mscstaticccssugsullabus13012011

8/2/2019 mscstaticccssugsullabus13012011

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UNIVERSITY OF CALICUT(Abstract)

MSc programme in Statistics- under Credit Semester System (PG)-Scheme and Syllabus

approved-implemented-with effect from 2010 admission onwards-Orders issued

GENERAL & ACADEMIC BRANCH-IV J SECTION

No. GA IV/J2/4230/10 Dated, Calicut University PO, 26.07.2010

Read:1. U.O.No. GAIV/J1/1373/08 dated, 23.07.2010.2. Item no. 1 of the minutes of the meeting of the Board of Studies in Statistics PG,

held on 09.06.2010

3.Item no. III a. 30 of the minutes of the meeting of the Academic Council held on03.07.2010

O R D E R

Credit Semester System was implemented to the PG programmes in affiliated Arts

and Science Colleges of the University with effect from 2010 admission onwards,as perpaper read as first,above.

The Board of Studies in Statistics PG vide paper read as second above decided torecommend the implementation of Calicut University regulations for for Credit SemesterSystem for PG curriculum 2010 in affiliated Colleges{CUCSS-PG-2010} and to apply the

Credit Semester System to the MSc Statistics programme conducted by the affiliated

colleges of the University with effect from 2010 11 academic year onwards.. The Vice Chancellor,due to exigency, approved the minutes of the meeting of the

Board of Studies in Statistics PG held on 09.06.2010 subject to ratification by the

Academic Council and the same was ratified by the Academic Council vide paper read as3 above.

Sanction has therefore been accorded for implementing the scheme and Syllabus of

MSc programme in Statistics under CSS (PG) with effect from 2010 admission onwards.Orders are issued accordingly. Scheme and syllabus are appended.

Sd/-

DEPUTY REGISTRAR (G & A-IV)For REGISTRAR

ToThe Principals of affiliated Colleges offering MSc programme in Statistics

Copy to:

P.S to V.C PAtoRegistrar/Chairman,B/S,Statistics,PG/CE/EX/DRIII/DR,PG/EGI/

Enquiry/System Administrator with a request to upload in the Universitywebsite/Information Centres/G&A I `F``G`Sns/GAII,III

Forwarded/By Order

Sd/-SECTION OFFICER.


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UNIVERSITY OF CALICUT(Abstract)

M.Sc Programme in Statistics under Credit Semester System PG 2010 Scheme and

Syllabus of II semester approved implemented with effect from 2010 admission

onwards Orders issued.GENERAL & ACADEMIC BRANCH-IV J SECTION

No. GA IV/J2/4230/2010 Dated, Calicut University PO, 04.01.2011

Read: 1. U.O.No.GAIV/J1/1373/08 dated 23.07.2010.2. U.O.No.GAIV/J2/4230/10 dated 26.07.2010.3. Letter dated 16.12.2010 from the Chairman, Board of Studies in Statistics PG.

O R D E R

As per paper read as (1) above, Credit Semester System PG was introduced for all

PG courses in affiliated Arts and Science Colleges of the University.

Vide paper read as (2) above, the syllabus of M.Sc Programme in Statistics for the1st semester was implemented.

The Chairman, Board of Studies in Statistics vide paper read as (3) above hasforwarded the syllabus for the II semester of M.Sc programme in Statistics as per the

decision of Board of Studies held on 09.06.2010.

The Vice-Chancellor, due to exigency, exercising the powers of the AcademicCouncil approved the syllabus subject to ratification by the Academic Council.

Sanction has therefore been accorded for implementing the syllabus of II semesterof M.Sc Programmes in Statistics under Credit Semester System PG for the affiliated

colleges with effect from 2010 admissions.

Orders are issued accordingly. Syllabus appended.

Sd/-

DEPUTY REGISTRAR(G&A IV)For REGISTRAR

To

The Principals of all affiliated colleges offering M.Sc Programmes in Statistics.

Copy to:PS to VC/PA to Registrar/Chairman Board of Studies in Statistics PG/CE/

DR III /EX section/ DR-PG/EG-I/ Information centres/Enquiry/System Administrator (with a request to upload in the University website)GAI F G Sections/GAII/GAIII/SF/FC

Forwarded/By Order

Sd/-SECTION OFFICER


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M. Sc. (Statistics) Degree Programme under the Credit Semester System(CSS) forthe Affiliated Colleges of University of Calicut

Programme Structure & Syllabi(With effect from the academic year 2010-2011 onwards)

Duration of programme: Two years - divided into four semesters of not less than 90 workingdays each.

