Discrete Mathematics

Syllabus:

i) BASIC COUNTING TECHNIQUES: Double-counting, Averaging principle, Inclusion- Exclusion principle. Euler indicator, Mobius function and inversion formula. Recursions and generating functions.
ii) PIGEONHOLE PRINCIPLE: The Erdos-Szekeres theorem. Mantels theorem. Turans theorem, Dirichlets theorem, Schurs theorem, Ramsey theory.
iii) GRAPHS: Eulers theorem and Hamilton Cycles. Spanning Trees. Cayleys theorem and Spanning trees.
iv) SYSTEMS OF DISTINCT REPRESENTATIVES: Halls marriage theorem, Applications to latin rectangles and doubly stochastic matrices, Konig-Egervary theorem, Dilworths theorem, Sperners theorem.
v) FLOWS IN NETWORKS: Max-flow min-cut theorem, Ford-Fulkerson theorem, Integrality theorem for max-flow.
vi) LATIN SQUARES AND COMBINATORIAL DESIGNS: Orthogonal Latin squares, Existence theorems and finite projective planes. Block designs. Hadamard designs, Incidence matrices. Steiner triple systems.

Reference Texts:

(a) S. Jukna: Extremal Combinatorics.
(b) J. H. van Lint & R. M. Wilson: A Course in Combinatorics.
(c) D. B. West: Introduction to Graph Theory.
(d) R. A. Beeler: How to Count: An Introduction to Combinatorics and Its Applications.
(e) H. J. Ryser: Combinatorial Mathematics.

https://www.isibang.ac.in/~adean/infsys/database/Bmath/DM1.html

Introduction to Statistical Inference

Syllabus:

For Prob III: All limit theorems used in this course will be stated in context with applications. These can be proved in Probability III rigorously.
i) Sufficiency, Exponential family, Bayesian methods, Moment methods, Maximum likelihood estimation.
ii) Criteria for estimators; UMVUE, Fisher Information.
iii) Multivariate normal distribution: Marginals, Conditionals; Distribution of linear forms.
iv) Order statistics and their distributions.
v) Large sample theory: Consistency, asymptotic normality, asymptotic relative efficiency.
vi) Elements of hypothesis testing; Neyman-Pearson Theory, UMP tests, Likelihood ratio and related tests, Large sample tests.
vii) Confidence intervals.

Reference Texts:

(a) George Casella and Roger L Berger: Statistical Inference.
(b) Peter J Bickel and Kjell A Doksum: Mathematical Statistics.
(c) Erich L Lehmann and George Casella: Theory of Point Estimation.
(d) Erich L Lehmann and Joseph P Romano: Testing Statistical Hypotheses.

https://www.isibang.ac.in/~adean/infsys/database/Bmath/ISI.html