– 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, in probability, in r-th mean, and almost everywhere.

– Criteria and inter-relations for convergence, Chebyshev’s inequality.

– Khintchine’s weak law of large numbers, strong law of large numbers, and Kolmogorov’s theorems.

– Probability generating function, moment generating function, characteristic function.

– Inversion theorem, uniqueness theorem, Lindeberg-Levy form of central limit theorem.

– Standard discrete and continuous probability distributions: Bernoulli, binomial, Poisson, geometric, negative binomial, hypergeometric, multinomial, rectangular, normal, exponential, Cauchy, gamma, beta, and Pareto.

– Bivariate normal distribution, marginal and conditional distributions.

– Factorization theorem, exponential family of distribution and its properties.

– Uniformly Minimum Variance Unbiased (UMVU) estimation, Rao-Blackwell and Lehmann-Scheffe theorems.

– Cramer-Rao inequality for a single parameter.

– Estimation methods: method of moments, method of maximum likelihood, method of least squares, method of minimum chi-square, modified minimum chi-square.

– Properties of maximum likelihood and other estimators, asymptotic efficiency.

– Prior and posterior distributions, loss function, risk function, minimax estimator, Bayes estimators.

– Non-randomized and randomized tests, critical function, Most Powerful (MP) tests, Neyman-Pearson lemma.

– Uniformly Most Powerful (UMP) tests, monotone likelihood ratio, similar and unbiased tests.

– Uniformly Most Powerful Unbiased (UMPU) tests for a single parameter.

– Likelihood ratio test and its asymptotic distribution.

– Confidence bounds and their relation with tests.

– Kolmogorov’s test for goodness of fit and its consistency.

– Sign test and its optimality, Wilcoxon signed-ranks test and its consistency.

– Kolmogorov-Smirnov two-sample test, run test, Wilcoxon-Mann-Whitney test, median test, their consistency and asymptotic normality.

– Wald’s Sequential Probability Ratio Test (SPRT) and its properties, Operating Characteristic (OC) and Average Sample Number (ASN) functions for tests regarding parameters for Bernoulli, Poisson, normal, and exponential distributions.

– Wald’s fundamental identity.

– Gauss-Markoff theory, normal equations, least squares estimates and their precision.

– 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, orthogonal polynomials, multiple regression, multiple and partial correlations.

– Estimation of variance and covariance components.

– Multivariate normal distribution, Mahalanobis’s D² and Hotelling’s T² statistics and their applications and properties.

– Discriminant analysis, canonical correlations, principal component analysis.

– Probability sampling designs: simple random sampling with and without replacement, stratified random sampling, systematic sampling.

– Cluster sampling, two-stage and multi-stage sampling.

– Ratio and regression methods of estimation involving one or more auxiliary variables.

– Two-phase sampling, probability proportional to size sampling with and without replacement.

– Hansen-Hurwitz and Horvitz-Thompson estimators, non-negative variance estimation with reference to the Horvitz-Thompson estimator.

– Non-sampling errors.

– Fixed effects model (two-way classification), random and mixed effects models (two-way classification with equal observations per cell).

– Completely Randomized Design (CRD), Randomized Block Design (RBD), Latin Square Design (LSD) and their analyses.

– Incomplete block designs, concepts of orthogonality and balance, Balanced Incomplete Block Design (BIBD).

– Missing plot technique, factorial experiments (including 2⁴ and 3²), confounding in factorial experiments, split-plot and simple lattice designs.

– Transformation of data, Duncan’s multiple range test.

– Different types of control charts for variables and attributes: X̄ (X-bar), R (Range), s (Standard deviation), p (Proportion defective), np (Number of defectives), and cumulative sum charts.

– Single, double, multiple, and sequential sampling plans for attributes.

– Operating Characteristic (OC), Average Sample Number (ASN), Average Outgoing Quality (AOQ), and Average Total Inspection (ATI) curves.

– Concepts of producer’s and consumer’s risks, Acceptable Quality Level (AQL), Lot Tolerance Percent Defective (LTPD), and Average Outgoing Quality Limit (AOQL).

– Sampling plans for variables, use of Dodge-Romig tables.

– Concept of reliability, failure rate, and reliability functions.

– Reliability of series and parallel systems and other simple configurations.

– Renewal density and renewal function.

– Failure models: exponential, Weibull, normal, lognormal.

– Problems in life testing, censored and truncated experiments for exponential models.

– Simulation and Monte-Carlo methods.

– 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.

– 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, transition probability matrix, classification of states, and ergodic theorems.

– Homogeneous continuous-time Markov chains, Poisson process.

– Elements of queuing theory: M/M/1, M/M/K, G/M/1, and M/G/1 queues.

– Solution of statistical problems on computers using well-known statistical software packages like SPSS.

– Box-Jenkins method, tests for stationary 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 numbers of wholesale prices, consumer price, agricultural production, and industrial production.

– Tests for index numbers: proportionality, time-reversal, factor-reversal, and circular.

– General linear model, ordinary least square, and generalized least squares methods of estimation.

– Problem of multi-collinearity, consequences, and solutions of multi-collinearity.

– Autocorrelation and its consequences.

– Heteroscedasticity of disturbances and its testing.

– Test for independence of disturbances.

– Concept of structure and model for simultaneous equations.

– Problem of identification: rank and order conditions of identifiability.

– Two-stage least squares method of estimation.

– Present official statistical system in India relating to population, agriculture, industrial production, trade, and prices.

– Methods of collection of official statistics, their reliability and limitations.

– Principal publications containing such statistics.

– Various official agencies responsible for data collection and their main functions.

– Definition, construction, and uses of vital rates and ratios.

– Measures of fertility, reproduction rates, morbidity rate, standardized death rate.

– 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 a logistic curve.

– Population projection, stable population, quasi-stable population.

– Techniques in the estimation of demographic parameters.

– Standard classification by cause of death, health surveys, and use of hospital statistics.

– Methods of standardization of scales and tests: Z-scores, standard scores, T-scores, percentile scores.

– Intelligence quotient and its measurement and uses.

– Validity and reliability of test scores and its determination.

– Use of factor analysis and path analysis in psychometry.