For a UPSC CSE aspirant, the optional subject is also an important subject. In the UPSC mains exam, optional marks have two papers, Paper 1 and Paper 2. Each paper is of 250 marks which makes a total of 500 marks. The UPSC optional subject list contains 48 subjects in total, one of which is Sociology.

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**SYLLABUS FOR PAPER I**

**Probability:**

- Sample space and events
- Probability measure and probability space
- Random variable as a measurable function, Vector-valued random variable, distribution function of a random variable, Conditional expectation, the convergence of a sequence of a random variable in distribution, in probability, in path mean and almost everywhere, their criteria and inter-relations, discrete and continuous-type random variable
- The probability density function, probability mass function
- Marginal and conditional distributions
- Stochastic independence of events and of random variables
- Expectation and moments of a random variable
- Continuous probability distributions
- Chebyshev’s inequality and Khintchine’s weak law of large numbers, strong law of large numbers, and Kolmogoroff’s theorems
- Inversion theorem, Linderberg and Levy forms of the central limit theorem
- Probability generating function, moment generating function, the characteristic function
- Standard discrete

**Statistical Inference:**

- Consistency, unbiasedness, sufficiency, efficiency, completeness, factorization theorem, ancillary statistics, exponential family of distribution and its properties, uniformly minimum variance unbiased (UMVU) estimation, Lehmann-Scheffe theorems, and Rao-Blackwell, Cramer-Rao inequality for a single parameter. Bayes estimators. Estimation by methods of moments, least squares, maximum likelihood, minimum chi-square, and modified minimum chi-square, properties of maximum likelihood and other estimators, prior and posterior distributions, asymptotic efficiency, loss function, risk function, and minimax estimator
- Randomized and non-randomized tests, critical function, UMP tests, MP tests, Neyman-Pearson lemma, similar and unbiased tests, monotone likelihood ratio, UMPU tests for single parameter likelihood ratio test, and its asymptotic distribution. Confidence bounds and their relation with tests
- 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
- Kolmogorov’s test for goodness of fit and its consistency, sign test, and its optimality. Kolmogorov-Smirnov two-sample test, run test, Wilcoxon signed-ranks test, and its consistency, Wilcoxon-Mann-Whitney test, and median test, their consistency, and asymptotic normality

**Linear Inference and Multivariate Analysis:**

- Linear statistical models’
- The theory of least squares and analysis of variance
- Gauss-Markoff theory, normal equations, least-squares estimates, and their precision
- Regression analysis, curvilinear regression, linear regression, and orthogonal polynomials, multiple regression, multiple and partial correlations, estimation of covariance, and variance components, multivariate normal distribution
- A test of significance and interval estimates based on least-squares theory in one-way, two-way, and three-way classified data
- Mahalanobis-D2, and Hotelling’s T2 statistics and their applications and properties, canonical correlations, principal component analysis, discriminant analysis

**Sampling Theory and Design of Experiments:**

- An outline of fixed-population and super-population approaches
- Distinctive features of finite population sampling, simple random sampling with and without replacement, probability sampling designs, stratified random sampling, systematic sampling, and its efficacy, cluster sampling, two-stage, and multi-stage sampling, probability proportional to size sampling with and without replacement, two-phase sampling, ratio and regression methods of estimation involving one or more auxiliary variables, 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, LSD, RBD, and their analyses, concepts of orthogonality and balance, incomplete block designs, BIBD, missing plot technique, factorial experiments, and 2n and 32, confounding in factorial experiments, the transformation of data Duncan’s multiple range test, split-plot and simple lattice designs

**SYLLABUS FOR PAPER II**

**Industrial Statistics:**

- Process and product control, general theory of control charts, different types of control charts for variables and attributes, cumulative sum chart, X, R, s, p, np, and c charts
- AQL, LTPD, and AOQL, Sampling plans for variables
- Single, double, multiple, and sequential sampling plans for attributes
- OC, AOQ, ASN, and ATI curves, concepts of producer’s and consumer’s risks
- Use of Dodge-Roaming tables
- Censored and truncated experiments for exponential models, problems in life testing
- 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.

**Optimization Techniques:**

- Different types of models in Operations Research, their general methods of solution, and construction
- Simulation and Monte-Carlo methods, sensitivity analysis,
- 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, transportation and assignment problems, rectangular games, methods of solution (graphical and algebraic), two-person zero-sum games
- Replacement of failing or deteriorating items, the concept of scientific inventory management and analytical structure of inventory problems, group and individual replacement policies, simple models with deterministic and stochastic demand with and without lead time, storage models with particular reference to dam type.
- The solution of statistical problems on computers using well-known statistical software packages like SPSS.
- Homogeneous discrete-time Markov chains, transition probability matrix, homogeneous continuous-time Markov chains, classification of states and ergodic theorems, M/M/1, M/M/K, G/M/1, and M/G/1 queues, Poisson process, elements of queuing theory

**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 numbers, 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.
- 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.
- 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, concept of structure and model for simultaneous equations, problem of identification-rank and order conditions of identifiability, two-stage least square method of estimation.

**Demography and Psychometry:**

- Demographic data from the census, NSS other surveys, registration, their limitations and uses, definition, use of vital rates and ratios, construction, measures of fertility, morbidity rate, reproduction rates, standardized death rate, complete and abridged life tables, construction of life tables from vital statistics and census returns, use of life tables, logistic and other population growth curves, fitting a logistic curve, stable population, population projection, 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, standard scores, Z-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.