**PAPER – I**

**Section – A**

**Probability :**

**Unit – I**

Sample space and events, probability measure and probability space, random variable as a measurable function, distribution function of a random variable, discrete and continuous-type random variable, probability mass function, probability density function, vector-valued random variable, marginal and conditional distributions, stochastic independence of events and of random variables, expectation and moments of a random variable, conditional expectation, convergence of a sequence of random variables in distribution and in probability almost everywhere, their criteria and inter-relations, BorelCantelli lemma, Chebyshev’s and Khinchine‘s weak law of large numbers, strong law of large numbers and Kolmogorov’s theorem, Glivenko-Cantelli theorem.

** Unit – II**

Probability generating function, characteristic function, inversion theorem, Laplace transform, determination of distribution by its characteristic function, Lindberg and Levy forms of central limit theorem, standard discrete and continuous probability distributions, their inter-relations and limiting cases. (Bernoulli, Binomial, Negative binomial, Poisson, Normal, Cauchy, Beta and Gamma), Exponential family of distributions and their

properties.

**Linear Models and Multivariate Analysis**

**Unit – III**

Linear statistical models, theory of least squares and analysis of variance, Gauss-Markov theorem, normal equations, least squares estimates and their properties, test of significance and interval estimates based on least squares theory in one-way, two-way and three-way classified data, regression analysis, linear regression, curvilinear regression and orthogonal polynomials, multiple regression, multiple and partial correlations, estimation of variance and covariance components, MINQUE theory.

** Unit – IV**

Multivariate normal distribution, Marginal and conditional distributions, Distributions of linear and quadratic functions of multivariate normal, Independence of the distribution of quadratic functions. Wishart’s distribution, Mahalanobis D2 and Hotelling’s T2 statistics and their applications and properties, discriminant analysis, canonical correlatons, principal component analysis, elements of factor analysis.

**Section – B**

**Statistical Inference**

**Unit-I**

Consistency, unbiasedness, efficiency, sufficiency, minimal sufficiency, completeness, ancillary statistic, factorization theorem, derivation of sufficient statistics for the exponential family of distribution, uniformly minimum variance unbiased (UMVU) estimation, Rao-Blackwell and Lehmann-Scheffe theorems, Cramer-Rao inequality for single and several-parameter family of distributions, minimum variance bound estimator and its properties, Chapman-Robbins inequality, Bhattacharya’s bounds, estimation by methods of moments, maximum likelihood, least squares, minimum chi-square and modified minimum chi-square, properties of maximum likelihood estimator, idea of asymptotic efficiency, Loss and Risk functions, idea of prior and posterior distributions, Bayes’ and minimax estimators.

** Unit – II**

Non-randomised and randomised tests, critical functions, MP tests, Neyman-Pearson lemma, UMP tests, monotone likelihood ratio, generalised Neyman-Pearson lemma, similar regions and unbiased tests, UMPU tests for single and several-parameter families of distributions, likelihood ratio and its large sample properties, chi-square goodness of fit test and its asymptotic distribution. Confidence bounds and its relation with tests. Kolmogorov’s test for goodness of fit and its consistency, sign test, Wilcoxon signedrank test and their consistency, Kolmogorov-Smirnov two-sample test, run test, WilcoxonMann-Whitney U-test and median test, their consistency and asymptotic normality. Wald’s SPRT and its properties, OC and ASN functions, Wald’s fundamental identity, application to Binomial, Poisson and Normal distributions only.

**Sampling Theory and Design of Experiments**

**Unit – III**

An outline of fixed-population and super-population approaches, distinctive features of finite population sampling, sampling designs, simple random sampling with and without replacement, stratified random sampling, systematic sampling and its efficiency for structural populations, cluster sampling, two-stage and three-stage sampling, ratio, product and regression methods of estimation involving one or more auxiliary variables, two-phase sampling, probability proportional to size sampling with and without replacement, the Hansen-Hurwitz and the Horvitz-Thompson estimators, non-negative variance estimation with reference to the Horvitz-Thompson estimator, non-sampling errors, Warner’s randomised response technique.

