Mathematical Science Syllabus Paper 1 Section B Unit 10 To 14,17,18
10. Data Analysis Basic Concepts: Graphical representation, measures of central tendency and dispersion. Bivariate data, correlation and regression. Least squares - polynomial regression, Applications of normal distribution.
11. Probability: Axiomatic definition of probability. Random variables and distribution functions (univariate and multivariate); expectation and moments; independent events and independent random variables; Bayes' theorem; marginal and conditional distribution in the multivariate case, covariance matrix and correlation coefficients (product moment, partial and multiple), regression.
Moment generating functions, characteristic functions; probability inequalities (Tchebyshef, Markov, Jensen). Convergence in probability and in distribution; weak law of large numbers and central limit theorem for independent identically distributed random variables with finite variance.
12. Probability Distribution: Bernoulli, Binomial, Multinomial, Hypergeomatric, Poisson, Geometric and Negative binomial distributions, Uniform, exponential, Cauchy, Beta, Gamma, and normal (univariate and multivariate) distributions Transformations of random variables; sampling distributions. t, F and chi-square distributions as sampling distributions, Standard errors and large sample distributions. Distribution of order statistics and range.
13. Theory of Statistics: Methods of estimation: maximum likelihood method, method of moments, minimum chi-square method, least-squares method. Unbiasedness, efficiency, consistency. Cramer-Rao inequality. Sufficient Statistics. Rao-Blackwell Theorem. Uniformly minimum variance unbiased estimators. Estimation by confidence intervals. Tests of hypotheses: Simple and composite hypotheses, two types of errors, critical region, randomized test, power function, most powerful and uniformly most powerful tests. Likelihood-ratio tests. Wald's sequential probability ratio test.
14. Statistical methods and Data Analysis: Tests for mean and variance in the normal distribution: one-population and two- population cases; related confidence intervals. Tests for product moment, partial and multiple correlation coefficients; comparison of k linear regressions. Fitting polynomial regression; related test. Analysis of discrete data: chi-square test of goodness of fit, contingency tables. Analysis of variance: one-way and two-way classification (equal number of observations per cell). Large-sample tests through normal approximation. Nonparametric tests: sign test, median test, Mann-Whitney test, Wilcoxon test for one and two-samples, rank correlation and test of independence.
17. Finite Population: Sampling Techniques and Estimation: Simple random sampling with and without replacement. Stratified sampling; allocation problem; systematic sampling. Two stage sampling. Related estimation problems in the above cases.
