation of parameters of Distributions by

i. Method of moments.ii. Method of least squares.iii. Method of Modified minimum χ2.iv. Maximum likelihood estimators.

9. Fitting of the

(i) Binomial distribution(ii) Poisson(iii) Normal distribution and their test of goodness of fit using χ2 test.

10. Correlation and regression

(i) Pearson’s coefficient of correlation(ii) Spearman’s rank correlation coefficient (with ties and without ties).(iii) Fitting of the lines of regression.

11. Multiple and partial correlations

(i) Multiple correlation coefficients(ii) Partial correlation coefficients(iii) Fitting of regression plane for three variates

12. Time series and SQC

a. To obtain trends by using(i) Method of Semi-Averages(ii) Method of curve fitting(iii) Method of moving average.(iv) Spencer’s 15 - point and 21 point - formulas.

b. To obtain seasonal variation indices by using(i) Ratio to trend method.(ii) Ratio to moving average method.(iii) Link relative method.

c. To construct __

(i) X and R–chart (ii) p–chart (iii) c–chart and comment on the State of control of the process.

M.Sc. Statistics Semester - IIPaper –I Demography (ST-201)

Course Objectives:

The course describes current population trends, in terms of fertility, mortality and population

growth. It describes the structure and composition of populations. Study the kind of events and

phenomena that affects the size and composition of population.

Develop different measures used to keep track of the phenomena that affect populations.


Understand the basic concept of demographic transition.

Understand the consequences of current population trends for future well being.

Understand how different factor affect mortality, fertility and migration.

M.Sc. Statistics Semester - IIPaper –I Demography(ST-201) (4 Credits) Max Marks: 75+25*

*Internal Assessment Time: 3 hrs.Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same.

Unit –IMethods of obtaining demographic data, Rates and ratios, measurement of population at a given time , measurement of mortality : crude death rate , specific rates ,infant mortality rate , perinatal mortality rate , standard death rates . Graduation of mortality rates: Makehams and Gompertz graduation formula, Life table : Construction of a complete life table and its uses.

Unit –IIAbridged life tables: Kings method, Reed and Merrell’s method, Greville’s method, Keyfitz and Frauenhal’s method and Chiang’s method . Measurement of fertility: Crude birth rate , general fertility rate , age specific fertility rate , total fertility rate , gross reproduction rate and net reproduction rate .

Unit –IIIMeasurement of Migration: Crude migration rate, age specific migration rate, internal migration and its measurement, Structure of populations: Stable and quasi-stable population, Fundamental equation of stable population theory , intrinsic rate of growth , intrinsic birth and death rates , intrinsic age distribution , Leslie’s model of population growth .

Unit –IVPopulation Projection: Projected values & estimates, Inter-censal and post-censal estimates, population composition: methods of Projection, survival rates : UN model life table, model life tables of Coale and Demeny, Ledermann’s model life tables, Brass model.

References:1. Ramakumar, R. : Technical Demography, Wiley, Eastern Limited.2. Gupta, S.C. & Kapoor, V.K. : Fundamental of applied Statistics, 1990. Sultan Chand and Sons, Ch. 9 only.3. Benjamin,B. (1969) : Demographic Analysis, George, Allen and Unwin.4. Cox, P.R. (1970). : Demography, Cambridge University Press.5. Keyfitz, N (1977). : Applied Mathematical Demography; Springer Verlag.6. Spiegelman, M. (1969). : Introduction to Demographic Analysis; Harvard University

M.Sc. Statistics Semester-II

Paper-II Operations Research

(ST-202)

Course Objectives:

Operations Research aims to introduce students to use quantitative methods and techniques for

effective decisions–making; model formulation and applications that are used in solving business

decision problems.


Identify and develop operational research models from the verbal description of the real

system.

Understand the characteristics of different types of decision-making environments and

the appropriate decision making approaches and tools to be used in each type.

Understand the mathematical tools that are needed to solve optimization problems.

Build and solve Transportation Models and Assignment Models.

M.Sc. Statistics Semester-II

Paper-II Operations Research (ST-202) (4 Credits) Max Marks: 75+25* *Internal Assessment

Time: 3 hrs.


Unit –I

Linear Programming : Basic concepts convex sets, Linear Programming problem (LPP), Examples of LPP, Hyperplane, Open and Closed half spaces. Feasible, basic feasible and optimal solutions, Extreme points and graphical method. Simplex method. Duality in linear programming

Unit –II

Transportation and Assignment problems. (Computational Techniques only). Decision Theory : Algorithm for decision based problems, Types of decision making, Decision making under uncertainty : Criterion of optimism , Criterion of pessimism and Hurwicz criterion . Decision making under risks: EVM, EOL and decision tree techniques.

Unit-III

Game Theory : Terminology , two person zero sun game; game of pure strategy , game of mixed strategy , reducing game by dominance, linear programming method. Replacement models: replacement of items whose efficiency deteriorates with time and(i) The value of the money remains same during the period (ii) The value of the money also changes with time. Criterion of present value for comparing replacement alternatives.

Unit – IV

Inventory models: Deterministic inventory models ( D.I.M ) with no shortages: Basic EOQ model , EOQ with several runs of unequal lengths , EOQ with finite replenishment D.I.M. with shortages : E O Q with instantaneous production and variable (fixed) order cycle time., EOQ with finite production EOQ with price breaks : E.O.Q with one price break. Simple Multi-item deterministic inventory model, model with limitation on space, model with limitation on investment. Queueing models : Introduction of queueing models, steady state solution of M/M/1 , M/M/1/N , M/M/C and M/M/C/N and their measures of effectiveness. CPM (Critical path method) to solve the network problems and PERT.

References:1. Hadley, G. : Linear Programming, Narosa Publications House.2. Churchman, C.W.. : Introduction to Operations Research John Wiley& Sons New York.3. Goel, B.S. & Mittal, S.K.: Operations Research, Pragater Prekshlen, John & Sons.4. Gross, D. & : Fundamentals of Queuing Theory Wiley. Harris, C.M.5. Allen, A.O : Probability Statistics & Queuing Theory With Computer Science Applications (Academic Press) INC. Elsevier Direct.

M.Sc. Statistics Semester – II

Paper – III Inference -II (ST-203)

Course Objectives:

This course deals with concepts and techniques to arrive at decisions in certain situations where there is lack of certainty on the basis of a sample whose size is fixed or regarded as a random variable. Interpret statistical analysis and draw conclusions in context and in the presence of uncertainty. Define how to develop Null and Alternative Hypotheses and how Type I Error and Type II Error relate to a hypothesis test. To acknowledge students the use of testing hypotheses for different parameter(s). Similar to estimation, the process of hypothesis testing is based on probability theory and the Central Limit Theorem. Nonparametric test are appropriate when sample size is small and distribution of the outcome is not known. Nonparametric tests deal with concepts of Empirical distribution function and order statistics .


Conceptually map the theoretical basis of tests of simple and composite hypotheses.

Define null and research hypothesis, test statistic, level of significance and decision rule.

Distinguish between Type I and Type II errors and discuss the implications of each.

Explain the difference between one-tailed and two-tailed statistical tests.

Know the concepts of critical value, critical region, and region of rejection.

Understand and apply the basic principles of nonparametric tests (distribution free tests).

Know the procedure based on type of outcome variable and number of samples.

Compare and contrast parametric and nonparametric tests.

Identify multiple applications where nonparametric approaches are appropriate.

Perform and interpret various nonparametric tests.

Compare and contrast the different nonparametric tests.

Identify the appropriate nonparametric hypothesis testing.


Paper – III Inference -II (ST-203) (4 Credits)


Time: 3 hrs.Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same.

Unit – IElements of Statistical decision theory. Neyman - Pearson lemma (with emphasis on the motivation of theory of testing of hypothesis) BCR and sufficient statistics. Testing a simple hypothesis against a class of alternatives. Uniformly most powerful test and sufficient statistics, power function. One and two sided tests.