Course Code Type Course Title Credits

I SEMESTER(Total Credits: 20)

ST1C01 Core Measure Theory and Probability 4ST1C02 Core Analytical Tools for Statistics I 4ST1C03 Core Analytical Tools for Statistics II 4ST1C04 Core Linear Programming and Its Applications 4ST1C05 Core Distribution Theory 4

II SEMESTER(Total Credits: 18)

ST2C06 Core Estimation Theory 4ST2C07 Core Sampling Theory 4ST2C08 Core Regression Methods 4ST2C09 Core Design and Analysis of Experiments 4ST2C10 Core Statistical Computing I 2

III SEMESTER(Total Credits: 16)

ST3C11 Core Stochastic Processes 4

ST3C12 Core Testing of Statistical Hypotheses 4ST3E-- Elective Elective-I 4ST3E-- Elective Elective-II 4

IV SEMESTER(Total Credits: 18)

ST4C13 Core Multivariate Analysis 4ST4E-- Elective Elective-III 4ST4C14 Core Project/Dissertation and General Viva-Voce 8ST4C15 Core Practical II 2

Total Credits: 72 (Core courses-52, Elective courses-12 andProject / Dissertation -8)


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The courses Elective I, Elective II, and Elective III shall be chosen from the following list.

LIST OF ELECTIVES

Sl. No. Course Title Credits

E01 Advanced Operations Research 4

E02 Econometric Models 4

E03 Statistical Quality Control 4

E04 Reliability Modeling 4

E05 Advanced Probability 4

E06 Time Series Analysis 4

E07 Biostatistics 4

E08 Computer Oriented Statistical Methods 4

E09 Lifetime Data Analysis 4

E10 Statistical Decision Theory and Bayesian Analysis 4

E11 Statistical Ecology and Demography 4

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SYLLABI OF COURSES OFFERED IN SEMESTER -I

ST1C01: Measure Theory and Probability (4 Credits)

Unit- 1 . Sets, Classes of sets, Measure space, Measurable functions and Distribution functions:

Sets and sequence of sets, set operations, limit supremum, limit infimum and limit of sets, Indicaterfunction, fields ,sigma fields, monotonic class, Borel field on the real line, set functions,Measure,measure space, probability space, examples of measures, properties of measures, measurable functions,

random variables and measurable transformations, induced measure and distribution function, Jordandecomposition theorem for distribution ,multivariate distribution function, continuity theorem foradditive set functions and applications, almost everywhere convergence, convergence in measure,convergence in probability, convergence almost surely, convergence in distribution.

Unit-2. Integration theory , expectation, types of convergence and limit theorems:

Definition of integrals and properties, convergence theorems for integrals and expectations-Fatouslemma, Lebesgue monotonic convergence theorem, Dominated convergence theorem, Slutskys theorem,

convergence in (convergence in mean), inter relations between different types of convergenceand counter examples.

Unit-3. Independence and Law of Large numbers:

Definition of independence, Borel Cantelli lemma, Borel zero one law, Kolmogrov s zero one law,Weak law of large numbers(WLLN), Convergence of sums of independent random variables-Kolmogrovconvergence theorem, Kolmogorovs three-series theorem. Kolmogorovs inequalities, Strong law oflarge numbers (SLLN), Kolmogorovs Strong law of large numbers for independent random variables,Kolmogorovs strong law large numbers for iid random variables.

Unit-4. Characteristic Function and Central limit Theorem:

Characteristic function, Moments and applications, Inversion theorem and its applications, Continuitytheorem for Characteristic function (statement only),Test for characteristic functions, Polyastheorem(statement only), Bochners theorem(statement only). Cenral limit theorem for i.i.d randomvariables, Liapounovs Central limit theorem, LindebergFeller Central limit theorem(statement only).

Book for study:-

1. A.K. Basu.(1999). Measure theory and probability. Prentice Hall of India private limited New

Delhi.

References:-

1. A.K.Sen.(1990), Measure and Probability .Narosa.

2. Laha and Rohatgi (1979).Probability Theory. John Wiley New York.

3. B.R.Bhat (1999),Modern Probability theory .Wiley Eastern ,New Delhi.

4. Patrick Billingsly,(1991),Probability and Measure ,Second edition ,John Wiley .

.


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ST1C02: Analytical Tools for Statistics I (4 Credits)Unit-1 .Multidimensional Calculus

Limit and continuity of a multivariable function, derivatives of a multivariable function, Taylorstheorem for a multivariable function. Inverse and implicit function theorem, Optima of a multivariablefunction, Method of Lagrangian multipliers, Riemann integral of a multivariable funtion.

Unit-2. Analytical functions and complex integration

Analytical function, harmonic function, necessary condition for a function to be analytic, sufficientcondition for function to be analytic, polar form of Cauchy- Riemann equation, construction of analyticalfunction. Complex line integral, Cauchys theorem, Cauchys integral formula and its generalized form.Poisson integral formula, Moreras theorem. Cauchys inequality, Liovilles theorem, Taylors theorem,Laurents theorem.

