Calculus: Finite, countable and uncountable sets, Real number system as a complete ordered field, Archimedean property; Sequences and series, convergence; Limits, continuity, unfirom continuity, differentiability, mean value theorems; Riemann integration, Improper integrals; Functions of two or three variables, continuity, directional derivatives, partial derivatives, total derivative, maxima and minima, saddle point, method of Lagrange's multipliers; Double and Triple integrals and their applications; Line integrals and Surface integrals, Green's theorem, Stokes' theorem, and Gauss divergence theorem.
Linera Algebra: Finite dimensional vector spaces over real or complex field; Lineaer transformations and their matrix representations, rank; systems of linear equations, eigenvalues and eigenvectors, minimal polynomial, Vayley-Hamilation Theorem, diagonalization, Jordan canonical form, symmetric, skew-symmetric, Hermitian, skew-Hermitian, orthogonal and unitary matrices; Finite dimensional inner product spaces, Gram-Schmidt orthonormalization process, definite forms.
Probability: Classical, relative frequency and axiomatic definitions of probability, conditional probability, Bayes' theorem, independent events; Random variables and probability distributions, moments and moment generating functions, quantiles; Standard discrete and continuous univariate distributions; Probability inequalities (Chebyshev, Markov,Jensen); Function of a random variable; jointly distributed random variables, marginal and conditional distributions; product moments, joint moment generating functions, independence of random variables; Transformations of random variables, sampling distributions, distribution of order statistics and range; Characteristics functions; Modes of convergence; Weak and strong laws of large numbers; Central limit theorem for i.i.d random variables with existence of higher order moments.
Stochastic Processes: Markov chains with finite and countable state space, classification of states, limiting behaviour of n-step transition probabilties, stationary distribution, Poisson and birth-and-death processes.
Inference: Unbiasedness, consistency, sufficiency, completeness, uniformly minimum variance unbiased estimation, method of moments and maximum likelihood estimations; Confidence intervals' Tests of hypotheses, most powerful and uniformly most powerful tests, likelihood ratio tests, large sample test, Sign test, Wilcoxon signed rank test, Mann-Whitney U test, test for independence and Chi-square test for goodness of fit.
Regression Analysis: Simple and multiple linear regression, polynomial regression, estimation, confidence intervals and testing for regression coefficients; Partial and mutliple correlation coefficients.
Multivariate Analysis: Basic properties of multivariate normal distribution; Multinomial distribution; Wishart distribution; Hotellings T2 and related tests; Prinicipal component analysis; Discriminant analysis; Clustering.
Design of Experiments: One and two-way ANOVA, CRD, RBD, LSD, 2 and 2 Factorial experiments.