Syllabus

Basic Probability Theory: Orientation, Elementary concepts: experiments, outcomes, sample space, events. Discrete sample spaces and probability models. Combinatorial probability and urn models; Conditional probability and independence; Random variables – discrete and continuous; Expectations, variance and moments of random variables; Markov inequality, Chebycheff inequality, First and second moment method (in tutorial with applications), Transformations of univariate random variables; Jointly distributed random variables; joint density and change of variables formula, independence and conditional distribution; probability transforms (if time permits), Modes of convergence, Borel-Cantelli lemma, WLLN under 2nd moment assumption and SLLN under 4th moment assumption, CLT (statement with applications)

Stochastic Processes: Simple symmetric Random Walk – reflection princliple, return to origin, Arc-sine law, Discrete Markov chains with countable state space, example of two state Markov chain, Recurrence, transience, criterion for recurrence, positive and null recurrence; Irreducibility and decomposition of state space, class property, stationary distributions, ratio limit theorem, periodicity, limit theorems, reversible chains. Generating functions; Several illustrations including the Gambler's ruin problem, queuing chains, birth and death chains etc., Branching process, Poisson process, continuous time Markov chain with countable state space, continuous time birth and death chains.

Analysis: Sequences, limits, limsup and liminf; infinte series, convergence and absolute convergence, Limit of a function at a point, continuity, intermediate value theorem, differentiation, Rolle's theorem, Taylor's theorem Power series – radius of convergence, term by term differentiation and integration, Abel's theorem, Cauchy-Schwarz Inequality for infinite series and integrals.

Linear Algebra: Matrices, Matrix Operations, Determinants, Matrix Representation of system of linear Equations, Vector Spaces and Subspaces, Basis of a Vector Space, Linear transformations, matrix representation, Inner product spaces and orthogonality, Cauchy-Schwarz Inequality for finite sum Eigenvalues, Classification and Transformation of Quadratic Forms, Diagonalisation of real symmetric matrices.

References:

• A. Ramachandra Rao and P. Bhimasankaram: Linear Algebra.
• G. F. Simmons: Introduction to Topology and Modern Analysis
• S. M. Ross: A first course in Probability
• Jacod & Protter: Probability Essentials
• W. Feller: Introduction to the Theory of Probability and its Applications, Vol. 1.
• P.G. Hoel, S.C. Port and C.J. Stone: Introduction to Stochastic Processes.
• (Lecture notes from the speakers, if available)

Accommodation

Participants

• Sleeper class train fare will be reimbursed to all the outstationed participants.
• The competent authority of NISER has approved free hostel accommodation from June 25, 2023 to July 22, 2023 for participants. Participants may bring their bed cover and other personal use materials.
• Food will be arranged for the participants from 25th June evening till 22nd July Morning.

Sponsors

[Parent] : Department of Atomic Energy, Govt. of India
[Host] : National Institute of Science Education and Research
[In collaborartion with ISI Kolkata]

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