Paper - VI Mathematical Statistics - II
Unit - 1 17%
Chebychev's inequality and its uses, Law of large number with known variance, characteristics function and its elementary properties, Inversion theorem with proof, Statement of Central limit law, Central limit theorem. Proof of Lendberg. - Levy form only, and statement of Liapounoff's theorem.
Unit - 2 17%
Review of general properties of bivariate distributions and regression as conditional expectation. Detailed study of bivariate normal distribution with properties, Definition of compound distribution, properties of compound distributions (Possion ^ Gamma, Binomial ^ Poisson).
Unit - 3 17%
Power series distribution : moments, recurrence relations for moments and cumulants, unique determination of a distribution form the first moments of Power series distribution, Determination of Binomial, Poisson, Negative binomial and Geometric distributions as a particular case of Power series distribution.
Unit - 4 17%
Use of orthogonal transformation and distribution of sample mean of
univeriate normal distribution, Chisquare,
Student's F distributions and their properties, distribution of sample
correlation coefficient bases on sampling from bivariate normal distribution
when g = 0.
Unit - 5 16%
Distribution of Ordered statistics : Definition of ordered statistics, distribution of largest and smallest ordered statistics, Distribution of sample range from rectangular and exponential distributions.
Unit - 6 16%
Definition and example of Stochastic Processes, Classification of general
stochastic processes into discrete /
continuous state, types of stochastic Processes, Elementary problems,
Definition and examples on Markov chain.
Reference Book :
1. Mood A.M.,
Grabill F.A. & Boes D.C. Introduction to theory of Statistics (3rd
edition)
2. Hogg R.V. & Grain A.J. Introduction to Mathematical Statistics
3. Kenny J.F. & Keeping E.S. Mathematical Statistics - II
4. C.E., WeathernburnFirst course in Mathematical Statistics
5. Mark FiszProbability theory and Mathematical Statistics (3rd edition)
Unit - 1 Estimation : 16%
Point Estimation; Properties of estimators, consistency, unbiased ness,
efficiency and sufficiency Rao-Cramer
Inequality and its use to obtain minimum variance unbiased estimators (MVUE).
Factorization theorem (for discrete case only), Raw-Blackwell theorem and
its applications.
Unit - 2 Methods of Estimation : 17%
Method of moment and method of maximum likelihod, propeties of M.L.E. (Statement only and proof of consistency property). Solving of likelihood equations by iterative procedure (Method of scoring).
Unit - 3 Testing of hypothesis : 17%
Statistical hypothesis, types of hypothesis, two kind of errors. critical region, level of significance, power and size of the test, Neyman - Person's theory of testing simple hypothesis versus sample alternative for regular family of distributions (Binomial, Possion, Geometric, Normal & Exponential distributions) Elementary idea of likelihood ratio test, confidence interval for mean and variance of a normal distribution using pivots.
Unit - 4 Theory of linear estimation : 16%
Full rank mode with uncorrelated variables, Definition of Best Linear Unbiased Estimators (BLUE) with properties and statement of Gauss - Markoff Theorem.
Unit - 5 Analysis of variance and Basic designs : 17%
One way and two way classification with fixed effect model, Basic principles of design of experiments, Completely Randomized Design (CRD), Randomixed Block Design (RBD), Latin Square design (LSD) and their relative efficiencies Missing plot techniques upto two missing observations.
Unit - 6 Factorial Experiments : 17%
Idea of Factorial design involving at the most four factors at two levels only. Yate's method of analysis, Simple idea of total and partial confounding.
Reference Books :
1. Freund J.S.
Mathematical Statistics (5th ed. 1994)
2. Mood A.M., Graybill F.A. & Boes D.C. Introduction to the at
Statistics (3rd edition)
3. Rohatgi V.K. An Introduction to Probability theory and Mathematical
Statistics
4. Dudewitz E. & Misra S.M. Modern Mathematics Statistics (1988. Wiley
E')
5. Silvery S.D. Statistical inference. (Penguin books)
6. Das & GiriDesign and analysis of experiment (Wiley East, 79)
7. Kempthorne . O. Design and analysis of experiment (Wiley East)
8. Joshi D.D. Linear estimation and design of experiments (Wiley East)
Unit - 1 Sampling Techniques : 17%
Need for Stratification and its principle, Estimation of population mean total, proportion and their variances, Different types of allocation; Proportional and Neyman allocation for fixed cost and for fixed size of sample, Gain in precision due to stratification.