Unit-3. Singularities and Calculus of Residues.

Zeroes of a function, singular point, different types of singularities. residue at a pole, residue at infinity,Cauchys residue theorem, Jordans lemma, integration around a unit circle, poles lie on the real axis,

integration involving many valued function.

Unit- 4. Laplace transform and Fourier Transform

Laplace transform, Inverse Laplace transform. Applications to differential equations, The infinite Fouriertransform, Fourier integral theorem. Different forms of Fourier integral formula, Fourier series.

Book for study

1. Andres I. Khuri(1993) Advanced Calculus with applications in statistics. Wiley & sons (Chapter 7)

2. Pandey, H.D, Goyal, J. K & Gupta K.P(2003) Complex variables and integral transforms Pragathi

Prakashan, Meerut.3. Churchill Ruel.V.(1975), Complex variables and applications .McGraw Hill.

References.

1. Apsostol (1974) Mathematical Analysis, Second edition Norosa, New Delhi.

2. Malik, S.C & Arora.S(2006) Mathematical analysis, second edition, New age international

.

ST1C03: Analytical Tools for Statistics II (4 Credits)

Unit-1.. Riemann-Stieltjes integral and uniform convergences.

Definition, existence and properties of Riemann-Stieltjes integral, integration by parts, change ofvariable, mean value theorems, sequence and series of functions, point wise and uniform convergences,


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test of uniform convergence, consequence of uniform convergence on continuity and integrability,Weirstrass theorem.

Unit- 2. Algebra of Matrices

Linear transformations and matrices, operations on matrices, properties of matrix operations, Matriceswith special structures triangular matrix, idempotent matrix, Nilpotent matrix, symmetric Hermitianand skew Hermitian matrices unitary matrix. Row and column space of a matrix, inverse of a matrix.Rank of product of matrix, rank factorization of a matrix, Rank of a sum and projections, Inverse of a

partitioned matrix, Rank of real and complex matrix, Elementary operations and reduced forms.

Unit- 3 Eigen values, spectral representation and singular value decomposition

Characteristic roots, Cayley-Hamilton theorem, minimal polynomial, eigen values and eigen spaces,spectral representation of a semi simple matrix, algebraic and geometric multiplicities, Jordan canonicalform, spectral representation of a real symmetric, Hermitian and normal matrices, singular valuedecomposition.

Unit -4 Linear equations generalized inverses and quadratic forms

Homogenous system, general system, Rank Nullity Theorem, generalized inverses, properties of g-inverse, Moore-Penrose inverse, properties, computation of g-inverse, definition of quadratic forms,classification of quadratic forms, rank and signature, positive definite and non negative definite matrices,extreme of quadratic forms, simultaneous diagonalisation of matrices.

Book for study:-

1. Ramachandra Rao and Bhimashankaran (1992).Linear Algebra Tata McGraw hill

2. Lewis D.W (1995) Matrix theory, Allied publishers, Bangalore.

3. Walter Rudin (1976).Principles of Mathematical Analysis, third edition, McGraw hill

international book company New Delhi.

References:-

1. Suddhendu Biswas (1997) A text book of linear algebra, New age international.

2. Rao C.R (2002) Linear statistical inference and its applications, Second edition, John Wiley

and Sons, New York.

3. Graybill F.A (1983) Matrices with applications in statistics.

..

ST1C04: Linear Programming and Its Applications (4 Credits)

Unit-1. Some basic algebraic concepts.

Definition of a vector space, subspaces, linear dependence and independence, basis and dimensions,direct sum and complement of subspaces, quotient space, inner product and orthogonality. Convex setsand hyperplanes.

Unit-2. Algebra of linear programming problems .


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Introduction to linear programming problem(LPP), graphical solution, feasible, basic feasible andoptimal basic feasible solution to an LPP, analytical results in general LPP, theoretical development ofsimplex method. Initial basic feasible solution, artificial variables, big-M method, two phase simplexmethod, unbounded solution, LPP with unrestricted variables, degeneracy and cycling, revised simplexmethod.

Unit- 3. Duality theory and its applications.

Dual of an LPP, duality theorems complementary slackness theorem, economic interpretation of duality,dual simplex method. Sensitivity analysis and parametric programming, integer programming, Gomeryscutting plane algorithm and branch and bound techniques.

Unit- 4. Transportation problem and game theory.

Transportation problem, different methods of finding initial basic feasible solution, transportationalgorithm, unbalanced transportation problem, assignment problem, travelling salesman problem. Gametheory, pure and mixed strategies. Conversion of two persons zero sum game to an Linear programmingproblem. Fundamental theorem of game. Solution to game through algebraic, graphical and Linearprogramming method.

