IIT JAM Mathematical Statistics Syllabus
Subject: The Mathematical Statistics (MS) test paper comprises of Mathematics (40% weightage) and Statistics (60%weightage).
Marking Scheme: 1/4 mark will be deduct for wrong answer in objective type section.
Sequences and Series:
Convergence of sequences of real numbers, Comparison, root and ratio tests for convergence of series of real numbers.
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Limits, continuity and differentiability of functions of one and two variables. Rolle’s theorem, mean value theorems, Taylor’s theorem, indeterminate forms, maxima and minima of functions of one and two variables.
Fundamental theorems of integral calculus. Double and triple integrals, applications of definite integrals, arc lengths, areas and volumes.
Rank, inverse of a matrix. systems of linear equations. Linear transformations, eigenvalues and eigenvectors. Cayley-Hamilton theorem, symmetric, skew- symmetric and orthogonal matrices.
Ordinary differential equations of the first order of the form y’ = f(x,y). Linear differential equations of the second order with constant coefficients.
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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 distribution functions, distribution of a function of a random variable. Mathematical expectation, moments and moment 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.
Joint, marginal and conditional distributions. Distribution of functions of random variables. Product moments, correlation, simple linear regression. Independence of random variables.
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).
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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. Confidence intervals for the parameters of univariate normal, two independent normal, and one parameter exponential distributions.
Testing of Hypotheses:
Basic concepts, applications of Neyman-Pearson Lemma for testing simple and composite hypotheses. Likelihood ratio tests for parameters of univariate normal distribution.