M.A./M.Sc.(prev.)
Paper I: MATHEMATICAL ANALYSIS
Paper II: PROBABILITY & MEASURE THEORY
Paper III: DISTRIBUTION THEORY
Paper IV:SAMPLE SURVEY AND DESIGN OF EXPERIMENT
Paper V:STATISTICAL INFERENCE
Paper VI:PRACTICAL BASED ON PAPER IV AND V
Paper VII(a):COMPUTER PROGRAMMING (THEORY)
Paper VII (b):PRACTICAL BASED PAPER III AND COMPUTER PROGRAMME


Paper I: MATHEMATICAL ANALYSIS
3 hrs. duration
100 Marks

Section-A
    Real Valued functions. Continuous Functions, Uniform Continuity,Sequences of Functions, Uniform Cojnvergence.power series and Radijus of convergence.
    Differentitatiojn, maxima-minima of functions, functions of several variables,constrained maxima-minima of functions, Multiple integrals and their evaluation by repeated integration,change of variables in multiple integration, uniform convergence in improper integrals, differentiation under the sign of integrals-Leibnitz rule.

Section-B
    Linear transformations, algebra of matrices, row and column spaces of matrix, elementary matrices, determinants, rank and inverse of a matrix,null space and nullity, partitioned matrices, Krojnecker product.
    Hermite ca onical form, generalized inverse, Moore-Penrose generalized inverse, idempotent matricesm solution of matrix equations.
    Real quadratic forms, reduction and classification of quadratic forms, index and signature, triangular reduction of a positive definite matrix.
    Characteristic roots and vectors, Cayley-Hamilton theorem, minimal polynomial, similar matrices, algebraic and geometric multiplicity of a characteristic roots, spectural decomposition of a real symmetric matrix. reduction of a pair of real symmetric matrices, hamilton matrices
    Singular values and singular value decompositions, Jordon decomposition, extrema of quadratic forms, vector and matrix differentiation.

Section-C
    Interpolation formulae (with reminder terms) due to langrange, Newton-Gregory, Newton divided difference, Guass and stirling
    Eular-Macalurin's summation formula. Inverse interpolation, Numerical integration and differentiation. Difference equations of the first order, linear difference equation with constant coefficients.
Books Recommended:
  1. Apostol, T.M.(1985) Mathematical Analysis
  2. Courant. R. and John, F.(1965) Introduction to calculus and Analysis, Wiley
  3. Miller, K.S.(1957) Advanced Real Calculus, Harper, New York
  4. Graybill, F.A.(1983) Matrices with applications in statistics, 2ndEd. Wadsworth
  5. Rao, C.R.(1973) Linear Stastical and its applications, 2ndEd. John Wiley and Sons
  6. Searle,S.R.(1982) Matrix Algebra useful for statistics John Wiley and Sons
  7. Rajaraman, V.(1981) Computer Oriented Numerical Methods, Prentice Hall
  8. Sastry, S.S.(1987) Introductory Methods of Numerical Analysis Prentice Hall

Paper II: PROBABILITY & MEASURE THEORY
3 hrs. duration
100 Marks

Section-A
    General Probability space, various definitions of Prabability, Axiomatic Approach of Probability, Combination of Events, Laws of Total and Compound Probability, Conditional Probability, Bayes Theorem
    Concept of Random Variables, Camulative distribution Function and Probability Density functions, Joint Marginal and Conditional distribution, Mathematical Expectations. Moments and conditional Expectations. Chevyshev's Inequality, Moment generating functions, Cumulant Generating Functions and their Applications