Book for study:-

1. Ramachandra Rao and Bhimashankaran (1992).Linear Algebra Tata McGraw hill.

2. Cooper and Steinberg (1975). Methods and Applications of Linear Programming, W.B.Sounders Company, Philodelphia, London.

References:-

1. J.K.Sharma(2001).Operations Research Theory and Applications.McMillan New Delhi.

2. Hadley,G.(1964).Linear Programming,Oxford &IBH Publishing Company,New Delhi.3. Kanti Swaroop,P.K. Gupta et.al,(1985),Operation Research,Sultan Chand & Sons.4. Taha.H.A.(1982).Operation Research and Introduction ,MacMillan.

.

ST1C05: Distribution Theory (4 Credits)

Unit- 1. Discrete distributionsRandom variables ,Moments and Moment generating functions, Probability generating functions,Discrete uniform, binomial, Poisson, geometric, negative binomial, hyper geometric and Multinomialdistributions, power series distributions.

Unit- 2. Continuous distributions


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Uniform , Normal, Exponential, Weibull, Pareto, Beta, Gama, Laplace, Cauchy and Log-normaldistribution. Pearsonian system of distributions, location and scale families.

Unit-3. Functions of random variables.

Joint and marginal distributions, conditional distributions and independence, Bivariate transformations,covariance and correlations, bivariate normal distributions, hierarchical models and mixture distributions,multivariate distributions, inequalities and identities. Order statistics.

Unit -4 .Sampling distributions

Basic concept of random sampling, Sampling from normal distributions, properties of sample mean andvariance .Chi-square distribution and its applications, t-distribution and its applications ,F-distributions-properties and applications. Noncentral Chi-square, t, and F-distributions.

Books for study:-

1. Rohatgi, V.K.(1976).Introduction to probability theory and mathematical statistics. John Wileyand sons.

2. George Casella and Roger L. Berger(2003). Statistical Inference. Wodsworth & brooks PaceficGrove, California.

References:-

1. Johnson ,N.L.,Kotz.S. and Balakrishna.N.(1995). Continuous univariate distributions, Vol.I&Vol.II, John Wiley and Sons, New York.

2. Johnson ,N.L.,Kotz.S. and Kemp.A.W.(1992).Univarite Discrete distributions, John Wiley andSons, New York.

3. Kendall, M. and Stuart, A. (1977). The Advanced Theory of Statistics Vol I: Distribution Theory,4th Edition

..

SYLLABI OF COURSES OFFERED IN SEMESTER-II

ST2C06: Estimation Theory (4 Credits)

Unit-1: Sufficient statistics and minimum variance unbiased estimators.

Sufficient statistics, Factorization theorem for sufficiency (proof for discrete distributions only), jointsufficient statistics, exponential family, minimal sufficient statistics, criteria to find the minimalsufficient statistics, Ancillary statistics, complete statistics, complete statistics, Basus theorem (proof fordiscrete distributions only), Unbiasedness, Best Linear Unbiased Estimator(BLUE), Minimum VarianceUnbiased Estimator (MVUE), Fisher Information, Cramer Rao inequality and its applications, Rao-Blackwell Theorem, Lehmann- Scheffe theorem, necessary and sufficient condition for MVUE.


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Unit-2: Consistent Estimators and Consistent Asymptotically Normal Estimators.

Consistent estimator, Invariance property of consistent estimators, Method of moments and percentiles todetermine consistent estimators, Choosing between consistent estimators, Consistent AsymptoticallyNormal (CAN) Estimators.

Unit-3:Methods of Estimation.

Method of moments, Method of percentiles, Method of maximum likelihood (MLE), MLE in exponential

family, One parameter Cramer family, Cramer-Huzurbazar theorem, Bayesian method of estimation.

Unit-4: Interval Estimation.

Definition, Shortest Expected length confidence interval, large sample confidence intervals, Unbiasedconfidence intervals, Bayesian and Fiducial intervals.

Books for Study:-

1. Kale,B.K.(2005). A first course in parametric inference, Second Edition, Narosa PublishingHouse, New Delhi.

2. George Casella and Roger L Berger (2002). Statistical inference, Second Edition, Duxbury,Australia.

References:-1. Lehmann, E.L (1983). Theory of point estimation, John Wiley and sons, New York.2. Rohatgi, V.K (1976). An introduction to Probability Theory and Mathematical Statistics, John

Wiley and sons, New York.3. Rohatgi, V.K (1984). Statistical Inference,4. Rao, C.R (2002). Linear Statistical Inference and its applications, Second Edition,

.

ST2C07: Sampling Theory (4 credits)

Unit-I: Census and Sampling-Basic concepts, probability sampling and non probability sampling,

simple random sampling with and without replacement- estimation of population mean and total-

estimation of sample size- estimation of proportions. Systematic sampling-linear and circular

systematic sampling-estimation of mean and its variance- estimation of mean in populations with

linear and periodic trends.

