## STATISTICS (Medium –English)

#### PAPER–I (subject code 1056)

**1. Probability :**

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 variable in

distribution, in probability, in path mean and almost everywhere, their criteria and inter-relations,

Chebyshev’s inequality and Khintchine’s weak law of large numbers, strong law of large numbers

and Kolmogoroffs theorems, probability generating function, moment generating function,

characteristic function, inversion theorem, Linderberg and Levy forms of central limit theorem,

standard discrete and continuous probability distributions.

**2. Statistical Inference :**

Consistency, unbiasedness, efficiency, sufficiency, completeness, ancillary statistics, 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

single Parameter. Estimation by methods of moments, maximum likelihood, least squares, minimum

chisquare and modified minimum chisquare, properties of maximum likelihood and other estimators,

asymptotic efficiency, prior and posterior distributions, loss function, risk function, and minimax

estimator. Bayes estimators.

Non-randomised and randomised tests, critical function, MP tests, Neyman-Pearson lemma, UMP

tests, monotone likelihood ratio: similar and unbiased tests, UMPU tests for single paramet likelihood

ratio test and its asymptotic distribution. Confidence bounds and its relation with tests.

Kolmogorov’s test for goodness of fit and its consistency, sign test and its optimality. Wilcoxon

signedranks test and its consistency, Kolmogorov-Smirnov two sample test, run test, Wilcoxon-

Mann-Whitney test and median test, their consistency and asymptotic normality.

Wald’s SPRT and its properties, Oc and ASN functions for tests regarding parameters for Bernoulli,

Poisson, normal and exponential distributions. Wald’s fundamental identity.

**3. Linear Inference and Multivariate Analysis : **

Linear statistical models, theory of least squares and analysis of variance, Gauss-Markoff theory,

normal equations, least squares estimates and their precision, test of significance and interval

estimates based on least squares theory in oneway, 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, multivariate

normal distribution, Mahalanobis’s D2 and Hotelling’s T2 statistics and their applications and

properties, discriminant analysis, canonical correlations, principal component analysis.

**4. Sampling Theory and Design of Experiments : **

An outline of fixed-population and super-population approaches, distinctive features of finite

population sampling, propability sampling designs, simple random sampling with and without

replacement, stratified random sampling, systematic sampling and its efficacy, cluster sampling,

twostage 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, the Hansen-Hurwitz and the HorvitzThompson 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 observation per cell), CRD, RBD, LSD and their analyses, incomplete block

designs, concepts of orthogonality and balance, BIBD, missing plot technique, factorial experiments

and 24 and 32, confounding in factorial experiments, split-plot and simple lattice designs,

transformation of data Duncan’s multiple range test.

#### PAPER II (subject code 1057)

**1. Industrial Statistics : **

Process and product control, general theory of control charts, different types of control charts for

variables and attributes, X, R, s, p, np and charts, cumulative sum chart. 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-Romin

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.

**2. Optimization Techniques :**

Different types of models in Operations Research, their construction and general methods of solution,

simulation and Monte-Carlo methods formulation of Linear Programming (LP) problem, simple LP

model and its graphical solution, the simplex procedure, the two-phase metbod and the M-technique

with artificial variables, the duality theory of LP and its economic interpretation, sensitivity analysis,

transpotation and assignment problems, rectangular games, two-person zerosum 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/MI, M/M/K, G/M/l and M/G/1 queues. Solution of statistical

problems on computers using wellknown statistical software packages like SPSS.

**3. Quantitative Economics and Official Statistics :**

Determination of trend, seasonal and cyclical components, Box-Jenkins method, tests for stationary

series, ARIMA models and determination of orders of autoregressive and moving average

components, fore-casting.

Commonly used index numbers – Laspeyre’s, Paasche’s and Fisher’s ideal index numbers, cham-base

index number, uses and limitations of index numbers, index number of wholesale prices, consumer

price, agricultural production and industrial production, test fot index numbers -proportionality, timereversal,

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 identificationrank

and order conditions of identifiability, two-stage least sauare 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.

**4. Demography and Psychometry :**

Demographic data from census, registration, NSS other surveys, their limitations. and uses, 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

estimation of demographic parameters, standard classification by cause of death, health surveys and

use of hospital statistics.

Methods of standardisation 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.