Unit –IIComposite hypotheses, An optimum property of sufficient statistics. Similar regions, Elementary ideas of complete statistics, Completeness of sufficient statistics. . Likelihood ratio test and its applications, asymptotic distribution of LR statistic and asymptotic power of LR tests.Sequential Analysis. Concept of ASN and OC functions. Wald's sequential probability ratio test and its OC and ASN functions.

Unit –IIINon - parametric tests and their applications: Empirical distribution function and its properties(without Proof), Test of randomness( Test based on the total number of runs). One-sample and paired-sample techniques: The Ordinary Sign test and Wilcoxon Signed-rank test. Tests of Goodness of Fit: Chi-square Goodness of Fit, Kolmogrov- Smirnov tests, Independence in Bivariate sample: Kendall’s Tau coefficient and Spearman’s rank correlation.

Unit –IVGeneralized two-sample problem: The Wald-Wolfowitz Runs test, Kolmogrov-Smirnov two sample Test, Median Test, Mann-Whitney U Test, Linear Ranked tests for the Location and Scale problem: Wilcoxon Test, Mood Test, Siegel-Tukey Test, Klotz Normal-scores Test, Sukhatme Test.

References:1. Kendall and Stuart : Advanced Theory of Statistics Vol.-II, Charles Griffin & Co. Ltd, London2. Rohtagi,V.K. : Introduction to probability Theory and Mathematical Statistics(for Numerical and Theoretical Applications).3. Wald, A : Sequential Analysis Dover Publications, INC. New York.4. Gibbons, Jean Dickinson : Nonparametric Statistical Inference ( For Unit - IV only ). McGraw – Hill Book Co. New York.5. Rao, C.R. : Advanced Statistical Methods in Biometric Research, John Wiley& Sons, INC, New York.


Paper –IV COMPUTER FUNDAMENTALS AND PROBLEM SOLVING USING C(ST-204)

Course Objectives: The aim of this course is to provide adequate knowledge of fundamentals of computer along with problem solving techniques using C programming. This course provides the knowledge of writing modular, efficient and readable C programs. Students also learn the utilization of arrays, structures, functions, pointers and implement these concepts in memory management. This course also teaches the use of functions and file systems.Learning Outcomes: On completion of this course students will be able to:

Identify and understand the working of key components of a computer system.

Understand computing environment, how computers work and the strengths and

limitations of computers.

Identify and understand the various kinds of input-output devices and different types of

storage media commonly associated with a computer.

Write programs related to mathematical and logical problems in ‘C’.

Uunderstand data structures, use of pointers, memory allocation and data handling

through files in ‘C’.


Paper –IV COMPUTER FUNDAMENTALS AND PROBLEM SOLVING USING C(ST-204)

(4 credits) Max Marks: 75+25*

*Internal Assessment Time: 3 hrs.


UNIT – I

Computer Fundamentals: Definition, Block Diagram along with Computer components, characteristics & classification of computers, hardware & software, types of software. Techniques of Problem Solving: Flowcharting, decision table, algorithms, Structured programming concepts.

UNIT – II

Overview of C: History of C, Importance of C, Structure of a C Program. Elements of C: Character set, identifiers and keywords, Data types, Constants and Variables. Operators: Arithmetic, relational, logical, bitwise, unary, assignment and conditional operators and their hierarchy & associativity. Input/output: Unformatted & formatted I/O function in C.

UNIT – III

Control statements: Sequencing, Selection: if and switch statement; alternation, Repetition: for, while, and do-while loop; break, continue, goto statement. Functions: Definition, prototype, passing parameters, recursion. Storage classes in C: auto, extern, register and static storage class, their scope, storage and lifetime. Arrays: Definition, types, initialization, processing an array, passing arrays to functions.

UNIT – IV

Pointers: Declaration, operations on pointers, use of pointers. String handling functions Structure & Union: Definition, processing, Structure and pointers, passing structures to functions. Data files: Opening and closing a file, I/O operations on files, Error handling during I/O operation.

-References1. Sinha, P.K. & Sinha, Priti, Computer Fundamentals, BPB

2. Dromey, R.G., How to Solve it By Computer, PHI3. Gottfried, Byron S., Programming with C, Tata McGraw Hill4. Balagurusamy, E., Programming in ANSI C, McGraw-Hill5. Jeri R. Hanly & Elliot P. Koffman, Problem Solving and Program Design in C, Addison Wesley.6. Yashwant Kanetker, Let us C, BPB7. Norton, Peter, Introduction to Computer, McGraw-Hill8. Leon, Alexis & Leon, Mathews, Introduction to Computers, Leon Tech World


Paper-V Practical (Computer based)(ST-205)

Course Objectives: The objective of this course is to provide an understanding for the student on statistical concepts to include measurements of location, dispersion, probability distributions , regression, correlation analysis and testing the significance of small sample size with the help of C Programming. The main objective of the course is to provide an understanding of C programming for calculation of statistical data.


Calculate measures of location, dispersion, regression, correlation analysis and testing the significance of small sample with the help of C Programming.

Compare different sets of data using C Programming.

Able to fit probability distributions using C Programming.


Paper-V Practical (Computer based)(ST-205) (4 Credits)

Max Marks : 75**+25* ** Practical : 60

Class Record : 10 Viva-Vice : 10

* Internal Assessment Time: 4 hrs.

Note: There will be 4 questions, candidates will be required to attempt any 3 questions.

List of Practicals based on C

1. Finding the mean and standard deviation for discrete and continuous data.

2. Computation of Moments, Skewness and Kurtosis of given data.

3. Computation of Karl Pearson’s, partial & multiple correlation coefficient and Spearman’s rank correlation coefficient.

4. Curve fitting, fitting of lines of regression.

5. Fitting of distribution: Binomial, Poisson and Normal.

6. Testing the significance of the mean of a random sample from a normal population.

7. Testing the significance of difference between two sample means,

8. Testing the significance of an observed correlation coefficient.

9. Testing the significance of an observed partial correlation coefficient.

10. Testing the significance of an observed regression coefficient.

11. Testing the significance of the ratio of two independent estimates of the population variance.

M.Sc. Statistics Semester – III

Paper –I Sampling Theory(ST-301) Course Objectives: The main objectives of the sampling theory are:

(1) Sampling Theory emphasize to understand the advanced techniques of sample surveys and

related issues which would be beneficial for the students to their further research

(2) To obtain the optimum results, i.e., the maximum information about the characteristics of the

population with the available sources at our disposal in terms of time, money and manpower

by studying the sample values only.

(3) To obtain the best possible estimates of the population parameters.


Understand how data to be collected for related study

Understand different methods of sample selection and analyzing data.

Understand to find more efficient results on the basis of sample selection.

Compare the results obtained under different sampling designs.


Paper –I Sampling Theory (ST-301) (4 Credits) Max Marks: 75+25*


Note:There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same.

Unit -IBasic finite population sampling techniques: Simple random sampling with replacement, Simple random sampling without replacement, Stratified sampling, Systematic sampling and related results on estimation of population mean/total, Relative precision of different sampling techniques, Allocation problem in stratified sampling.

Unit –IIUse of supplementary information: Ratio estimation, bias and mean square error, estimation of variance, comparison with SRS, ratio estimator in stratified sampling, unbiased ratio-type estimators, regression and difference estimators, comparison of regression estimator with SRS and ratio estimator.

Unit –IIISampling and sub-sampling of clusters. Equal size cluster sampling. Estimation of proportions, estimation of efficiency of clustering, two stage sampling, Multistage sampling, Stratified multi stage sampling. Double Sampling: for unbiased ratio estimation, for regression estimation, for stratification.

Unit –IVRepetitive surveys: Sampling over two occasions, probability proportionate to sampling (PPS) with replacement and without replacement methods [Cumulative total and Lahiri’s method] and related estimators of a finite population mean[ Horvitz Thompson estimator, Yates and Grundy estimator, Desraj estimators for a general sample size and Murthy’s estimator for a sample of size two].