Unit-II: Stratification and stratified random sampling. Optimum allocations , comparisons of

variance under various allocations. Auxiliary variable techniques. Ratio method of estimation-

estimation of ratio, mean and total. Bias and relative bias of ratio estimator. Mean square error of

ratio estimator. Unbiased ratio type estimator. Regression methods of estimation. Comparison of ratio


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and regression estimators with simple mean per unit method. Ratio and regression method of

estimation in stratified population.

Unit-III: Varying probability sampling-pps sampling with and without replacements. Des-Raj

ordered estimators, Murthys unordered estimator, Horwitz-Thompson estimators, Yates and Grundy

forms of variance and its estimators, Zen-Midzuno scheme of sampling, PS sampling.

Unit-IV: Cluster sampling with equal and unequal clusters. Estimation of mean and variance, relative

efficiency, optimum cluster size, varying probability cluster sampling. Multi stage and multiphase

sampling. Non-sampling errors.

References

1. Cochran W.G (1992) Sampling Techniques, Wiley Eastern, New York

2. D. Singh and F.S. Chowdhary Theory and Analysis of Sample Survey Designs, Wiley

Eastern(New Age International), NewDelhi.

3. P.V.Sukhatme et.al. (1984) Sampling Theory of Surveys with Applications. IOWA State

University Press, USA

.

ST2C08: Regression Methods (4 Credits)

Unit-1: Simple and multiple regression.

Introduction to regression. Simple linear regression- least square estimation of parameters, Hypothesistesting on slope and intercept, Interval estimation, Prediction of new observations, Coefficient ofdetermination, Regression through origin, Estimation by maximum likelihood, case where x is random.

Multiple Linear Regression- Estimation of model parameters, Hypothesis testing in multiple linearregression, Confidence interval in multiple regression, Prediction of new observations.

Unit- 2: Model Adequacy Checking, Transformation and weighting to correct model Inadequacies.

Residual analysis, the press statistics, detection of treatment of outliers, lack of fit of the regressionmodel. Variance -stabilizing transformations, Transformation to linearize the model, Analytical methodsfor selecting a transformation, Generalized and weighted least squares.

Unit- 3: Polynomial regression model and model building.


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Polynomial models in one variable, Nonparametric regression, Polynomial models in two or morevariables, orthogonal variables. Indicator variables, Regression approach to analysis of variance. Modelbuilding problem, computational techniques for variable selection.

Unit-4: Generalized Linear Models.

Logistic regression model, Poisson regression, The generalized linear models- link function and linearpredictors, parameter estimation and inference in GLM, prediction and estimation in GLM, residualanalysis in GLM over dispersion.

Books for Study:-

1. Montgomery ,D.C., Peck, E.A., Vining G Geofferey (2003). Introduction toLinear Regression Analysis. John Wiley & Sons.

References:-

1. Chatterjee, S & B. Price (1977). Regression analysis by example, Wiley, NewYork.

2. Draper, N.R & H. Smith (1988). Applied Regression Analysis. 3rd Edition, Wiley,New York.

3. Seber, G.A.F (1977). Linear Regression Analysis. Wiley, New York.4. Searle , S.R (1971). Linear Model. Wiley, New York.

..

ST2C09: Design and Analysis of Experiments (4 credits)

Unit- 1: Linear Model, Estimable Functions and Best Estimate, Normal Equations, Sum of Squares,Distribution of Sum of Squares, Estimate and Error Sum of Squares, Test of Linear Hypothesis, BasicPrinciples and Planning of Experiments, Experiments with Single Factor-ANOVA, Analysis of FixedEffects Model, Model Adequacy Checking, Choice of Sample Size, ANOVA Regression Approach, Nonparametric method in analysis of variance.

Unit- 2: Complete Block Designs, Completely Randomized Design, Randomized Block Design, LatinSquare Design, Greaco Latin Square Design, Analysis with Missing Values, ANCOVA,

Unit- 3: Incomplete Block Designs-BIBD, Recovering of Intra Block Information in BIBD, Constructionof BIBD, PBIBD, Youden Square, Lattice Design.

Unit- 4: Factorial Designs-Basic Definitions and Principles, Two Factor Factorial Design-GeneralFactorial Design, 2k Factorial Design-Confounding and Partial Confounding, Two Level FractionalFactorial, Split Plot Design.

Books for study


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1) Joshi D.D. (1987) Linear Estimation and Design of Experiments. Wiley Eastern Ltd., New Delhi.

2) Montgomery D.C. (2001) Design and Analysis of Experiments. 5th edition, John Wiley & Sons-NewYork.

References

1) Das M.N. & Giri N.S. (2002) Design and Analysis of Experiments. 2th edition , New AgeInternational (P) Ltd., New Delhi.

