The Mathematical Statistics (MS) test paper comprises of Mathematics (40% weightage) and Statistics (60% weightage).
Convergence of sequences of real numbers, Comparison, root and ratiotests for convergence of series of real numbers.
Limits, continuity and differentiability of functions of one and two variables.Rolle's theorem, mean value theorems, Taylor's theorem, indeterminate forms, maxima and minimaof functions of one and two variables.
Fundamental theorems of integral calculus. Double and triple integrals, applicationsof definite integrals, arc lengths, areas and volumes.
Rank, inverse of a matrix. Systems of linear equations. Linear transformations, eigenvaluesand eigenvectors. Cayley-Hamilton theorem, symmetric, skew-symmetric and orthogonal matrices.
Ordinary differential equations of the first order of the form y'=f(x,y), Bernoulli's equation, exact differential equations, integrating factor, orthogonal trajectories, homogeneous differential equations, variable separable equations, linear differential equations of second order with constant coefficients, method of variation of parameters, Cauchy-Euler equation.
Axiomatic definition of probability and properties, conditional probability, multiplication rule. Theorem of total probability. Bayes' theorem and independence of events.
Probability mass function, probability density function and cumulative distributionfunctions, distribution of a function of a random variable. Mathematical expectation, moments andmoment generating function. Chebyshev's inequality.
Binomial, negative binomial, geometric, Poisson, hypergeometric, uniform, exponential, gamma, beta and normal distributions. Poisson and normal approximations of a binomial distribution.
Chi-square, t and F distributions, and their properties.
Weak law of large numbers. Central limit theorem (i.i.d.with finite variance case only).
Unbiasedness, consistency and efficiency of estimators, method of moments and method of maximum likelihood. Sufficiency, factorization theorem. Completeness, Rao-Blackwell and Lehmann-Scheffe theorems, uniformly minimum variance unbiased estimators. Rao-Cramer inequality. Confidenceintervals for the parameters of univariate normal, two independent normal, and one parameter exponentialdistributions.
Basic concepts, applications of Neyman-Pearson Lemma for testing simpleand composite hypotheses. Likelihood ratio tests for parameters of univariate normal distribution.