Preliminary
Probability
Random experiment, sample space, event, algebra of events, probability on a
discrete sample space, basic theorems of probability and simple examples based
there on, conditional probability of an event, independent events, Bayes
theorem and its application, discrete and continuous random variables and their
distributions, expectation, m
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oments, moment generating function, joint
distribution of two or more random variables, marginal and conditional
distributions, independence of random variables, covariance, correlation,
coefficient, distribution of function of random variables. Bernoulli, binomial,
geometric, negative binomial, hypergeometric, Poisson, multinomial, uniform,
beta, exponential, gamma, Cauchy, normal, longnormal and bivariate normal
distributions, reallife situations where these distributions provide
appropriate models, Chebyshevs inequality, weak law of large numbers and
central limit theorem for independent and identically distributed random
variables with finite variance and their simple applications.
Statistical Methods
Concept of a statistical population and a sample, types of data, presentation
and summarization of data, measures of central tendency, dispersion, skewness
and kurtosis, measures of association and contingency, correlation, rank
correlation, intraclass correlation, correlation ratio, simple and multiple
linear regression, multiple and partial correlations (involving three variables
only), curvefitting and principle of least squares, concepts of random sample,
parameter and statistic, Z, X2, t and F statistics and their properties and
applications, distributions of sample range and median (for continuous
distributions only), censored sampling (concept and illustrations).
Statistical Inference
Unbiasedness, consistency, efficiency, sufficiency, Completeness, minimum
variance unbiased estimation, RaoBlackwell theorem, LehmannScheffe theorem,
CramerRao inequality and minimum variance bound estimator, moments, maximum
likelihood, least squares and minimum chisquare methods of estimation,
properties of maximum likelihood and other estimators, idea of a random
interval, confidence intervals for the paramters of standard distributions,
shortest confidence intervals, largesample confidence intervals.
Simple and composite hypotheses, two kinds of errors, level of significance,
size and power of a test, desirable properties of a good test, most powerful
test, NeymanPearson lemma and its use in simple example, uniformly most
powerful test, likelihood ratio test and its properties and applications.
Chisquare test, sign test, WaldWolfowitz runs test, run test for
randomness, median test, Wilcoxon test and WilcoxonMannWhitney test.
Wals sequential probability ratio test, OC and ASN functions, application to
binomial and normal distributions.
Loss function, risk function, minimax and Bayes rules.
Sampling Theory and Design of Experiments
Complete enumeration vs. sampling, need for sampling, basic concepts in
sampling, designing largescale sample surveys, sampling and nonsampling
errors, simple random sampling, properties of a good estimator, estimation of
sample size, stratified random sampling, systematic sampling, cluster sampling,
ratio and regression methods of estimaton under simple and stratified random
sampling, double sampling for ratio and regression methods of estimation,
twostage sampling with equalsize firststage units.
Analysis of variance with equal number of observations per cell in one, two
and threeway classifications, analysis of covariance in one and twoway
classifications, basic priniciples of experimental designs, completely
randomized design, randomized block design, latin square design, missing plot
technique, 2n factorial design, total and partial confounding, 32 factorial
experiments, splitplot design and balanced incomplete block design.
Mains
PaperI
Probability :
Sample space and events, probability measure and probability space, random
variable as a measurable function, distribution function of a random variable,
discrete and continuoustype random variable probability mass function,
probability density function, vectorvalued 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 pth mean and almost everywhere, their criteria and
interrelations, BorelCantelli lemma, Chebyshev’s and Khinchine‘s weak laws of
large numbers, strong law of large numbers and kolmogorov’s theorems,
GlivenkoCantelli theorem, probability generating function, characteristic
function, inversion theorem, Laplace transform, related uniqueness and
continuity theorems, determination of distribution by its moments. Linderberg
and Levy forms of central limit theorem, standard discrete and continuous
probability distributions, their interrelations and limiting cases, simple
properties of finite Markov chains.
Statistical Inference
Consistency, unbiasedness, efficiency, sufficiency, minimal sufficiency,
completeness, ancillary statistic, factorization theorem, exponential family of
distribution and its properties, uniformly minimum variance unbiased (UMVU)
estimation, RaoBlackwell and LehmannScheffe theorems, CramerRao inequality
for single and severalparameter family of distributions, minimum variance bound
estimator and its properties, modifications and extensions of CramerRao
inequality, ChapmanRobbins inequality, Bhattacharyya’s bounds, estimation by
methods of moments, maximum likelihood, least squares, minimum chisquare and
modified minimum chisquare, properties of maximum likelihood and other
estimators, idea of asymptotic efficiency, idea of prior and posterior
distributions, Bayes estimators.