2) Angola Dean & Daniel Voss (1999) Design and Analysis of Experiments. Springer-Verlag, NewYork.

..

ST2C10: Statistical Computing-I (2 credits)(Practical Course)

Teaching scheme: 6 hours practical per week.

Statistical Computing-I is a practical course. The practical is based on the following FIVE

courses of the first and second semesters.

1. ST1C05: Distribution Theory

2. ST2C06: Estimation Theory

3. ST2C07: Sampling Theory

4. ST2C08: Regression Methods

5. ST2C09: Design and Analysis of Experiments

Practical is to be done using R programming / R software. At least five statistical data

oriented/supported problems should be done from each course. Practical Record shall be maintained by

each student and the same shall be submitted for verification at the time of external examination.

Students are expected to acquire working knowledge of the statistical packages SPSS and SAS.


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The Board of Examiners (BoE) shall decide the pattern of question paper and the duration of the

external examination. The external examination at each centre shall be conducted and evaluated on the

same day jointly by two examiners one external and one internal, appointed at the centre of the

examination by the University on the recommendation of the Chairman, BoE. The question paper for

the external examination at the centre will be set by the external examiner in consultation with the

Chairman, BoE and the H/Ds of the centre. The questions are to be evenly distributed over the entiresyllabus. Evaluation shall be done by assessing each candidate on the scientific and experimental skills,

the efficiency of the algorithm/program implemented, the presentation and interpretation of the results.

The valuation shall be done by the direct grading system and grades will be finalized on the same day.

.


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MODEL QUESTION PAPER

FIRST SEMESTER M.Sc. DEGREE EXAMINATION (CSS), 2010

Branch: Statistics

ST1C01: Measure Theory and Probability

Time :3hrs MaximumWeightage:36

Part A

(Answer all the questions , Weightage 1 for each question)

1. Define sigma field of subsets of a set.2. Define a measurable function .3. If P(A)=0.6,P(B)=0.4,A and B are independent then find P(AUB).4. State Dominated Convergence theorem.5. With the help of an example show that convergence in distribution need not imply convergence in

probability.

6. State Fatous lemma.7. When do you say that two classes of events are independent.8. Verify whether

is a distribution function.

9. When do you say that a sequence { } of random variables obey the strong law of largenumbers.

10. Obtain the characteristic function of an exponential random variable .11. State Bochners theorem for characteristic functions.12. State Lindeberg-Feller central theorem.

Part B(Answer any eight questions .Weightage 2 for each question)

13 .Prove that -field is a monotone field. Verify whether the converse is true.

14 LetB be the class of subsets of N ,the set of natural numbers . Define : B R by

(A) is a measure on B15.Prove or disprove the statement almost sure convergence implies convergence in probability.16. Define integral of an arbitrary function starting from that of a simple function.17. State and prove monotone convergence theorem.

18. Show that convergence in


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19. Show that P (lim sup )=0 if < .20. Establish Kolmogrovs Strong law of large numbers for independent random variables.21. Prove that tail events of a sequence of independent random variables have their probability eitherzero or one.22. Show that characteristic function is uniformly continues on the real line.23.Obtain the probability density function corresponding to the characteristic function

(t)= , t R.24.Establish Lindeberg -Levy central limit theorem.

Part C

(Answer anytwo questions ,Weightage 4 for each question)25.(a) If f and g are measurable functions then show that (i)f+g (ii)Max (f,g ) (iii) Min(f,g) aremeasurable.

(b) If { } is sequence of measurable functions .Show that lim sup{ } is measurable.

26.(a)If X then show that If X.

(b) prove or disprove the statement :convergence in probability implies convergence in mean.

27.(a) Establish Borel zero one law.(b) State and prove Kolmogrovs three series criterion

28.(a)State and prove inversion theorem on characteristic functions .(b) Examine if central limit theorem holds for the sequence of independent random va riables {

}with P( = P( and P(

.



Branch: Statistics

ST1C02: Analytical Tools for Statistics- I

Time :3 hrs Maximum Weightage:36

Part A

(Answer all the questions , Weightage 1 for each question)

1. Investigate for continuity at(1,2) of the function f(x,y)


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2. Define the concept of directional derivative of a function.3. State implicit function theorem.4. When do you say that a function is analytic in a domain D.5. State a necessary condition that W=f(z) =u+iv be analytic in a domain D.6. Write down the Cauchys integral formula.7. When do you say that Z=a is an isolated singularity of function f(z).8. State residue theorem.9. Define pole of order m of a function f(z).10. Define Laplace transform of a function.

11. If L{F(t)}=f(s),then find L{F(at)}12. Define a periodic function, Give an example.

Part B

(Answer any eight questions. Weightage 2 for each question)

13. If f(x,y)= y+ .Find and .

14. Find the value of where E is the triangle bounded by the strightlinesy=x ,y=0 and y=1.

15. Find maxima and minima of the function f(x,y)= -3x-12y+20.16.Show that real and imaginary part of an analytic function satisfies the Laplaceequation.

