**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,
Borel-Cantelli lemma, Chebyshev's and
Khinchine's weak laws of large numbers,
strong law of large numbers and
kolmogorov's theorems, Glivenko-
Cantelli 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 inter-relations 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, Rao-Blackwell and
Lehmann-Scheffe theorems, Cramer-
Rao inequality for single and severalparameter
family of distributions, minimum
variance bound estimator and its
properties, modifications and extensions
of Cramer-Rao inequality,
Chapman-Robbins 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.
Non-randomised and randomised tests,
critical function, MP tests, Neyman-
Pearson lemma, UMP tests, monotone
likelihood ratio, generalised Neyman-
Pearson lemma, similar and unbiased
tests, UMPU tests for single and several-
parameter families of distributions,
likelihood rotates and its large sample
properties, chi-square 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 signed-ranks test
and its consistency, Kolmogorov-
Smirnov two-sample test, run test,
Wilcoxon-Mann-Whiltney 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 modesl, theory of least
squares and analysis of variance,
Gauss-Markoff theory, normal equations,
least squares estimates and their
precision, test of signficance and interval
estimates based on least squares
theory in one-way, two-way 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 distributin,
Mahalanobis;' D2 and Hotelling's T2
statistics and their applications and
properties, discriminant analysis,
canonical correlations, one-way
MANOVA, principal component analysis,
elements of factor analysis.

**Sampling Theory and Design of
Experiments**:
An outline of fixed-population and
super-population 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, 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,
the Hansen-Hurwitz and the Horvitz-
Thompson estimators, non-negative
variance estimation with reference to
the Horvitz-Thompson estimator, nonsampling
errors, Warner's randomised
response technique for sensitive characteristics.
Fixed effects model (two-way classification)
random and mixed effects models
(two-way 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,
split-plot and simple lattice
designs.

**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, 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 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
bath-tub), different types of redundancy
and use of redundancy in reliability
improvement, problems in life-testing,
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 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
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 discrete-time Markov
chains, transition probability matrix,
classification of states and ergodic theorems,
homogeneous continous-time
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, 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, Paashe'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, tests, for mdex numbers lve
proportonality test, time-reversal test,
factor-reversal test, circular test and
dimensional invariance test.
General linear model, ordinary least
squares and generalised least squires
methods of estimation, problem of multicollineaity,
consequences and solutions
of multicollinearity, autocorrelation
and its consequences, heteroscedasticity
of disturbances and its testing, test
for independence of disturbances,
Zellner's seemingly unrelated regression
equation model and its estimation,
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 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, fitting a logistic
curve, population projection, stable
population theory, uses of stable population
and 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.
Methods of standardisation of scales
and tests, Z-scores, 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.