IIT JAM 2019 Syllabus for Mathematical Statistics (MS)

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To M.Sc. (Four Semesters), Joint M.Sc.-Ph.D., M.Sc.-Ph.D. Dual Degree, etc, to provide admissions Joint Admission Test for M.Sc. (JAM) is being conducted from 2004. At the IITs and Integrated PhD Degree Programmes, for bright students, programmes at IISc for consolidating Science as a career option are being conducted.

Apart from developing academic talent leading to a challenging and rewarding professional life, To provide opportunities to the students, the curricula for these programmes are designed. At IITs and IISc, in their respective disciplines, comparable to the best in the world, these postgraduate programmes offer high-quality education.

Syllabus of Mathematical Statistics – IIT JAM 2019

Mathematics

Sequences and Series

Comparison, the convergence of sequences of real numbers, root and ratio tests for convergence of series of real numbers;

Differential Calculus

Rolle’s theorem, mean value theorems, Taylor’s theorem; Limits, continuity and differentiability of functions of one and two variables; indeterminate forms, maxima and minima of functions of one and two variables;

Integral Calculus

Double and triple integrals, applications of definite integrals; Fundamental theorems of integral calculus; arc lengths, areas and volumes;

Matrices

Cayley-Hamilton theorem, symmetric, skew-symmetric and orthogonal matrices; Rank, the inverse of a matrix; Systems of linear equations; linear transformations, eigenvalues and eigenvectors;

Statistics 

Probability

Theorem of total probability; Axiomatic definition of probability and properties, conditional probability, multiplication rule;  Bayes’ theorem and independence of events;

Random Variables

Mathematical expectation, moments and moment generating function. Chebyshev’s inequality;
Probability mass function, probability density function and cumulative distribution functions, distribution of a function of a random variable;

Standard Distributions

Gamma, beta and normal distributions; Binomial, negative binomial, geometric, Poisson, hypergeometric, uniform, exponential, Poisson and normal approximations of a binomial distribution;

Joint Distributions

Product moments, correlation, simple linear regression; Joint, marginal and conditional distributions; Distribution of functions of random variables; Joint moment generating function; Independence of random variables;

Sampling Distribution

Chi-square, t and F distributions, and their properties;

Limit Theorems

Central limit theorem (i.i.d.with finite variance case only); Weak law of large numbers;

Estimation

Sufficiency, factorization theorem; Completeness, Rao-Blackwell and Lehmann-Scheffe theorems, uniformly minimum variance unbiased estimators; Unbiasedness, consistency and efficiency of estimators, a method of moments and method of maximum likelihood; Rao-Cramer inequality. Confidence intervals for the parameters of univariate normal, two independent normals, and one parameter exponential distributions;

Testing of Hypotheses

Likelihood ratio tests for parameters of the univariate normal distribution; Basic concepts, applications of Neyman-Pearson Lemma for testing simple and composite hypotheses;

Reference books for Mathematical Statistics- IIT JAM 2019

MATHEMATICS

  • Mathematical Analysis by Apostol
  • Mathematical Analysis by S.C.Malik
  • The principle of Mathematical Analysis by Rudi
  • Ordinary Differential Equation by Peter J. Collins, G.F. Simmons, M.D. Raisinghania
  • Vector Calculus by Murray R. Spiegel (Schaum’s), A.R.Vasishtha
  • Integral Calculus by F. Ayres (Schaum’s), Gorakh Prasad

STATISTICS                      

  • An introduction to probability and statistics by V.K. Rohatgi
  • Introduction to Mathematical statistics by Hogg & Craig
  • Introduction to the theory of statistics by Mod & Graybill
  • Fundamentals of Mathematical statistics by S.C. Gupta & V.K. Kapoor

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