Syllabus for UPSC Statistics Optional Subject:

Paper I:
1. Probability
- Sample space and events
- Probability axioms and properties
- Conditional probability and Bayes' theorem
- Random variables and probability distributions
- Joint distributions and marginal and conditional distributions
- Independence of random variables
- Mathematical expectation
- Moment generating function and characteristic function
- Standard probability distributions
- Random sampling and sampling distributions
- Central limit theorem

2. Statistical Methods
- Collection, classification, and presentation of data
- Measures of central tendency and dispersion
- Correlation and regression analysis
- Index numbers
- Time series analysis
- Vital statistics
- Design of experiments
- Sampling techniques and sampling distributions
- Estimation and testing of hypotheses
- Analysis of variance

3. Statistical Inference
- Concept of statistical inference
- Estimation of parameters
- Testing of hypotheses
- Non-parametric tests
- Analysis of categorical data
- Regression and correlation analysis
- Analysis of variance

Paper II:
1. Linear Models
- Simple linear regression analysis
- Multiple linear regression analysis
- Analysis of covariance
- Multiple and partial correlation analysis
- Principles of design of experiments
- Analysis of variance for simple and factorial experiments
- Designs for comparing treatments
- Fractional factorial experiments

2. Multivariate Analysis
- Multivariate normal distribution
- Inference about mean vector and covariance matrix
- Principal component and factor analysis
- Discriminant analysis
- Cluster analysis
- Canonical correlation analysis
- Structural equation modeling

3. Sampling Techniques
- Concepts of population and sample
- Parameters and statistics
- Sampling and census methods
- Simple random sampling
- Stratified random sampling
- Systematic sampling
- Cluster sampling
- Multi-stage sampling
- Sampling with probability proportional to size
- Ratio and regression methods of estimation

4. Optimization Techniques
- Linear programming
- Simplex method
- Duality and sensitivity analysis
- Transportation and assignment problems
- Integer programming
- Goal programming
- Non-linear programming
- Kuhn-Tucker conditions
- Convex programming
- Quadratic programming
- Dynamic programming

5. Demography and Vital Statistics
- Measures of fertility, mortality, and migration
- Life tables and their applications
- Stable population models
- Measures of disease frequency and morbidity
- Standardization of rates
- Decomposition of demographic rates
- Vital statistics and their sources
- Census and registration data

Overall, the syllabus for the UPSC Statistics Optional Subject covers various topics in probability, statistical methods, statistical inference, linear models, multivariate analysis, sampling techniques, optimization techniques, and demography and vital statistics.