References:1. Chaudhuri A and Mukerjee R.(1988) : Randomized Response, Theory and Techniques, New York : Marcel; Dekker Inc.2. Cochran W.G. : Sampling Techniques (3rd Edition, 1977), Wiley.3. Des Raj and Chandak ( 1998) : Sampling Theory, Narosa Publications House.4. Murthy.M.N (1977) : Sampling Theory & Statistical Methods Publishing Society, Calcutta.5. Sukhatme et atl (1984). : Sample Theory of Surveys with Applications. lowa State University Press & IARS.


Paper –II OBJECT-ORIENTED PROGRAMMING WITH C++(ST-302)

Course objective: Understand fundamentals of object oriented programming in C++. To help students understand fundamentals of programming such as variables, conditional and iterative execution, methods, etc.Learning Outcomes: On completion of this course students will be able to:

Understand algorithmic thinking and apply it to programming.

Write program in C++ language.

Understand problem-solving techniques.

Code with C++ arithmetic, increment, decrement, assignment, relational, equality and

logical operators.

Code C++ control structures (if, if/else, switch, while, do/while, for) and use built-in data

types.

Use standard library functions.

Write user-defined function definitions.

Understand and manipulate arrays.


Paper –II OBJECT-ORIENTED PROGRAMMING WITH C++ (ST-302) (4 Credits)


Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same.Unit-I

Object-Oriented Programming: Basic Concepts of OOP, Principles of OOP, Benefits of OOP, Applications of OOP, Structure of C++ Program, Identifier and key-words, Data types, variables, Operators and expressions in C++, Input and output operations, Control statements: If statement, if else statements, Nested if statements and switch statement. Loops: Do loop, while loop, Do-while loop and FOR loop. Functions: Prototypes, reference arguments, Inline functions. Arrays : one dimensional, multidimensional arrays, Structures.

Unit –IIClasses and Objects: Classes, member function, inline member function, objects, access specifiers: public, private and protected, Static class members, friend function, Constructor and destructor: Constructor, parameterized constructor, copy constructor, destructor. Operator overloading: Function overloading, operator overloading, type conversions.

Unit – IIIInheritance in C++: types of inheritance, container classes, virtual base class, abstract class, constructor in derived classes. Pointers, Virtual functions and polymorphism: polymorphism, early binding, polymorphism with pointers, virtual functions, late binding, pure virtual functions.

Unit-IVStreams and File processing: Streams classes, opening and closing a file, file modes, file pointers and their manipulation, sequential files, random files. Errors handling, Exception handling features.

References

1. Bjarne Stroustrup, The C++ Programming Language, Pearson2. Herbert Scildt, C++, The Complete Reference, Tata McGraw-Hill3. Robert Lafore, Object Oriented Programming in C++,4. Lippman, C++ Primer, 3/e, Addison-Wesley5. Balaguruswami, E., Object Oriented Programming In C++, Tata McGraw-Hill


Paper –III & IV Opt. (i) Theory of Queues

(ST-303 & ST-304)

Course Objectives:

The objective of this course is to introduce the concept of probabilistic tools and concepts which

are useful in modeling, such as Markov models and queueing theory. In last two sections of the

course, the students also study the various advanced queueing models with their practical utility

in communication systems, computer networks, traffic control and other related fields. The

queuing models are used to find out the optimum service rate and the number of servers so that

the average cost of being in queuing system and the cost of service are minimized.


Acquire skills in handling situations involving more than one random variables and

functions of random variables.

Understand to analyze the performance of computer systems and queues by applying

basic concept of probability techniques and models.

Learn how to analyze a network of queues with Poisson external arrivals, exponential

service requirements and independent routing.


Paper –III & IV Opt. (i) Theory Of Queues (ST-303 & ST-304) (4 Credits)


Time: 3 hrs.

Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same. Unit-I

Queueing system. Components of a queueing system, measures of effectiveness, notations, exponential distribution and its various properties, stochastic processes, definition and examples, Poisson process and its some important properties related to queues. Markov chains and its properties (without proof). Concepts of steady state and transient state, K-Erlang distribution. Birth and death process.

Unit-IIM/M/1 queueing system steady state and time dependent solutions. measures of effectiveness, busy period distribution, 'waiting time distribution, Little’s formula. State probability generating function for M/M/1/N queueing system and its steady state probabilities measures of effectiveness, Time dependent solutions of M/M/ queuing system and M/M/ queueing system with time dependent input parameter, measures of effectiveness.

Unit-IIIM/M/1 queueing system with phase type service, busy period time distribution, waiting time distribution, Mu1tiple channel queueing system with Poisson input and constant service time (M/D/C), Measures of effectiveness. Erlang service model M/Ek/1, Erlang arrival model Ek/M/1.

Unit-IVDeparture point steady state system size probabilities for M/G/1 queueing system, special cases M/Ek/1 and M/D/1 Pollaczek-Khintchine formula, waiting time, busy period analysis. Arrival point steady state system size probabilities for GI/ M/1 queueing system. Machine interference Model

References:1. Gross, D. & : Fundamental. of queuing theory, John Wiley and Son.. Harris C.M2. Saaty, T.L. : Elements of queueing theory with app lications. McGraw Hill Book Company Inc.3. Allen, A.O. : Probability, Statistics and Queuing Theory with Computer Science Applications, Academic Press4. Kashyap, B.R.K & : An Introduction to Queueing Theory, AARKAY Publications, Chaudhary, M.L. Calcutts.


Paper –III & IV Opt. (ii) Linear Programming

(ST-303 & ST-304)

Course Objectives:

The course is designed to expose the students to the concepts of optimization. The course

emphasizes the solution of linear programming problems by using powerful techniques of linear

programming at a not too abstract level , and their applications to both practical and theoretical

problems.

Learning Outcomes:

The students are expected to acquire the following skills in this course:

Develop a fundamental understanding of linear programming models,

Able to develop a linear programming model from problem description,

Apply the Simplex method for solving linear programming problems,

Apply the revised simplex method to solve linear programming problems,

Apply different methods to resolve degeneracy problem in linear programming problems,

Express the dual of a linear programming problem and solve the resulting dual problem

using the dual simplex method, interpret the results and obtain solution to the primal

problem from the solution of the dual problem,

Apply the stepping stone algorithm to solve transportation problems,


Paper –III & IV Opt. (ii) Linear Programming (ST-303 & ST-304) (4 Credits)


Time: 3 hrs.Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same.Unit-I

Introduction : Theory of Simplex method; Slack and surplus variables, Reduction of any feasible solution to Basic feasible solution, Improving a BFS, Unbounded solutions, Optimal conditions, Computational aspects of the Simplex method :Selection of the vector to enter the basis , Degeneracy and breaking the ties, Transformation formulas, Artificial Variables, Inconsistency and Redundancy, Tableau format. The two phase method for artificial variables, Unrestricted variables.

Unit-IIResolution of degeneracy problem: Charne's perturbation method, use of perturbation technique and Simplex tableau format. The generalized linear programming problems, Generalized Simplex method. The revised Simplex method: Standard from I and II. Computational procedure for standard from I and II. Comparison of Simplex and revised Simplex method.

Unit-IIIDuality theory : Dual linear programming problem, Fundamental properties of dual problems, complementary Slackness, Unbounded solution in the primal , Dual algorithm , Alternative derivation of dual Simplex algorithm, Initial solution for dual Simplex algorithm, a primal dual algorithm.

Unit –IVTransportation problem, Simplex method and transportation problem, transportation problem tableau, the stepping stone algorithm , Degeneracy and the transportation tableau, Determination of an initial basic feasible solution, Generalised transportation problems and their solutions. Applications of linear programming to industrial problems. Production allocation and transportation, Machine assignment problem , Regular time and over time production , optimal product mix and activity levels, Petroleum - refinery operations, Blending problems.

References:1. Hadley, G. : Linear programming, Narosa Publications House.2. Vejda, S. : Mathematical Programming, Dover Publications.3. Saul E.Gauss. : Linear programming Methods and Applications, Dover Publications.4. Kambo, N. S. : Mathematical Programming Techniques, East –West Press Pvt. Ltd.5. Mittal, K.V. : Optimization Methods, New Age International (P) Ltd.


