# UPSC Mains Exam Syllabus : Statistics Optional

**UPSC Mains Exam Syllabus : Statistics Optional**

## 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,
twophase 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 2nand 32, confounding in factorial experiments, split-plot and
simple lattice designs, transformation of data Duncan’s multiple range test.

## IAS Mains General Studies Study Kit

## 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.

**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, twoperson zero-sum 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/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- Laspeyre’s, Paasche’s and Fisher’s ideal index numbers, chain-base
index number, uses and limitations of index numbers, 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, problem of multicollinearity, consequences and solutions of
multicollinearity, 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 identification-rank
and order conditions of identifiability, twostage 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 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.