Nonrandomised and randomised tests, critical function, MP tests,
NeymanPearson lemma, UMP tests, monotone likelihood ratio, generalised
NeymanPearson lemma, similar and unbiased tests, UMPU tests for single and
severalparameter families of distributions, likelihood rotates and its large
sample properties, chisquare goodness of fit test and its asymptotic
distribution.
Confidence bounds and its relation with tests, uniformly most accurate (UMA)
and UMA unbiased confidence bounds.
Kolmogorov’s test for goodness of fit and its consistency, sign test and its
optimality. wilcoxon signedranks test and its consistency, KolmogorovSmirnov
twosample test, run test, WilcoxonMannWhitney test and median test, their
consistency and asymptotic normality.
Wald’s SPRT and its properties, OC and ASN functions, Wald’s fundamental
identity, sequential estimation.
Linear Inference and Multivariate Analysis
Linear statistical models’, theory of least squares and analysis of variance,
GaussMarkoff theory, normal equations, least squares estimates and their
precision, test of signficance and interval estimates based on least squares
theory in oneway, twoway and threeway classified data, regression analysis,
linear regression, curvilinear regression and orthogonal polynomials, multiple
regression, multiple and partial correlations, regression diagnostics and
sensitivity analysis, calibration problems, estimation of variance and
covariance components, MINQUE theory, multivariate normal distribution,
Mahalanobis;’ D2 and Hotelling’s T2 statistics and their applications and
properties, discrimi nant analysis, canonical correlatons, oneway MANOVA,
principal component analysis, elements of factor analysis.
Sampling Theory and Design of Experiments
An outline of fixedpopulation 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 for structural populations, cluster
sampling, twostage and multistage sampling, ratio and regression, methods of
estimation involving one or more auxiliary variables, twophase sampling,
probability proportional to size sampling with and without replacement, the
HansenHurwitz and the HorvitzThompson estimators, nonnegative variance
estimation with reference to the HorvitzThompson estimator, nonsampling
errors, Warner’s randomised response technique for sensitive characteristics.
Fixed effects model (twoway classification) random and mixed effects models
(twoway classification per cell), CRD, RBD, LSD and their analyses, incomplete
block designs, concepts of orthogonality and balance, BIBD, missing plot
technique, factorial designs : 2n, 32 and 33, confounding in factorial
experiments, splitplot and simple lattice designs.
PaperII
I. 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, Vmask, 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 DodgeRomig and Military Standard tables.
Concepts of reliability, maintainability and availability, reliability of
series and parallel systems and other simple configurations, renewal density and
renewal function, survival models (exponential), Weibull, lognormal, Rayleigh,
and bathtub), different types of redundancy and use of redundancy in
reliability improvement,
problems in lifetesting, censored and truncated experiments for exponential
models.
II. Optimization Techniques
Different, types of models in Operational Research, their construction and
general methods of solution, simulation and MonteCarlo methods, the structure
and formulation of linear programming (LP) problem, simple LP model and its
graphical solution, the simplex procedure, the twophase method and the
Mtechnique with artificial variables, the duality theory of LP and its economic
interpretation, sensitivity analysis, transportation and assignment problems,
rectangular games, twoperson zerosum games, methods of solution (graphical and
algerbraic).
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 discretetime Markov chains, transition probability matrix,
classification of states and ergodic theorems, homogeneous continoustime Markov
chains, Poisson process, elements of queueing 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.
III. Quantitative Economics and Official Statistics
Determination of trend, seasonal and cyclical components, BoxJenkins method,
tests for stationery of series, ARIMA models and determination of orders of
autoregressive and moving average components, forecasting.
Commonly used index numbersLaspeyres, Paashes and Fishers ideal index
numbers, chainbase 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, timereversal test, factorreversal test, circular test
and dimensional invariance test.
General linear model, ordinary least squares and generalised least squires
methods of estimation, problem of multicollinearlity, consequences and solutions
of multicollinearity, autocorrelation and its consequeces, heteroscedasticity of
disturbances and its testing, test for independe of disturbances, Zellners
seemingly unrelated regression equation model and its estimation, concept of
structure and model for simulaneous equations, problem of identificationrank
and order conditions of identifiability, twostage 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 limitation and the principal
publications containing such statistics, various official agencies responsible
for data collection and their main functions.
IV. Demography and Psychometry
Demographic data from census, registration, NSS and other surveys, and their
limitation 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, fifting a logistic curve, population projection,
stable population quasistable population techniques in estimation of
demographic parameters, morbidity and its measurement, standard classification
by cause of death, health surveys and use of hospital statistics.
Methods of standardisation of scales and tests, Zscores, standard scores,
Tscores, 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.