Paper –III & IV Opt. (iii) Stochastic Processes (ST-303 & ST-304)

Course Objectives:

The course aims at:

Developing an awareness of the use of stochastic processes to build adequate mathematical

models for random phenomena evolving in time. Acquainting students with notions of long-time

behavior including transience, recurrence, and equilibrium to answer basic questions in several

applied situations including branching processes and random walk. Introducing students to basic

concepts, techniques and results associated primarily with the elementary theory of Markov

processes.


Construct transition matrices in Markov chains and calculate various types of transition

probabilities.

Classify states and Markov chains according to their long term behavior.

Use Poisson processes for modeling various phenomena.

Derive the probabilities for the birth death process and Polya processes.


Paper –III & IV Opt. (iii) Stochastic Processes (ST-303 & ST-304) (4 Credits) Max Marks: 75+25*


Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same..Unit-I

Introduction to Stochastic processes, Classification of Stochastic processes according to state, space and time domain. Generating function, Convolutions, Compound distribution, Partial fraction expansion of generating functions.

Unit-IIRecurrent events, recurrence time distribution: necessary and sufficient condition for persistent and transient recurrent events & its illustrations and Notion of delayed recurrent event. Random walk models : absorbing, reflecting and elastic barriers, Gambler's ruin problem, probability distribution of ruin at nth trial.

Unit-IIIMarkov chains: transition probabilities, classification of states and chains, evaluation of the nth power of its transition probability matrix. Discrete branching processes, chance of extinction, means and variance of the nth generation.

Unit-IVNotions of Markov processes in continuous time and Chapman-Kolmogorov equations. The Poisson process: The simple birth process, the simple death processes. The simple birth and death process: The effect of immigration on birth and death process. The Polya Processes: Simple non-homogeneous birth and death processes.

References:1. Bailey, N.T : The Elements of Stochastic Processes.(1964 Ed.)2. Medhi , J. : Stochastic Processes, New Age International (P) Limited3. Karlin , S. : Introduction to Stochastic Processing, Vol. I, Academic Press.4. Karlin, S. : Introduction to Stochastic Modeling, Academic Press.5. Basu, A.K. : Introduction to Stochastic Process, Narosa Publishing House.


Paper –III & IV Opt. (iv) Bio-Statistics(ST-303 & ST-304)

Course Objectives:

Biostatistics is one area of Applied Statistics that concerns itself with the application of statistical

methods to medical, biological and health related problems. This course is designed to teach

public health students the basic principles of biostatistics.


Able to discuss and explain what biostatistics is and how it is used in the field of public

health.

Able to understand the basic principles of probability and how they relate to biostatistics.

Apply descriptive techniques commonly used to summarize public health data..

Apply basic informatics techniques with vital statistics and public health records in the

description of public health characteristics and in public health research and evaluation.

Interpret results of statistical analyses found in public health studies.


Paper –III & IV Opt. (iv) Bio-Statistics(ST-303 & ST-304) (4 Credits)


Time: 3 hrs.Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same..Unit-I

Bioassays : Quantitative and quantal response, dose response relation. estimation of median effective dose, estimation of unknown concentration or potency, probit and logit transformations, Parallel line and slope ratio assays , potency, ratio, Feller's theorem. Tests for non-validity, symmetric and asymmetric : assays, Toxic action of mixtures.

Unit-IITypes of mating: Random mating, Hardy-Weinberg equilibrium, Random mating in finite population. Inbreeding (Generation Matrix Approach) Segregation and linkage. Estimation of segregation and linkage parameters

Unit-IIIConcept of gen frequencies. Estimation of gene frequencies Quantitative inheritance, Genetic parameters heritability, genetic correlation and repeatability methods of estimation. Selection and its effect, Selection Index, dialled and partially dialled Crosses.

Unit-IVGenotype environment interactions. Components of variance and Genotypic variance, Components of Covariance, Correlations between relatives, Genetic parameters; Heritability, Repeatability

References:1. Kempthorne, O : An Introduction to Genetical Statistics, Wiley Eastern2 Jain, l.R. (1982) : Statistical techniques in quantitative genetics. Tata-McGraw Hill3. Poti, S.J. (1983) : Quantitative study in life sciences, Vikas Publishing Ltd.4. Prem Narain; Bhatia : Handbook of Statistical Genetics, V.K. and Malhotra, I.A.S.R.I.P.K. (1979)5. Daniell, W.W : Bio Statistics – A foundation for analysis in health sciences , 3rd ed. John wiley6. Falconer, D.S. : Introduction to quantitative Genetics (Longman Group Ltd.


Paper –III & IV Opt. (v) Statistical Methods in Epidemiology(ST-303 & ST-304)

Course Objectives:

This is a course on bio statistical methods in the design and analysis of epidemiological studies.

Epidemiology is the study of the distribution and determinants of human disease and health

outcomes, and the application of methods to improve human health. Epidemiological studies are

typically concerned about the health of populations, while clinical medicine is concerned with

the health of individual persons. The course develops basic statistical inference for risk measures

according to the nature of the outcome variables (binary and ordinal, continuous, rate, time-to-

event).


Understand epidemiologic hypotheses, concepts and measures

Describe epidemiologic study designs.

Understand the strengths and limitations of various epidemiologic study designs.

Design an epidemiological study.

Identify sources of bias, confounding and effect modification in epidemiological studies.

Analyze epidemiologic data using multivariable methods.


Paper –III & IV Opt. (v) Statistical Methods in Epidemiology (ST-303 & ST-304) (4 Credits) Max Marks: 75+25*



Measures of disease frequency: Mortality/Morbidity rates, incidence, rates, prevalence rates. Sources of mortality/Morbidity statistics-hospital records, vital statistics records. Measures of accuracy or validity, sensitivity index, specificity index.

Unit-IIEpidemiologic concepts of diseases, Factors which determine the occurrence of diseases, models of transmission of infection, incubation period, disease spectrum and herd immunity. Observational studies in Epidemiology: Retrospective and prospective studies. Measures of association :Relative risk, odds ratio, attributable risk.

Unit-IIIStatistic techniques used in analysis: Cornfield and Garts’ method, Mantel.Haenszel method. Analysis of data from matched samples, logistic regression approach. Experimental Epidemiology: clinical and community trials. Statistical Techniques: Methods for comparison of the two treatments. Crossover design with Garts and Mcnemars test. Randomization in a clinical trial, sequential methods in clinical trials. Clinical life tables,

Unit-IVAssessment of survivability in clinical trials. Mathematical Modelling in Epidemiology: simple epidemic model, Generalized epidemic models, Reed First and Green wood models, models for carrier borne and host vector diseases.

References:1. Lilienfeld and LiJenfeld : Foundations of Epidemiology, Oxford University Press.2. Lanchaster, H.O. : An Introduction to Medical Statistics, John Wiley & Sons Inc.3. FIeiss, J.L. : Statistical Methods for Rates and Proportions, Wiley Inter Science.4. Armitage : Sequential Medical Trials, Second Edition,Wiley Blackwell.5. Bailey, N.T.J. : The mathematical theory of infectious disease and Applications, Griffin.


Paper –III & IV Opt. (vi) Statistical Ecology(ST-303 & ST-304)

Course Objectives:

The objective of this course is to study Students how to design, plan, carry out, analyze and

present ecological field and lab research projects. The emphasis is on the formulation of a

challenging and testable hypothesis, the experimental design, and the appropriate choice and

application of statistical techniques for ecological data analysis. Students will carry out their own

field research project, combining experimental and correlative studies, based on a prior written

research proposal. These research projects are hypothesis-driven, and serve as a good preparation

for an M.Sc. thesis projects.

Learning Outcomes: After successful completion of this course students will be able to:

Formulate ecological hypotheses;

Judge the variation, replication and sampling in experiments

Design field and lab experiments for testing hypotheses;

Assess the measurements that best test the hypotheses;

Create a research proposal;

Apply statistics properly for a given hypothesis and data set;

Summarize the ecological consequences of experimental results.


