The very nature of Statistics as a subject ensures that it is chosen as an optional paper in the UPSC CSE by very few aspirants. It is not the kind of subject that any layman can choose and score well enough to clear the UPSC CSE Mains examination.

Statistics as an optional subject for UPSC requires a thorough comprehension as well as the application of its theories. The UPSC CSE syllabus for this subject includes, among other things, probability theory, inferences, sampling theory, and quantitative economics.

The curriculum also tends to focus on a more significant number of topics in comparison to certain other papers. As a result, predicting the topic from which questions will be asked is quite tricky. Also, it does not have anything in common with the General Studies paper of UPSC CSE. So, it would make sense to opt for this optional paper only if you are interested in the subject and have decent knowledge levels before you start your preparation.

## 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 con-tinuous-type random variable, probability mass function, prob-ability density function, vector-valued random variable, mar-ginal 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 path 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 Kolmogoroffs theorems, probability generating function, moment generating function, characteristic function, inver-sion theorem, Linderberg and Levy forms of central limit theo-rem, standard discrete and continuous probability distribu-tions.
2. Statistical Inference: Consistency, unbiasedness, efficiency, sufficiency, com-pleteness, ancillary statistics, factorization theorem, exponen-tial family of distribution and its properties, uniformly mini-mum variance unbiased (UMVU) estimation, Rao Blackwell and Lehmann-Scheffe theorems, Cramer-Rao inequality for single Parameter. Estimation by methods of moments, maxi-mum likelihood, least squares, minimum chisquare and modi-fied minimum chisquare, properties of maximum likelihood and other estimators, asymptotic efficiency, prior and posterior distributions, loss function, risk function, and minimax esti-mator. Bayes estimators. Non-randomised and randomised tests, critical function, MP tests, Neyman-Pearson lemma, UMP tests, monotone like-lihood ratio: similar and unbiased tests, UMPU tests for single paramet likelihood ratio test and its asymptotic distribution. Confidence bounds and its relation with tests. Kolmogorov’s test for goodness of fit and its consis-tency, sign test and its optimality. Wilcoxon signedranks 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, Pois-son, normal and exponential distributions. Wald’s fundamen-tal 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 signifi-cance and interval estimates based on least squares theory in oneway, two-way and three-way classified data, regression analysis, linear regression, curvilinear regression and orthogo-nal polynomials, multiple regression, multiple and partial cor-relations, estimation of variance and covariance components, multivariate normal distribution, Mahalanobis’s D2 and Hotelling’s T2 statistics and their applications and properties, discriminant analysis, canonical correlations, principal com-ponent analysis.
4. Sampling Theory and Design of Experiments: An outline of fixed-population and super-population approaches, distinctive features of finite population sampling, propability sampling designs, simple random sampling with and without replacement, stratified random sampling, sys-tematic 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 ob-servation per cell), CRD, RBD, LSD and their analyses, incom-plete block designs, concepts of orthogonality and balance, BIBD, missing plot technique, factorial experiments and 24 and 32, confounding in factorial experiments, split-plot and simple lattice designs, transformation of data Duncan’s multiple range test.

## Statistics Optional 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 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-Romin 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 metbod and the M-technique with artificial variables, the du-ality theory of LP and its economic interpretation, sensitivity analysis, transpotation and assignment problems, rectangu-lar games, two-person zerosum games, methods of solution (graphical and algebraic). Replacement of failing or deteriorating items, group and individual replacement policies, concept of scientific inven-tory management and analytical structure of inventory prob-lems, 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 theo-rems, homogeneous continuous-time Markov chains, Pois-son process, elements of queuing theory, M/MI, M/M/K, G/ M/l 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, fore-casting. Commonly used index numbers – Laspeyre’s, Paasche’s and Fisher’s ideal index numbers, cham-base index number, uses and limitations of index numbers, index number of wholesale prices, consumer price, agricultural production and industrial production, test fot index numbers -proportionality, time-reversal, factor-reversal and circular. General linear model, ordinary least square and generalized least squares methods of estimation, problem of multi-collinearity, consequences and solutions of multi-collinearity, 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, two-stage least sauare 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.