Section-B
    Classes of sets, fields, sigma-fields, minimal sigma-field, Borel sigma-field in R_K. sequence of sets , limsup and liminf of a sequence of sets, Measure, Probability measure, Properties of a measure, Caratheodory extension theorum (statement only), Lebsegue and Lebsegue-Stieltjes measures on R_K
    Measurables functions, random variables, sequences of random variables, almost sure convergence, convergence in Probability (and in measure), Integration of a measureable function with respect to a measure, Monotone convergence theorum. Fatou's lemma, dominated convergence theorum
    Borel-Cantelli Lemna. Independence. Weak law and strong law of large numbers for Id sequences. Definition and examples of Markov dependence, Exchangeable sequences. m-dependent sequences, stationary sequences
    Convergence in distribution. Characteristic function. uniqueness theorum. Levy's continuity theorum (statement only). CLT-for a sequence of independent random variables under Lindeberg's condition, CLT for iid random variables.
References:
  1. Kingman J.F. & Taylor, S.J.: Introduction to measure and Probability
  2. Loeve: Probability Theory
  3. Bhatt, B.R.: Probability
  4. Feller,W: Introduction to Probability theory and its Applications, Vol. I and II
  5. Rohatgi, V.K.: Introduction to Probability Theory and Mathematical statitics
  6. Billingsley, P.: Probability and measure, Wiley
  7. Dudley, R.M. Real analysis and Probability

Paper III: DISTRIBUTION THEORY
3 hrs. duration
100 Marks

Section-A
    Brief review of joint, marginal and conditional probability density functions, standard discrete and continuous probability distributions Bernoulli, Uniform, Binomial, Poisson, Geomatric, Negative Binomial, Hypergeomatric, Normal, Exponential, Cauchy, Beta, Gemma, Lognormal

Section-B
    Bivariate normal, Bivariate Exponential and multinomial Distributions. Functions of random variables and their distributions, compound, truncated and mixture distribution, Correlation and their regrassion analysis, Concept of sampling distribution: chi-square, t,f distribution (central and non-central) and their properties and applications. Order Statistics their distribution and properties, Joint and Marginal Distribution of Order Statistics, Range and Median Distribution
References:
  1. Kendall, M.G. and Stuart, A. Advanced Theory of Statistics, Vol. I, II
  2. Rohatgi, V.K., Introduction to Probability Theory and Mathematical Statistics
  3. Goon, Gupta & Das Gupta: outline of statistical Theory, vol.I
  4. Rao,C.R., Linear Statistical Inference and its Applications, Wiley Eastern
  5. Johnson,S. and Kotz(1972) Distribution in statistics, Vol.I,II and III
  6. Mood, Grabill and Boes: Introduction to the Theory of statistics
  7. Mogg and Craig: Introduction to the Theory of Statistics
  8. Arnold B.C., Balakrishnan., Nand Nagaraga,H.N.: A first Course on order Statistics, Wiley

Paper IV:SAMPLE SURVEY AND DESIGN OF EXPERIMENT
3 hrs. duration
100 Marks

Section-A

 Sample Surveys: Planning execution and analysis of large scale sample surveys with illustrative examples. Detailed treatment of simple random sampling. strafied sampling Problem of allocation of sample sizes in different strata, comparison with simple random sampling and Estimation of the gain in Precision due to stratification. Cluster, two stage (equal and unequal first stage with unbiased estimators for mean) and systematic sampling. Ratio, Product and regression method of estimation, Hartley and Ross unbaised Ratio estimator.

Section-B

 Analysis of experimental model by least squares, Cochran's theorem and Regression analysis (case of full rank). Analysis of variance and Covariance. Transformations Principles of Experimentation, Uniformity trails, randomised blocks, latin squares, balanced incomplete block designs (intra block analysis). Missing plot technique. Factoral experiments, 2nd and 3rd, total and partial confounding, split plot designs. Construction of confounded factorial experiments belonging to series. References:
  1. Sukhatme,P.V.,Sukhatme,B.V.,Sukhatme,S., and Ashok,C.: Sampling Theory of Surveys with Applications
  2. Cochran,W.G.: Sampling Techniques
  3. Goon,A.M., Gupta,M.K.& Das Gupta,B (1986): Fundamentals of Statistics, Vol.II, World Press, Calcutta
  4. Das,M.N. and Giri,N.C. (1979): Design and Analysis of Experiments, Wiley Eastern
  5. Federer,W.T. (1975): Experimental designs-theory and applications Oxford & IBH