17.Obtain the Taylors series of the function f(z)= in the region |z|0 then =21. State and prove Cauchys residue theorem.

22.Find the Laplace transform of the function cos ax.

23. Find the inverse Laplace transform of .

24.Find the Fourier transform of

Part C

(Answer any two questions, Weightage 4 for each question)

25. State and prove Taylors theorem on two variables.

26. Obtain the Taylor and Laurents series expansion of the function

f(z)= in the region (i)0


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(-, ). Hence show that + +


FIRST SEMESTER M.Sc. DEGREE EXAMINATION (CSS), 2010Branch: Statistics

ST1C03: Analytical Tools for Statistics- II


Part A(Answer all the questions, Weightage 1 for each question)

1. State a necessary and sufficient condition for a function to be Riemann- Stieltjes integrable.2. Give an example of a sequence of function that converges pointwise but not uniformly.3. Define normal form of an m x n matrix.4. Define Eigen values and Eigen vectors of a square matrix.

5. Define minimal polynomial of a square matrix A.6. What is the reduced echelon form of a matrix with full column rank.7. Define image and kernel of a linear map.8. Give two definitions of rank of a matrix.9. Differentiate between nilpotent matrix and idempotent matrix.10. Give an example of a Hermitian matrix

11. Write down the symmetric matrix associated with the quadratic form q(x,y)=12. Define Moore Penrose g- inverse.

Part B


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(Answer any eight questions. Weightage 2 for each question)

13 .If is a refinement of P ,then prove that L( , f , ) L(P, f , )

14.If f is monotonic on [a,b],and if is continuous on [a.,b],then show that f ().15.Prove that the following series is uniformly convergent in [-1,1]

.

16. Prove that any two characteristic vectors corresponding to two distinctcharacteristic roots of a hermitian matrix are orthogonal

17. Find eigen values and eigen vectors of

18.State and prove Cayley-Hamilton theorem.19. Show that A is nilpotent if and only if all the characteristic roots of A are zero.

20. Prove or disprove : If and are n x n matrices in HCF , s in HCF

21. Explain the classification of quadratic form.22. Let A be symmetric and B = how that and have the samesignature.

23. Show that every matrix has a g inverse. If is the g- inverse of A then showthat is idempotent.

24.Find the Jordan form of

Part C


25. If a sequence { } converges uniformly to f on [ a, b] and each function is

integrable, then f is integrable on [a , b] ,and the sequence { }

converges uniformly to on [ a, b].

26.Define nullity of a matrix. If and are idempotent show that (i) =0(ii) rank of an idempotent matrix A is equal to its trace.

27. State and prove a necessary and sufficient condition for a quadratic form XA Xto be positive definite. Also classify the quadratic form

9 +1228. Compute a g-inverse

---------------


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Branch: Statistics

ST1C04: Linear Programming and Its Applications


Part A(Answer all the questions , Weightage 1 for each question)

1. Define linear independence of a set of vectors.

2. Verify whether S={(x,y) : x0}is a subspace of .3. What you mean by dimension of a vector space.4. Define a convex set.5. What are slack and surplus variables.6. What is the role of basic feasible solution in an LPP.

7. Indicate a situation where artificial variable is used to solve an LPP.8. How can we handle unrestricted variable in an LPP.9. Give an example of an unbalanced transportation problem.10. State weak duality theorem.11. What you mean by a two person zero sum game.12. Define the saddle point of a rectangular game.

Part B(Answer any eight questions. Weightage 2 for each question)

13. Let V be a finite dimensional vector space show that all bases of V have same number ofelements.

14. Determine whether or not the following set of vectors is linearly independent. {(1,2,6),(-

1,3,4),(-1,-4,-2)} 15. Explain the algorithm for Gram- Schmidt orthogonalization process16. Show that the set of feasible solution to an LPP is a convex set.17. Distinguish between simplex method and revised simplex method.18. Write the dual of the problem. Maximize Z = + 2

Subject to :


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- = 22 + 3 7

19. Why sensitivity analysis is important in an LPP.20. Explain Vogels approximation method for finding an initial basic feasible solution of a

transportation problem.21. Solve the game graphically

2 1 0 -2

1 0 3 2

22. Solve the assignment problem

A B C

I 19 28 31

II 11

17

16

III

12 15 13

23. Discuss the dominance property in rectangular game.

24. Briefly outline the procedure of branch and bound technique for solving an integer programmingproblem.

Part C


25. Let V be a vector space over R with dimension n. Show that V is isomorphic to .

Player B

26. Solve the following game by LPP

27. Given the LPP. Maximize Z = + 5Subject to + 2 18

8

6

a) Determine an optimum solution to the LPP.

b) Discuss effect on the optimum ability of the solution where the objective function ischanged to Z=

28. Discuss various steps involved in solving a transportation problem.

..