Paper –III & IV Opt. (vi) Statistical Ecology(ST-303 & ST-304) (4 Credits)


Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set

from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same.Unit-I

Population Dynamics: Single species -exponential logistic and Gompertz models, two species competition and competitive exclusion, Predator-pray interaction, Lotka- Volteria equations.

Unit-IIEstimation of Abundance: Capture-recapture method, Line transect methods, nearest neighbour and nearest individual distance methods.

Unit-IIIAnalysis of bird ring recovery data, open and closed populations. Survivorship Models: Discrete case-life table, Leslie matrix. Continuous case survivorship curve, hazard rate, life distribution with monotone and non-monotone hazard rates.

Unit-IVEcological community: Species abundance curve, broken stick model. Diversity and its measures. Renewable Resources': Maximum sustainable yield, maximum economic yield, optimal harvesting strategy.

References:1. Begin M, and Mortiner, M. : Population Ecology, Blackwell Science.2. Clark, C.W : Bioeconomic Modelling and Fisheries Management3. Hallan, T.G. and Levin, S.A. : Mathematical Ecology, Springer4. Kapur, J.N : Mathematical Models in Biology and Medicine, Affiliated East-West Press5. Pielou, E.C : Mathematical Ecology, John Wiley & Sons Inc.6. Clark, C.W. : Mathematical Bioeconomics--the Optimal Management of Renewable Resources, Wiley-Inter Science.7. Seber, G.A.F. : The Estimation of Animal,Abundance; The Blackburn Press.


Paper-V Practical (Computer based)

(ST-305)

Course Objectives: The objective of this course is to provide Knowledge to student about sampling theory with the help of C++ Programming. The main objective of the course is to provide an understanding of C++ programming for calculation and comparison of statistical data.


Understand different methods of sample selection and analyzing data using C++.

Estimate of population total, population mean and variance of estimator for different

sampling designs with the help of C++ Programming.


Paper-V Practical (Computer based)(ST-305) (4 Credits)

Max Marks : 75**+25* **Practical : 60

Class Record: 10 Viva-Vice : 10 * Internal Assessment

Time: 4 hrs.Note: There will be 4 questions, candidates will be required to attempt any 3 questions. List of Practicals based on C and C++:

1. Estimation of population mean, total, confidence limits and variance of estimator under simple random sampling.2. Estimation of population total, population mean and variance of estimator under stratified random sampling.3. Calculation of optimum, proportional allocation.4. Comparison of stratified sampling with different types of allocation with unstratified simple random sampling.5. Comparison of systematic sampling with simple random and stratified random sampling.6. Estimation of proportions and percentages, Variance of the estimators.7. Estimation of variance in double sampling for stratification.8. Estimation of gain in precision due to stratification from the results of stratified sample.9. Ratio estimator for mean and total of population, variance of the estimators. 10. Comparison of ratio estimator with mean per unit estimator under simple random sampling.11. Comparison of separate and combined ratio estimators under stratified random sampling with mean per unit estimate or under random sampling.12. Comparison of different types of allocation for ratio estimator under stratified random sampling.13. Comparison of two ratios.14. Comparison of regression, ratio and mean per unit estimates from a simple random sample.

M.Sc. Statistics Semester – IV

Paper –I Multivariate Analysis (ST-401)

Course Objectives:

The objective of the course is to introduce several useful multivariate techniques, making strong

use of illustrative examples. The course focuses on extensions of univariate techniques to

multivariate framework, such as multivariate normal distribution, hypothesis testing and

simultaneous confidence intervals. The course will also cover the techniques unique to the

multivariate setting such as principal component analysis, factor analysis, discrimination and

classification. The course introduces statistical tools used to analyze multivariate data and how

these tools can be applied to real life problems.


Understand the concept of Multivariate analysis and its usefulness.

Understand data requirements for Multivariate analysis.

Perform exploratory analysis of multivariate data, such as plot multivariate data,

calculating descriptive statistics, testing for multivariate normality;

Conduct statistical inference about multivariate means including hypothesis testing and

different types of confidence intervals estimation;

Undertake statistical analyses using appropriate multivariate techniques, which include

principal component, factor analysis and discriminant;


Paper –I Multivariate Analysis(ST-401) (4 Credits)


Time: 3 hrs.


Notion of multivariate distribution, multivariate normal distribution of linear combination of normal variates, Marginal and Conditional distributions, Multiple and partial correlation coefficients. Characteristic function of a random vector, characteristic function when the random vector is normally distributed. Moments and semi-invariants of multivariate normal distribution. Estimation of the mean vector and covariance matrix, maximum likelihood estimator of the parameters of multivariate normal distribution.

Unit-IIThe distribution of the sample mean vector and sample dispersion matrix. Sample correlation coefficient, maximum likelihood estimators of total, partial and multiple correlation coefficients; sampling distribution of simple, partial and multipal correlation coefficients when the corresponding population correlation coefficients are zero. Testing hypotheses of significance of these distributions.

Unit-IIIHotteling’s T2 and Mahalanobis D2-Statistic; Justification , distribution and uses . The multivariate Behren’s Fisher Problem and its solution. Classification Problem : Standards of good classification, Baye’s and minimax regions for classification into one of two known multivariate normal populations when the parameters are known and unknown. Fisher’s linear discriminator, Anderson’s discriminator.

Unit-IVWishart Distribution : Definition, Character function and properties. Sample generalized variance, asymptotic distribution of sample generalized variances. Principal components in the population, Canonical correlation in the population.

References:1. Anderson, T.W.(1983), : An Introduction to Multivariate Statistical analysis, Second Edition John Wiley.2. Narayan, C. Giri : Multivariate Statistical analysis, Marcel Dekker.3. Srivastava, M.S.& : An introduction to Multivariate Statistics, North Khatri C.G.(1979), Holland.4. Kshirsagar, A.M.(1972) : Multivariate Analysis, Marcell-Dekher5. Johnson, R.A : Applied Multivariate Statistical Analysis,PHI Learning & Wichern,D.W6. Bhuyan, K.C : Multivariate Analysis and its applications, New Central Book Agency(P) Ltd.


Paper – II Linear Estimation & Design of Experiments(ST-402)

Course objective:

This course provides an overview about linear models and design of experiments. To provide

orientation of statistics while designing statistical experiments. Exposure to various statistical

designs leading to the analysis of variance, eliminating heterogeneity of the data and construction

of designs will be provided.

Learning Outcomes: On successful completion of this course, students will be able to

Select and apply appropriate regression techniques to address research questions and

hypotheses;

Apply experimental design techniques in real problems.

Argue the necessity of experimental design to the task of collecting valid and relevant

data in order to draw the correct statistical evidence to support a hypothesis.

Understand to design and conduct experiments, as well as analyze and interpret data.

Understand to check the affects of different factors under study.

Understand different techniques for reducing the experimental error.

Understand to design a system, component or process to meet desired needs

Understand the effects of independence or dependence of different factor under study.


Paper – II Linear Estimation & Design of Experiments (ST-402) (4 Credits)


Time: 3 hrs.

Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same

Unit –I

Linear estimation : Least Square estimates of regression coefficients, Standard Gauss- Markov models, estimability of parameters, best linear unbiased estimators (BLUE), Method of least squares and Gauss Markov theorem; Variance-Covariance matrix of BLUES, Distributional Properties. Tests of General Linear hypothesis.

Unit –II

One-way and two way classifications: ANOVA for Fixed, random and mixed effects Models (One observation per cell). Terminology in experimental designs. Basic principles of design of experiments, General block design and its information matrix, balance and orthogonality, Layout and analysis of completely randomised, randomised blocks and Latin-square designs.

Unit – III

Factorial experiments: 22-experiment, 23-experiment and 2n-experiment in 2k blocks per replicate. Confounding in Factorial Experiments: Complete confounding for 22-experiment and 23-experiment, Partial confounding for 22-experiment and 23-experiment., Advantages and Disadvantages of Confounding. Split-plot design.

