**UPSC Mains Statistics Syllabus**

### Statistics Paper - I

**1. Probability:** Sample space and
events, probability measure and probability space, random variable as a
measurable function, the 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 random variables, expectation
and moments of a random variable, conditional expectation, convergence of
a sequence of a 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 a 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-randomized and randomized 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 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, 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, a 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, 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 concerning 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 2nand 32,
confounding in factorial experiments, split-plot and simple lattice designs, the transformation of data Duncan’s multiple range test.

### Statistics 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-Roaming 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 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, the 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.

**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, forecasting.
Commonly used index numbers- Laspeyres, Paasche’s and Fisher’s ideal index
numbers, chain-base index number, uses and limitations of index numbers, the index
number of wholesale prices, consumer prices, agricultural production and
industrial production, test for index numbers - proportionality, time-reversal,
factor-reversal and circular. General linear model, ordinary least square and
generalized least squares methods of estimation, the problem of multicollinearity,
consequences and solutions of multicollinearity, autocorrelation and its
consequences, heteroscedasticity of disturbances and its testing, test for
independence of disturbances, the concept of structure and model for simultaneous
equations, the problem of identification-rank and order conditions of
identifiability, two-stage least square 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 the 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 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.

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