The Mathematical Statistics (MS) test paper comprises of Mathematics (40% weightage) and Statistics (60% weightage).
Mathematics:
Sequences and Series: Convergence of sequences of real numbers, Comparison, root and ratio tests for convergence of series of real numbers.
Differential Calculus:
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.
Integral Calculus:
Fundamental theorems of integral calculus. Double and triple integrals,
applications of definite integrals, arc lengths, areas and volumes.
Matrices:
Rank, inverse of a matrix. systems of linear equations. Linear
transformations, eigenvalues and eigenvectors. Cayley-Hamilton theorem,
symmetric, skew-symmetric and orthogonal matrices.
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.
Statistics:
Probability:
Axiomatic definition of probability and properties, conditional
probability, multiplication rule. Theorem of total probability. Bayes's
theorem and independence of events.
Random Variables:
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.
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.
Limit Theorems: Weak law of large numbers. Central limit theorem (i.i.d.with finite variance case only).
Estimation:
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.
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