Unit – IV

Incomplete Block Design , Balanced incomplete block design, parameters relationship of Balanced incomplete block design, Symmetric Balanced incomplete block design, construction of Balanced incomplete block design by developing initial blocks, analysis of Balanced incomplete block design. Orthogonal Latin squares: construction of orthogonal Latin squares of order 4.

References:

1. Searle, S.R. (1971) : Linear Models , John Wiley & sons New York. 2. Aloke Dey, : Theory of Block Designs , Wiley Eastern Ltd. (Chapter First ) 3. Chakrabarti, M.C.(1962) : Mathematics of Design and Analysis of Experiments, Asia Publishing House,(First Chapter.)

4. Joshi, D.D (1990) : Linear Estimation and Design of Experiments , Wiley Eastern Ltd. ( Chapter Fourth)

5. Angela Dean and Daniel : Design and Analysis of Experiments,Voss (1999) Springer.

6. Das, M.N.and Giri, N (1979) : Design and Analysis of Experiments, Wiley Eastern.

7. Giri, N.(1986) : Analysis of Variance, South Asian Publishers. 8. John, P.W.M.(1971) : Statistical Design and Analysis of Experiments,

Machmillan. 9. Montogomery, C.D.(1976) : Design and Analysis of Expertiments,

Wiley, New York. 10. Meyers, R.H.(1971) : Response Surface Methodology, Allyn & Bacon. 11. Pearce, S.C.(1984) : Design of Experiments, Wiley, New York. 12. Rao, C.R.and Kleffe, J (1988) : Estimation of Variance

Components and applications, North Holland. 13. Goon, A.M., Gupta, M.K. : An Outline of Statstical Theory, Vol. II, World and Dasgupta. B. (1985). Press.


Paper-III & IVth Opt. (i) Reliability and Renewal Theory(ST-403 & ST-404)

Course Objectives:

This course provides the students the ability to use statistical tools to characterize the reliability

and the working knowledge to determine the reliability of a system/component. Also, suggest

various approaches to enhancing system reliability. This course also provide the students the

ability to select appropriate reliability validation methods and renewal processes.


Apply the appropriate methodologies and tools for enhancing the inherent and actual

reliability of components and systems, taking into consideration cost aspects.

Specify life test plans for reliability validation.

Compute various measures of reliability of products and systems.

Analyze failure data.

Identify component importance.

Use redundancy to achieve reliability.

Evaluate the impact of maintenance on reliability.

Derive the probabilities for renewal theory.


Paper-III & IVth Opt. (i) Reliability and Renewal Theory (ST-403 & ST-404) (4 Credits)


Time: 3 hrs.

Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the sameUnit-I

Concept of reliability , early age failures, wearout failures and chance failures. Derivation of general reliability function failure rate, failure density function and mean time between failures ( MTBF). System reliability evaluation: series system, parallel system, partially redundant system, standby system with perfect switching / imperfect switching. Effect of spare components (identical / non- identical) on the system reliability.

Unit-IIWearout and Component reliability, Combined effect of wearout and chance failures.Reliability of a two component system with single repair facility. Reliability evaluation Techniques : Conditional probability approach , cut set method, approximation evaluation, Deducing the minimal cut sets. Tie set method , connection matrix technique.

Unit-IIIGeneral Introduction. The distribution of the number of renewals: The asymptotic distribution of N. The asymptotic normality of Nt with mean t/ and variance t/3 The number of renewals in a random time, the renewal function , the asymptotic form of the renewal function. The renewal density, variance of the number of renewals.

Unit-IVBackward and forward recurrence times. Limiting distribution of recurrence times. Pooled output of p renewal processes when p is small and its general properties. The mean time upto rth renewal. The interval between successive renewals, study of the pooled output of p renewal processes when p is large. Alternating renewal processes: The renewal functions, the type of component in use at time t, equilibrium alternating renewal processes.

References:1. Cox D.R. & Miller H.D. : Theory of Stochastic Processes, Chapman and Hall Ltd.2. Billinton, R. : Reliability Evaluation of Engineering systems: Concepts and Techniques Plemum Press New York London.3. Cox, D.R. : Renewal Theory, Methuen & Co. Ltd.4. Medhi,J. : Stochastic Processes New Age International (P) Limited.5. Igor Bazovsky : Reliability Theory and Practice, 2nd ed. Prentice Hall.


Paper –III & IV Opt. (ii) Non –Linear and Dynamic Programming(ST-403 & ST-404)

Course Objectives:

The main objective of this course is to introduce the students to various classical optimization

techniques for solving unconstrained and constrained nonlinear programming problems. To

make concepts and algorithms clear to apply for solving quadratic programming problems,

integer programming problems, separable programming problems and fractional programming

problems used in different fields of interest. To use the powerful technique of dynamic

programming for optimizing multi-stage (or sequential) decision processes.


Understand the importance of convexity in nonlinear optimization problems

Understand the broad classification of optimization problems, and where they arise in

simple applications

Apply mathematical modeling techniques to different types of non linear programming

problems

Solve Quadratic programming problems, Integer Programming problems , Separable

programming problems, Fractional programming problems using standard techniques

Solve the real- life decision problems using Dynamic Programming.

Identify appropriate models, methods and evaluate solutions to decision problems.


Paper –III & IV Opt. (ii) Non –Linear and Dynamic Programming (ST-403 & ST-404) (4 Credits) Max Marks: 75+25*


Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set

from the four unit uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same. Unit – I

Introduction: Classical optimization Techniques, maxima and minima in the absence of constraints, constrained maxima and minima; Lagrangian multipliers, treatment of non-negative variables and inequality constraints. Convex and concave functions and their maxima and minima. Saddle point problems, Khun-Tucker theory : Necessary and sufficient conditions for saddle points, Nonlinear programming problem and Saddle points, Khun-Tucker's derivation of the necessary and sufficient conditions for saddle point correspondence.

Unit – IIQuadratic programming: Kuhn- Tucker conditions for Quadratic programming problem, Wolfe’s approach for treating the quadratic objective function, Beale's Technique, Separable programming and its reduction to LPP. Separable programming Algorithm, Fractional programming, fractional programming Algorithm and its computational procedure.

Unit – IIIInteger linear programming, Formulation of sequencing problems as integer linear programming problems, project planning and man-power scheduling, the traveling sales man problem , Capital budgeting in a firm. Solution of integer programming problems, Gomory's algorithm for all integer programming problems, Branch and Bound Algorithm.

Unit – IVDynamic Programming: Principle of optimality , Cargo Loading problem, Inventory Problem, Computational Technique , Dimensionality Problem , Approximation by piecewise linear functions, Optimal path Problem, Sequencing Problem, Control Problem, Optimal page allocation Problem , Serial Multi Stage system , Comparison of Linear & Dynamic Programming.

References:1. Hadley, G. : Non linear and Dynamic programming.2. Vejda,S. : Mathematical Programming, Dover Publications.3. Kambo, N. S. : Mathematical Programming Techniques, East-West Press Ltd.4. Mittal, K.V. : Optimization Methods, New Age International Pvt. Ltd.Publisher.


Paper –III & IV Opt. (iii) Information theory(ST-403 & ST-404)

Course Objectives:

The participants will learn the basic concepts of information theory, communication system and

coding theory, including information, source coding, channel model, channel capacity, channel

coding and so on. The main purpose of this course is to help students to enhance knowledge of

probabilities, entropy, measures of information and coding theory and to guide the student

through the implications and consequences of fundamental theories and laws of information

theory and coding theory with reference to the application in modern communication.


Define communication process.

Relate the joint, conditional, and marginal entropies of variables in terms of their

coupled probabilities.

Define channel capacities and properties using Shannon's Theorems.

Construct efficient codes for data on imperfect communication channels.

M.Sc. Statistics Semester - IVPaper –III & IV Opt. (iii) Information theory (ST-403 & ST-404) (4 Credits)


Time: 3 hrs.