Unit - 2 Systematic Sampling : 16%
Advantages of Systematic Sampling, Estimation of sample mean and variance, Different variance formulae for sample mean, relative comparison with simple random sampling.
Unit - 3 (A) Two Stage sampling : 17%
Efficiency with simple random sampling, Comparison of Stratified Systematic and Two sampling. Estimation of sample mean and its variance, Estimation of optimum sample size of second stage.
(B) Statistical Quality Control : 16%
Unit - 4 Introductory part of Control Charts : 16%
Importance of statistical methods in industrial research and practice, Specification of items and qualities, Determination of tolerance limits, General theory of control charts, Causes of variation in quality, Control limits, Summary of out of control criteria and theory of runs, Specification and modified control limits.
Unit - 5 (A) Control charts for variables : 9%
X and R charts, X and S charts
(B) Control charts for attributes : 8%
p,np, c and u-charts, comparison of control charts.
Unit - 6 Acceptance Sampling : 17%
Principles of acceptance sampling, Stipulation of good and bad lots, producer's & consumer's risks, single and double sampling plans for attributes and their O.C. functions, Concepts of AQL, LTPD, ASN, Rectification, AOQL, Average amount of inspection (AOI), Dodge - Romig Tables.
Reference Books :
1. Cochran W.G.
Sampling Techniques (3rd Edn.)
2. Duncan A.J. Quality Control & Industrial Statistics (3rd Edn.)
3. Burr I.W. Elementary Statistical Quality Control
4. Grant E.L. & Leavanworth R.S. Statistical Quality Control.
5. Ekambaranm & S.K. The Statistical Basis of Acceptance sampling.
Unit - 1 Algebra of real numbers : 17%
Algebra of real numbers, order and completeness in R, Upper and Lower bounds, least upper and greatest lower bounds, and root of a positive real number, Dense set and Archimedean principle, Decimal representation of real numbers, Algebra of complex numbers, Bernoulli and Cauchy - Schwarz inequality.
Unit - 2 Convergence : 17%
Convergence of real sequences, Limit theorems, monotonic sequences, Cantor's Completeness principle, Statement of Bolzano - Weiersrass theorem without proof, cluster points, Cauchy sequence and Cauchy's completeness principle, Series of real terms and its convergence, consideration test, Cauchy's root test and D'Alembert's ratio test, Definition of absolute convergence of series, Matric space, open and classed sets, limit points, isolated points.
Unit - 3 (A) Limit & Continuity : 17%
Limit of a function Cauchy criterion of limit of a function, left and right limit of a function, Left and right continuity of a function, Bounded function, Types of discontinuity of a function with examples, properties of continuity with reference to monotonic function, Infinite limits and limit at infinity, Limit of exponential and logarithmic functions, Definition of uniform continuity.
(B) Integration (50%)
Unit - 4 Remain Satieties Integrals : 16%
Definition and existence of Remain Satieties integral, Integration and differentiation, Fundamental theorem of Calculus and Integration by parts.
Unit - 5 Jacobian of transformations : 17%
Jacobian of transformation, Co-ordinate and polar transformation, implicit function theorem (Statement only), Existence theorem for inverse transformation, Fundamental dependence, Leibniz theorem without proof relating to differentiation under integral signs and its applications.
Unit - 6 Multiple Integration : 17%
Dirichlet's integral in n-dimension and its use in obtaining the volume and surface area of an n-dimensional sphare of redius r and sampling distribution of some statistics and examples based on multiple integration.
Reference Books :
1. G.Das and
S.PattanayakFundamentals of Mathematical Analysis "Tata MacGraw Hill,
New Delhi, 1987.
2. Walter RudinPrinciples of Mathematical Analysis" (3rd Edn.) McGraw
Hill International Edition.
