**SYLLABUS**

**MATHEMATICAL STATISTICS (MS)**

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.

** Probability**:
Axiomatic definition of probability and properties, conditional probability,
multiplication rule. Theorem of total probability. Bayes' 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. Joint
moment generating function. 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.