Joint Admission Test to M.Sc. - Indian Institute of Technology (IIT)

STATISTICS (weightage : 50%)

Probability: Axiomatic definition of probability and properties. Conditional probability, multiplication rule. Theorem of total probability. Bayes' theorem and independence of events.

Random Variables: Distribution functions. Probability mass function and probability density function. Distribution of a function of a random variable. Mathematical expectation. Moments and moment generating function. Chebyshev's inequality.

Standard Distributions: Binomial, negative binomial, geometric, Poisson, hypergeometric, uniform, exponential, gamma, beta and normal distributions. Poisson and normal approximations of a binomial distribution.

Joint Distributions: Joint, marginal and conditional distributions. Distribution of functions of random variables. Product moments, correlation, simple linear regression. Independence of random variables.

Sampling distributions: Chi-square, t and F distributions, and their properties

Estimation and Testing: Unbiased estimators. Method of moments, method of maximum likelihood. Tests of hypotheses for binomial proportion(s), mean(s) and variance(s) of normal populations(s) and associated confidence intervals.

MATHEMATICS (weightage : 25%)

Functions of two and three Variables: Limit, continuity and differentiability. Partial derivatives. Maxima and minima. Double and triple integrals. Surface areas and volumes.

Differential Equations: Ordinary differential equations of the first order of the form y’ = f (x,y). Linear differential equations of the second order with constant coefficients. Euler-Cauchy equation. Method of variation of parameters.

Linear Algebra and Algebra: Dimension of a vector space, linear transformations. Groups, normal subgroups. Lagrange’s theorem for finite groups.

SECTION C
STATISTICS (weightage : 25%)

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

Point Estimation: Minimum mean square error estimators. Sufficient statistic. Factorization theorem. Complete statistic. Rao-Blackwell theorem. Uniformly minimum variance unbiased estimators. Cramer-Rao inequatlity.

Testing of Hypotheses: Basic concepts and simple applications of Neyman-Pearson Lemma for testing simple and composite hypotheses. Likelihood ratio test for parameters of univariate normal distributions.

Interval Estimation: Concepts of confidence intervals and confidence coefficients. Confidence intervals for the parameters of univariate normal, two-independent normal and one parameter (mean) exponential distributions.