(e) Ethnic conflicts, communalism, religious revivalism.
(f) Illiteracy and disparities in education.
STATISTICS
PAPER - I
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 p-th 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 Kolmogoroff’s 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 chi-square and modified
minimum chi-square, 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 parameter likelihood ratio test and
its asymptotic distribution. Confidence bounds and its relation with tests.
Kolmogoroff’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 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 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, multivariate normal distribution, Mahalanobis-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 superpopulation approaches, distinctive features of finite population
sampling, probability 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 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 observation per cell), CRD, RBD, LSD and their analyses, incomplete block designs, concepts of
orthogonality and balance, BIBD, missing plot technique, factorial experiments and 2n and 32, confounding in
factorial experiments, split-plot and simple lattice designs, transformation of data Duncan’s multiple range test.
PAPER - II
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 c 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-Roming 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.
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