Introduction : communication process, communication system, measure of information, unit of information. Memoryless finite scheme: Measure of uncertainty and its properties, sources and binary sources. Measure of information for two dimensional discrete finite probability scheme: conditional entropies, Noise characteristics of a channel, Relations among different entropies

Unit-IIMeasure of Mutual information, Shanan's fundamental inequalities, Redundancy, Efficiency and channel capacity, capacity of channel with symmetric noise structures, BSC and BEC, capacity of binary channels, Binary pulse width communication channel, Uniqueness of entropy function.

Unit-IIIElements of enconding : separable binary codes, Shannon-Fano encoding, Necessary and sufficient conditions for noiseless coding. Theorem .of decodibility, Average length .of encoded messages; Shannon's Binary Encoding.

Unit-IVFundamental theorem. of discrete noiseless encoding, Huffman's minimum redundancy code, Gilbert-Moore encoding. Error detecting and Error correcting codes, Geometry of binary codes, Hammings single error correcting code.

References:1. Reza, F.M. : An Introduction to Information Theory, Mc Graw Hill Book:Company Inc.2. Feinstein, A. (I) : Foundations of Information Theory, McGraw Hill Book Company Ioc.3. Kullback, S. (I) : Information Theory and Statistic., John Wiley and Sons.4. Middleton, D. : An Introduction to Statistical Communication Theory, Mc Graw Hill Company.


Paper –III & IV Opt. (iv) Game Theory(ST-403 & ST-404)

Course Objectives:

This course introduces students to the basics of game theory and reviews different types of

games which are widely used in practice. It also considers the various solution concepts of games

and how they can be applied to solve problems occurring in economics and other scientific

disciplines.Through an in depth studies of different types of games students will develop

comprehensive knowledge of the many concepts prevalent in game theory, as well as possessing

a set of useful tools which will enable students to apply such knowledge in real world contexts.

Students will appreciate the impact that game theory has made, and will continue to make, in

many fields of scientific and other human endeavors. The course provides a strong foundation

for those students wishing to study more advanced level courses in game theory.

Learning Outcomes: On completion of this course, students will be able to:

Model competitive real world phenomena using concepts from game theory.

Discuss the theory which underlies games.

Possess a set of intermediate level game-theoretic skills which can be applied in real

world contexts.

Review and critically assess literature which deals with game theory and related

materials.

Elucidate the potential or proven relevance of game theory and its impact in many fields

of human Endeavour which involve conflict of interest between two or more participants.

Communicate game-theoretic ideas and concepts to non-specialist audiences in a

language which is accessible and comprehensible.


Paper –III & IV Opt. (iv) Game Theory (ST-403 & ST-404) (4 Credits) Max Marks: 75+25*



Rectangu1ar games, rectangular games with saddle points. Fundamental theorem of rectangular games: Mixed strategies, Geometrical background, Proof of the fundamental theorem for arbitrary rectangular games.

Unit-IIProperties of optimal strategies. Relations of dominance.A graphical method of solution. Applications of linear programming. The solution of a rectangular game . A method of approximating the value of a game.

Unit-IIIGame in extensive form.Normal form and extensive form. Graphical representation information sets. Chance moves. Games with more than two players. Re3trictions on information sets. General theory of games in extensive form.

Unit-IVGeneral definition of finite games with perfect information equilibrium points. Games with perfect recall and behaviour strategies. Games with infinitely many strategies. The fundamental theorem for continuous games.

References:Mackinsey, J.G.C. : Introduction to the theory of games McGraw Hill Book Company. Inc.. New Delhi, Toronto and London.


Paper –III & IV Opt. (v) Econometrics(ST-403 & ST-404)

Course Objectives:

The objective of this course is to study more advanced topics in econometrics. Students are

expected to have knowledge in statistics and multiple regression models. Topics typically

include linear regression models, instrument variables (IV) estimation, generalized method of

moment (GMM), Maximum likelihood estimation (MLE), limited dependent variable (LDV)

models, Treatment effect and sample selection corrections, panel data methods, Monte Carlo

simulations and bootstrap methods. The emphasis is on understanding the models and the related

theories. Through the course, students will apply the theories developed to real-world data and

interpret the estimation results in many different respects.

Learning Outcomes :On successful completion of this course students will be able to:

Acquire knowledge of various advanced econometric models, estimation methods and

related econometric theories

Learn how to apply the above theories to empirical data or be able to develop new

econometric theory

Able to work in groups when doing problem solving and computer exercises, and present

relevant research papers in the field of applied or theoretical econometrics

Learn how to write Matlab code and how to use statistical packages like STATA to

estimate econometric models using real world data

Able to conduct econometric analysis of data properly and understand the results


Paper –III & IV Opt. (v) Econometrics (ST-403 & ST-404) (4 Credits)


Time: 3 hrs.

Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four units uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same..Unit-I

Aitken's generalized least square (G.L.S) estimator, Heteroscedasticity, auto-correlation, test of auto-correlation. Multicollinearity, tools for handling multicollinearity, .idea of ridge regression and properties of ridge regression. Lagged variables end distributed lag models : Meaning. Avarage lag, Koyak lag model, Alen lag model, partial adjustment model, Adaptive expectation model.

Unit-IISimultaneous equation system : structure and models, typology of economic relations, structural form, reduced form and final form of an economic ,model. Problem of identification under linear homogeneous and Covariance restrictions. Rank and Order conditions of Identification, Restrictions on structural parameters.

Unit-IIIMethods of estimation: Limited information models, indirect least squares. Two stage least squares, limited information maximum likelihood (LIML), fuIl information methods. Three stage least square (3SLS) and full information maximum likelihood (FIML).

Unit-IVEstimation of demand common functional forms for estimation demand, estimation of demand from household budget and market data. Aggregated "problem, pooling of time series and cross section data. Estimation of production and cost function. Estimation of cabb- Dauglas, SMAC and translog production function estimation of cost curves and cost functions.

References:1. Johnston, J. : Economic Models, McGraw Hills2. Jan Kmenta : Elements of Econometrics, University of Michigan Press3. Intriligatore,M.D. : Economic models -techniques and applications, Prentice Hall4. Maddala, G.S. : Econometrics, North Holland5. Klein, L.R. : Applied Economics, Taylor and Francis6. Koytsoyiannis,A. : Theory of Econometrics, Barnes and Noble Books


Paper –III & IV Opt. (vi) Acturial Statistics(ST-403 & ST-404)

Course Objectives:

Actuary is charged with assessing the uncertainties involved in providing insurance. The

quantification of these uncertainties requires statistical methods with some mathematical

development. This course investigates modern actuarial modeling and examines the basic

techniques used in actuarial analysis. The aim of this course is to provide the students a firm

mathematical and statistical background so that they can apply mathematical and statistical

models to assess risk factors and to apply stochastic models appropriate to the representation of

the risk process.


Able to recognize the basic concepts and standard terms in actuarial sciences

Apply the typical long-tailed distributions representing claim size and those representing

claim numbers.

Understand how to use appropriate methods to extract relevant moment information and

derive consequential information about total claim size.

Use life tables, determine and manipulate the financial functions related to life insurance.

Use statistical models to analyze the risk factors for categories of policy holders.

Apply appropriate mathematical methods to get solutions for some problems in risk

theory.


Paper –III & IV Opt. (vi) Acturial Statistics (ST-403 & ST-404) (4 Credits)


Time: 3 hrs.Note: There will be nine questions in all. Question No.1 will be compulsory covering whole of the syllabus and comprising short answer type questions. Rest of the eight questions will be set from the four unit uniformly i.e. two from each unit. The candidate will be required to attempt five questions in all one from each unit and the compulsory one. The weightage of all the questions will be the same.Unit-I

Concepts of mortality rates and other indices, construction of mortality table from graduated data, determination and use of the functions in mortality table, graph of force of mortality, laws of mortality, mortality funds, Sources and collection of data for the continuous mortality investigation.