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Branch: Statistics

ST1C05 Distribution Theory

Time: 3 Hours Maximum Weightage: 36

Part A

(Answer all the questions. Weightage 1 for each question)

1) Define Hyper geometric distribution .

2) Write down the p.m.f of multinomial distribution and deduce that of Binomial.

3) Comment on the statement that, for a Binomial distribution mean is 2 and varianceis 3.

4) Give any two real life situations where Pareto distribution is employed.

5) What is the lack of memory property? Name a continuous distribution possessingthis property.

6) Describe lognormal distribution.

7) Let nXXX ,...,, 21 be i.i.d random variables from NkN

pk ,...,2,1,1

== . Find the marginal

distribution of the first order statistic )1(X .

8) If2

1)( =XE ,

12

1)( =XV , find an upper limit for

>

12

12

2

1XP using Tchebychev's

inequality.}

9) Write down Liapunov's inequality.

10) Define 2 distribution with ' r ' degrees of freedom.

11) Give any two applications ofFdistribution.

12) How are 2 and Fstatistics related?

Part B(Answer any eight questions. Weightage 2 for each question)

13) What is the class of Power series distribution? Mention any two major distributionsbelonging to this class.

14) Derive the p.g.f of Negative Binomial distribution. Hence deduce that of geometricdistribution.


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15) Show that for the Poisson distribution 1,11 += + rd

dr rrr

, where r is the thr

central moment.

16) State and prove the additive property of Gamma distribution.

17) If 2,~ NX obtain the moment measure of skewness and kurtosis.

18) IfX and Y are jointly distributed with p.d.f ( ) 10,22),(


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SECOND SEMESTER M.Sc DEGREE EXAMINATION (CSS), 2011

Branch: Statistics

ST2C06: Estimation TheoryTime: 3 hrs. Max. Weightage: 36

Part A(Answer all the questions. Weightage 1 for each question)

1. Define sufficient statistic. Give an example.

2. State factorization theorem for sufficiency.3. What is meant by ancillary statistic. Give an example.

4. Let T be an unbiased estimator of . Is 2T unbiased for2 ?

5. Define minimum variance unbiased estimator.6. Define one parameter exponential family of distributions.7. Discuss consistency of estimator with an example.8. Explain the criterion of choosing an estimator from a given class of consistent estimators.9. Describe the method of maximum likelihood estimation.

10. If nXXX ,....,, 21 be a random sample from ),0( N . Find the MLE of .

11. Discuss Bayesian method of estimation.12. Define (i) Unbiased confidence interval (ii) Shortest confidence interval.

Part B(Answer any eight questions. Weightage 2 for each question.)

13. Define minimal sufficient statistic. Based on a random sample from the distribution

( ).,

)(1

11),(

2


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20. Let nXXX ,...,, 21 be a random sample from gamma distribution

.0,),,( 1 = ,,),( )(

,

show that the class of linear unbiased estimators of is empty.

3. a) If nXXX ,...,, 21 is a random sample from ),(2N , show that 2

1

)(1 XXn

n

i

i = and

2

1

)(1

1XX

n

n

i

i = are both consistent for2

. Which one of the two is preferable?

b) Define marginal consistency and joint consistency. Establish their equivalence.4. State the Cramer regularity conditions. Show that with probability approaching one as ,n the

likelihood equation admits a consistent solution.

..


SECOND SEMESTER M.Sc. DEGREE EXAMINATION(CSS), 2011

Branch: Statistics

ST2C07: Sampling Theory

Time: 3 hours Maximum weightage: 36

Part A

(Answer all the questions. Weightage 1 for each question)

1. Distinguish between sampling and census.

2. Explain non probability sampling.


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3. Explain Linear systematic sample.

4. Explain proportional allocation in stratified sampling.

5. What do you mean by auxiliary variable?

6. Explain linear regression estimator.

7. Explain Lahiri's method of sample selection in pps sampling.

8. Define PS sampling.

9. What is meant by Zen-Midzuno scheme of sampling?

10. Distinguish between unit and element of a population.

11. Distinguish between cluster and stratum

12. What is double sampling? Explain.

Part B

(Answer any eight questions. Weightage 2 for each question.)

13. What are the principal steps in a statistical investigation? Explain.

14. Explain the method of estimating the proportion of units possessing an attribute in a finite

population of N units, using SRSWOR.

15. Illustrate the method of selecting a circular systematic sample of size n, using an example.

16. Point out the advantages of stratified random sampling over simple random sampling.

17. In ratio estimation, with usual notation, show that

xofcvRinbias

R

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