Paper V:STATISTICAL INFERENCE
3 hrs. duration
100 Marks

Section-A

 Elements of decision theory: Loss function, risk function, estimation and testing viewed as decision rules, bayes and minimax estimators, Pitman estimators, Adminissiblity of estimators. Point estimation, characteristics of good estimator, likelihood function, examples from standard discrete and continuous models (such as bernoulli Poisson, Negative Binomial Normal, exponential, Gamma, Pareto etc.), Plotting Likelihood Functions for these modelsupto two parameters. Methods of Estimation: Maximum likelihood method, method of moments and percentiles, choice of estimators based on unbiasedness, minimum variance, mean squared error, minimum variance unbaised estimators, Rao-Blackwell Theorem, Completeness, Lehmann Scheffe theorem, necessary and sufficient conditions for MVUE. Cramer-Rao lower bound approach.

Section-B

Tests of Hypothese, Concepts of Critical regions, test functions, level, MP and UMP tests wald's SPRT with perscribed errors of two types. Neymam Pearson Lemma, MP test for simple null against simple alternative hypothesis. UMP tests for simple null-hypothesis against one sided alternatives and for one sided null against one sided alternatives in one parameter exponential family. Introduction to standard one sample and two sample non-parametric confidence intervals for percentiles.

Interval estimation, confidence level, construction of confidence intervals using pivots, shortest expected length confidence interval and uniformally most accurate one sided confidence interval

References:
  1. Rohatgi,V.K.: Introduction to Probability Theory and Mathematical Statistics, Wiley
  2. Rao,C.R.: Linear Statistical Inference, Wiley
  3. Lehmann,E.L.(1986): Theory and Point Estimation, Wiley
  4. Lehmann,E.L., Testing Statistical Hypothesis
  5. Ferguson,T.S.: Mathematical Statistics, Academic Press
  6. Zocks,S.: Theory of Statistical Inference, John Wiley

Paper VI:PRACTICAL BASED ON PAPER IV AND V
4 hrs. duration
100 Marks

Paper VII(a):COMPUTER PROGRAMMING (THEORY)
3 hrs. duration
50 Marks

Introduction in computers, Operating Systems, MS-Office (MS-Excel, MS-Word, MS-Power Point, MS-Access).

Programming fundamentals: Computer Based problems solving Techniques, Flowcharts and Algorithms, Modular Programming Approach, Structured Programming.

Pogramming through C/C++ Programming Language: Introduction Basic Structure of a C Program. Executing a C Programs, Character set, C tokens, Keywords & Identifiers. Constant, Variables, Data Types. Input-Output Statements, Assignment Statements & Control Statements. Types of operators and their precedence. Arithmatic Expressions Arrays. Character Strings. Standard Library functions.

Modular Programming: User defined Functions, Returning Values. Parameter passing Mechanism. Structures.Pointers Concepts. Defining a Pointer. Arrays Vs Pointers. Dynamic Memory allocations. C-Preprocessors.

Basic idea of C++ Programming: Characteristics of C++ Programming, Classes 7 objects. Structure Vs Classes. Arrays of Objects, Pointer to Object, Arrays to Pointer to Objects. Generic Programming through Templates.

References:
  1. Norton, Peter: Guide to MS-DOS
  2. Mathur, Rajiv: Learning Windows-98, step by step, Galgotia
  3. Mathur, Rajiv: Learning Excel-97 for windows step by step, Galgotia
  4. sanders H.D.: Computer Today, McGraw Hill
  5. B.W. Kernighan and D.M.Ritchie (1988). The C Programming Language, Second Edition, Prentice hall
  6. W.H.Press, S.A.Teukolsky, W.T.Vellering and B.P.Flannery(1993), Numerical Recipies in C, Second Edition, Cambridge University Press.
  7. B.Ryan and B.L.Joiner (2001). MINITAB Handbook, Fourth Edition, Duxbury

Paper VII (b):PRACTICAL BASED PAPER III AND COMPUTER PROGRAMME
50 Marks
4 hours