Unit 1 - Analysis (Real & Complex) and Linear Algebra
Real number system; Sequences and series of real numbers and their convergence; Algebra
of continuous functions; Types of discontinuities; Differentiability of a function; Reimann
integration; Euclidean space Rn
; Metric on Rn
, Open and closed sets, Limit points of a set in a
metric space, Complete metric spaces, Compact sets. Complex numbers and their properties;
Analytic functions, Cauchy-Reimann equations; Line integrals; Cauchy’s theorem; Power
series; Singularities; Cauchy’s residue theorem; Contour integration.
Vector spaces, subspaces, linear independence, basis and dimension; Algebra of linear
transformations; Matrices and algebra of matrices, rank and determinant of matrices; Simultaneous
linear equations; Characteristic roots and vectors, Cayley-Hamilton theorem;
Classification of quadratic forms.
Unit 2 - Measure and Probability
Sequence of sets- limit superior and limit inferior, convergence; Field and σ-field of sets;
Measure, Lebesgue and Lebesgue-Stieltjes measures; Measurable functions and integration,
Basic integration theorems; Radon-Nikodym theorem-applications only; Measures in product
spaces.
Definitions of probability, addition theorem, independent events; Conditional probability,
multiplication theorem, Bayes’ theorem; Monotone and continuity properties of the probability
measure; Borel-Cantelli lemma, Borel zero-one law; Random variables, distribution
functions and their properties (univariate & bivariate), discrete and continuous random
variables, joint, marginal and conditional distributions, independent random variables; Moments,
Chebychev’s and Liapunov inequalities; Moment generating and characteristic functions;
Stochastic convergence of sequence of random variables; Law of large numbers and
central limit theorem.
Unit 3 - Ditribution Theory
Standard discrete and continuous distributions - binomial, Poisson, geometric, negative
binomial, hypergeometric; uniform, exponential, normal, Cauchy, Laplace, gamma, beta
and lognormal. Functions of random variables and their distributions, distributions of sum,
product and ratio of independent random variables; Sampling distributions of the mean and
variance of a random sample from the normal distribution; χ
2
, t, and F distributions; Order
statistics- basic distribution theory, order statistics arising from uniform and exponential distributions
and their properties; Multivariate normal distribution; Correlation and regression,
partial and multiple correlation coefficients.
Unit 4 - Statistical Inference
Estimation- Point estimation, properties of estimators; Cramer-Rao inequality, MVB
estimator, UMVU estimator, Rao-Blackwell and Lehmann-Scheffe theorems; Methods of
estimation; Basic concepts in interval estimation; Fundamental notions of hypothesis testing,
Neymann-Pearson lemma, most powerful and uniformly most powerful tests; Tests for the
mean and variance of normal distribution, tests for proportions, tests for simple, partial and
multiple correlation coefficients; Sequential probability ratio test; Non-parametric tests.
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Unit 5 - Sampling Theory and Design of Experiments
Sampling techniques- simple random sampling, stratified sampling, systematic sampling,
ratio and regression methods, cluster sampling; estimation problems in these sampling methods;
Sampling and non-sampling errors.
Linear models and estimation; Analysis of variance; Basic principles of experimental design;
Analysis of CRD, RBD, LSD and BIBD; Factorial experiments - 2n and 3n
experiments;
Confounding in 2n and 3n
experiments.
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