Name of the Speaker with affiliation

No. of Lectures

Detailed Syllabus

Teacher 1

5

Basic Probability Theory I: Orientation, Elementary concepts: experiments, outcomes, sample space, events. Discrete sample spaces and probability models. Combinatorial probability and urn models; Conditionalprobability and independence;

Teacher 2

5

Linear Algebra: Vector Spaces: Definition of Vector Spaces and Subspaces, Basis of a Vector Space, Linear Equations, Vector Spaces with an Inner Product

Theory of Matrices and Determinants: Matrix Operations, Elementary Matrices and Diagonal Reduction of a Matrix, Determinants, Transformations, Generalized Inverse of a Matrix, MatrixRepresentation of Vector Spaces, Bases, etc., Idempotent Matrices, Special Products of Matrices

Eigenvalues and Reduction of Matrices: Classification and Transformation of Quadratic Forms,

Roots of Determinantal Equations, Canonical Reduction of Matrices, Projection Operator,Further Results on g-Inverse, Restricted Eigenvalue Problem

Convex Sets in Vector Spaces: Definitions, Separation Theorems for Convex Sets

 Teacher 3

5

Markov Chain I:

Random Walk, Discrete Markov chains with countable state space.

Teacher 4

5

Real Analysis: Metric spaces, open/closed sets, Cauchy-Schwarz Inequality, Holder’s Inequality,

Hadamard’s Inequality, Inequalities Involving Moments, Convex Functions and Jensen’s Inequality, Inequalities in Information Theory, Stirling’s Approximation sequences, compactness,completeness, continuous functions and homeomorphisms, connectedness, product spaces, completeness of C[0, 1] and Lp spaces, Arzela-Ascoli theorem

Teacher 6

5

Markov Chain II: Classification of states - recurrence, transience, periodicity. Stationary distributions, limit theorems, positive and null

recurrence, ratio limit theorem, reversible chains. Several illustrations including the Gambler’s

ruin problem, queuing chains, birth and death chains etc.

Teacher 7

5

Basic Probability Theory III: Jointlydistributed random variables; Conditional expectation; Generating functions; Limit theorems;

Teacher 8

5

Markov Chain III: Poisson process, continuous time

markov chain with countable state space, continuous time birth and death chains.