** Unit – IV**

Fixed effect model (one-way and two-way classification), random and mixed effect models (one-way and two-way classification), Basic principles of design, CRD, RBD, LSD and their analyses and efficiencies, missing plot technique, factorial designs : 2n, 32 and 33, confounding in factorial experiments, split-plot, strip-plot and simple lattice designs, incomplete block designs, concepts of orthogonality and balance, BIBD.

**PAPER – II**

**Section – A**

**Industrial Statistics**

**Unit – I**

Process and product control, general theory of control charts, different types of control charts for variables and attributes, concept of 3s limits, X , R, s, p, np and c charts, cumulative sum chart, V-mask. Single, double, multiple and sequential sampling plans for attributes, OC, ASN, AOQ and ATI curves, concepts of producer’s and consumer’s risks, AQL, LTPD and AOQL, sampling plans for variables, use of Dodge-Romig table.

** Unit – II**

Concepts of reliability, maintainability and availability, reliability of series and parallel systems, Hazard functions, I.F.R. and D.F.R. distributions survival models (exponential, Weibull, lognormal, Rayleigh, and bath-tub), problems in life-testing, censored and truncated experiments for exponential models.

**Quantitative Economics and Official Statistics**

**Unit – III**

Concept of time series, additive and multiplicative models, Determination of trend, seasonal, cyclical and random components, Box-Jenkins method, tests for stationery of series, ARIMA models and determination of orders of autoregressive and moving average components, forecasting. Commonly used index numbers:- Laspeyre’s, Paasche’s and Fisher’s ideal index numbers, chain-base index number, uses and limitations of index numbers, index number of wholesale prices, consumer price index number, index numbers of agricultural and industrial production, test for index numbers like proportionality test, time-reversal test, factor-reversal test, circular test and dimensional invariance test. General linear model, ordinary least squares and generalised least squares methods of estimation, problem of multicollinearlity, consequences and solutions of multicollinearity, autocorrelation and its consequeces, heteroscedasticity of disturbances and its testing, test for independence of disturbances.

** Unit – IV**

Present official statistical system in India relating to population, agriculture, industrial production, trade and prices, methods of collection of official statistics, their reliability and limitation and the principal publications containing such statistics, various official agencies responsible for data collection and their main functions.

**Section – B**

**Optimization Techniques**

**Unit – I**

Different types of models in Operational Research, their construction and general methods of solution, simulation and Monte-Carlo methods, the structure and formulation of linear programming (LP) problem, simple LP model and its graphical solution, the simplex procedure, the two-phase method and the M-technique with artificial variables, the duality theory of LP and its economic interpretation, sensitivity analysis, transportation and assignment problems, rectangular games, two-person zero-sum games, methods of solution (graphical and algebraic). Replacement of failing or deteriorating items, group and individual replacement policies, concept of scientific inventory management and analytical structure of inventory problems.

** Unit – II**

Simple models with deterministic and stochastic demand with and without lead time, storage models with particular reference to dam type. Homogeneous discrete-time Markov chains, simple properties of finite Markov chains, transition probability matrix, classification of states and ergodic theorems, homogeneous continuous-time Markov chains, Poisson process, elements of queueing theory, M/M/1, G/M/1 and M/G/1 queues.

**Demography and Psychometry**

**Unit – III**

Demographic data from census, registration, NSS and other surveys, and their limitations and uses; definition, construction and uses of vital rates and ratios, measures of fertility, reproduction rates, morbidity rate, standardized death rate, Infant mortality rate, nuptiality, complete and abridged life tables, construction of life tables from vital statistics and census returns, uses of life tables, logistic and other population growth

curves, fitting of a logistic curve, population projection, stable population quasi-stable population techniques in estimation of demographic parameters, morbidity and its measurement, standard classification by cause of death, health surveys and use of hospital statistics, health statistics.

** Unit – IV**

Methods of standardization of scales and tests, Z-scores, standard scores, T-scores, percentile scores, intelligence quotient and its measurement and uses, validity of test scores and its determination, use of factor analysis and path analysis in psychometry.