Unit-IIModels of population dynamics: Lotka' theory. Relationship between the number of births and the number of women in the population. Population with unvarying age distribution. Nature of reserve, prospective and retrospective reserves, fractional premiums and fractional durations, modified reserves, (continuous reserves, surrender values and paid up policies, Industrial assurance; children's. deffered assurances, Joint life and last survivorship.

Unit-IIIPure endowments, Life Annuities; Single payment, continuous life annuities, discrete life annuities, life annuities with monthly payments, commutation functions, varying annuities, recursions, complete annuities-immediate and apportion able annuities-due. Accumulations, Assurances, family income benefits, capital sums on retirement and death.

Unit-IVWidows pensions, Sickness benefits, disability benefits. Orphan's benefits, Benefits dependent on marriage. Contingent probabilities, contingent assurances, reversionary annuities, multiple-decrement table, forces of decrement, construction of multiple decrement table.

References:1. King, G. : Institute of acutries text book of part II second edu. Charles and Edwin Layton London.2. Jordan, C.W. Jr : Life Contingencies, second edition, Chicago Society of Actuaries.3. Neil1, A. (1977) : Life Contingencies, Heinemann, London4. Donald DWA (1970) : Compound interest and annuities. Heinemann London


Paper-V Practical (Calculator and SPSS/SYSTAT based)(ST-405)

Course Objectives: The objective of the course is to introduce several useful multivariate techniques, making strong use of illustrative examples. During the course the students will train how to use Statistical Package for Social Studies (SPSS) and SYSTAT. It provides knowledge about different experimental designs and statistical analysis. To acknowledge students the use of testing hypotheses for different parameter(s).

Learning Outcomes: On completion of this course students will be able to: Perform exploratory analysis of multivariate data using SPSS and SYSTAT. Conduct statistical inference about multivariate means including hypothesis testing and

different types of confidence intervals estimation; Understand to design and conduct experiments, as well as analyze and interpret data. Understand to check the affects of different factors under study. Perform and Analysis Split-plot design and BIBD. Obtain experience in using Statistical Package for Social Studies (SPSS) and SYSTAT.


Paper-V Practical (Calculator and SPSS/SYSTAT based) (ST-405) (4 Credits)

Max Marks : 75**+25* **Practical : 60

Class Record : 10 Viva-Vice : 10

* Internal Assessment Time: 4 hrs.

Note: There will be 4 questions, candidates will be required to attempt any 3 questions.

List of Practicals:1. Estimating parameters of multinormal distribution.2. Calculation of multiple and partial correlation coefficients.3. Estimating the parameters of conditional distribution.4. Test based on total, partial and multiple correlations.5. Test based on Hotelling - T2 and Mahalanobis - D2 Statistics.6. Fisher’s linear discriminate function.7. Calculation of principal components.8. Analysis of three basic designs- Basic analysis and splitting of treatment S. S. for different contrasts.9. Analysis of 22 – factorial experiment.10. Analysis of 23 – factorial experiment.11. Analysis of completely confounded factorial experiment.12. Analysis of partially confounded factorial experiment.13. Analysis of split plot design.14. Analysis of BIB Design.


(OE-209) Statistics-1*

Course Objectives

This course is designed to provide the Social Sciences student a intense foundational introduction to the fundamental concepts in Statistics. It is aimed at students who need a basic background in statistics and its application. This course is to provide an understanding for the student on statistical concepts to include measurements of location and dispersion, probability, probability distributions, regression, and correlation analysis. It provides knowledge about basic probability theory and its significance in the real world. To analyze univariate and bivariate data, discrete and continuous random variables.


Understand the importance and scope of Statistics. Calculate and apply measures of location and measures of dispersion grouped and

ungrouped data cases. Understand the nature of statistical data. Organize and display data by means of diagrams and charts. Compute and interpret the results of Bivariate Regression and Correlation Understand positive (right) vs. negative (left) skew Understand the significance of statistics and probability in the real world Assess the nature of random variables and probability distributions (including binomial,

Poisson, normal ) through direct calculation. Analyze and/or compare different sets of data using charts, diagrams and numerical

measures. Calculate probabilities and derive the marginal and conditional distributions of bivariate

random variables. Understand and apply basic rules of probability and Central Limit Theorem.

*Would be offered to students of Faculty of Sciences outside the department subject to the condition that the teaching staff and infrastructure will be provided by the University.


(OE-209) Statistics-I* (2 Credits) Max Marks: 35+15*


Unit -1

Meaning, importance and scope of statistics, Types of statistical data: primary and secondary

data, qualitative and quantitative data, time series data, discrete and continuous data, ordinal,

nominal, ratio and interval scales, Frequency distributions, cumulative frequency distributions,

Diagrammatic representation of data: Bar diagrams, histogram, pie chart, measures of central

tendency, Measures of dispersion, moments, skewness, kurtosis, Correlation coefficient , rank

correlation, regression lines, partial correlation coefficient, multiple correlation coefficient.

Unit-II

Basic concepts of probability: Random experiment, sample space, events, different definitions of

probability, Additive law of probability, conditional probability, Random variables: discrete and

continuous random variables, Probability density function, distribution functions, mathematical

expectation, moment generating function and characteristic function, Bivariate probability

distributions: marginal and conditional distributions, Probability distributions: Binomial, Poisson,

Normal, exponential, uniform, Central limit theorem.

References:

1. Gupta, S.C. & Kapoor, V.K. : Fundamentals of Mathematical Statistics, Sultan Chand and

Sons.

2. Gupta, S.C. & Kapoor, V.K. : Fundamentals of Applied Statistics, Sultan Chand and Sons.

3. Goon, A.M., Gupta, : Fundamentals of Statistics, Vol. II, ed. VI, M.K. & Dasgupta, B. Word

Press Calcutta 1988



(OE-309) Statistics-II*

Course Objectives:

This is an introductory course in statistics. Students are introduced to the fundamental concepts involved in using sample data to make inferences about populations. The purpose of this course is to introduce students to the application of statistical methods in research. The intent is to enable students to read and understand current research literature in their respective fields of specialization especially with regard to the use of statistical methods. Among the statistical methods that students will be exposed to are the following: t-tests, F- distribution, Fisher’s z – distribution, Chi-square distribution and one-way and two-way ANOVA. It provides knowledge to perform hypothesis tests on means and proportions for one or two populations for small as well as large sample. Be able to perform hypothesis tests on means and proportions for one or two populations

Learning Outcomes: On successful completion of the course, Students will be able to:

Understand concepts of sample vs. population.

Analysis and also perform tests based on t, F, chi square and normal variate z..

Perform basic statistical inference tasks involving various forms of hypothesis test for one and two samples.

Apply inferential statistical methods via hypothesis testing.

Perform one way and two way analysis of variance and explain the assumptions involved in the technique.

Form Test of Hypothesis as well as calculate confidence interval for a population parameter for single sample and two sample cases.



(OE-309) Statistics-II* (2 Credits) Max Marks: 35+15*


Unit-1

Concept of population and sample, parameter and statistic, need of sampling, types of sampling

(definition only),sampling distribution of a statistic, standard error, Statistical hypothesis, null and

alternative hypotheses, simple and composite hypotheses, procedure in hypothesis testing, types

of errors, power of test and critical region, levels of significance, one and two tailed test, degrees

of freedom, Tests of hypothesis-large sample tests: test of hypothesis for single mean, difference

of means, proportions, difference of proportions, standard deviation and difference of standard

deviations.

Unit-II

Sampling distributions and their applications: Student - t distributions, F- distribution,

Fisher’s z – distribution and Chi-square distribution. Simple tests based on t, F, chi

square and normal variate z. Analysis of variance: for one-way classification, two-way

classification (for fixed effect model only).

References:

1. Gupta, S.C. & Kapoor, V.K. : Fundamentals of Mathematical Statistics, Sultan

Chand and Sons.

2. Gupta, S.C. & Kapoor, V.K. : Fundamentals of Applied Statistics, Sultan

Chand and Sons.

3. Goon, A.M., Gupta, : Fundamentals of Statistics, Vol. II, ed. VI,

M.K. & Dasgupta, B. Word Press Calcutta 1988


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