3. Apostol T.M. Mathematical Analysis" (Addition Wesley)
4. Gramer H. Mathematical Methods for Statistics
5. Olmesled John M.H. Advanced Calculus (Eurasia Pub. House, Delhi, 1970)
Unit - 1 17%
Scope of OR, Advantages of OR, Different models in OR, Linear Programming Problem (LPP), Formulation of LPP, General Solution of LPP, Basic, non basic, degenerate, non-degenerate and basic feasible solution of LPP, Convex & Concave Sets, Properties of Convex Sets, Convex function, Slack and Surplus Variables, LPP in standard matrix form, properties of solutions of LPP, Generating extreme point solutions.
Unit - 2 16%
Theory and application of Simplex method of solution of LPP, Computational
procedure, Big M Method, Artificial
basis techniques, Duality of LPP, Non-Symmetric and dual problems, Relation
between dual and prime problems,
methods for finding initial basic feasible solution.
Unit - 3 17%
Transportation problem, methods for finding initial basic feasible solution optimal solution of T.P. by Modi method, unbalanced TP, Assignment Problems, The Hungarian method, Balanced and unbalanced assignment problems.
Unit - 4 17%
Competitive games, Two person zero sum game, minimax & maximin
principle, Fundamental theorem of game
theory. Saddle point and the value of the game (based on pure strategies),
Mixed strategies, Solution of game
with / without saddle point.
Dominance rule, solution of m x 2 and 2 x n games using graphical method, Algebraic methods of solving games.
Unit - 5 17%
Basic Definition of parametric association with inventory problems, various costs associated with inventory problems, Inventory models with infinite rate of replacement., Finite rate of replacement with & without strategies.
Unit - 6 17%
Queuing Theory; Classification of queues, Poisson process and exponential distribution, Transient and steady states, Poisson queues with simple service channel, Finite and infinite capacity and its characteristics.
Reference Books :
1. J.K. Sharma
O R Theory & Applications; Macmillan India Ltd., 1998.
2. Kanti Swaroop, P.K. Gupta & ManmohanOperation Research; S.Chand &
Sons, New Delhi, 1998.
3. G.HadlyLinear Programming; Narosa Publishing House, New Delhi, 1995.
4. G.P. AriceLinear Programming Transportation Assignment Game; Books &
Allied Pvt. Ltd., KolKota-9.
5. K.V.Mital & L.MohanOptimization Methods in OR & System analysis;
(New, Age
International Publications)
6. Hilter & LeibemanOR; S.Chand & Sons, New Delhi.
(A) Computer Programming : (50%)
Unit - 1 16%
Preliminary concepts of algorithm, flowcharts & their execution traces,
a simplified model of a computer,
Elementary ideas about Compiler and Operating System,
Representation for characters and numbers.
Unit - 2 17%
Basic representation for integers and real numbers effect of finite representation of numbers on properties of arithmetic operations e.g. overflow, associatively and normalization, Some elementary methods overcoming these limitations.
Unit - 3 17%
Constants and Variables, Arithmetic expressions I/O Statements, Control
Statements, Subscribed examples on
programming.
(B) Mathematical Programming : (50%)
Unit - 4 17%
Elementary idea of convex sets, Formulation of a general linear programming problem Solution by graphical method, basic concepts and properties of simplex method, artificial variable techniques, concept of duality formulation of primal, dual pairs, Statement of fundamental theorem on duality and related problems.
Unit - 5 17%
Transportation problem and assignment problem mathematical formulation of the problem, finding initial basic feasible solution, degeneracy in transportation problem, transportation algorithm for obtaining optimum solution, Un balanced transportation problem, Mathematical formulation of an assignment problem, assignment algorithm unbalanced assignment problem.
Unit - 6 Game Theory : 16%
Two person
zero sum game, Maximin Minimax principles, game without saddle points mixed
strategies, solution of 2x2 rectangular games.
Reference Books :
1.
Balaguruswamy E. Programming in BASIC (3rd Edn.); TATA McGraw Hill.
2. Gotfried B.S. Theory and Problem of Programming with BASIC" (4th Edn,
Schaum's Outline
Series), McGraw Hill.
3. Hadley Linear Programming (Adison Wiley)
4. Sharma J.K. Mathematical Models in OR, TATA McGraw Hill
5. Grover P.S. Computer Programming in BASIC, (Allied Pub. Ltd.)
6. Kantiswaroop & ManmohanIntroduction to O.R.
7. Gass S.I. Linear Programming.