Course Archives Theoretical Statistics and Mathematics Unit

Course: Probability III Instructor: Vivek Kumar Room: G23 Level: Undergraduate Time: Currently offered

Syllabus Past Exams Syllabus: - Sigma-algebras, axioms of probability, pi - lamda theorem (proof can be skipped), uniqueness of extension for probability measures. Examples of countable probability spaces, Borel sigma-algebra on the real line and standard probability distributions on the real line. - Construction of Lebesgue measure (statement alone). Random variables and examples. Push-forward of a probability measure (sketch of proof) . Borel probability measures on Euclidean spaces as push-forward of Lebesgue measure (statement alone); Cumulative distribution function and properties. - General definition of expectation and properties. Change of variables. Review of conditional distribution and conditional expectation, General definition, Examples. - Limit theorems: Monotone Convergence Theorem (MCT) (without proof), Fatous Lemma, Dominated Convergence Theorem (DCT), Bounded Convergence Theorem (BCT), Cauchy-Schwartz, Jensen and Chebyshev inequalities. - Different modes of convergence and their relations, Weak Law of large numbers, First and Second Borel-Cantelli Lemmas, Strong Law of large numbers (proof under finite variance). - Characteristic functions, properties, Inversion formula and Levy continuity theorem (statements only), CLT in i.i.d. finite variance case. Slutskys Theorem. - Introduction to Finite Markov chains - Definition. Random mapping representation. Examples. Irreducibility and aperiodicity. Stationary distribution and reversibility. Random walks on graphs. Reference Texts: (a) N. Lanchier: Stochastic Modelling. (b) W. Feller: Introduction to Probability: Theory and Applications - Vol. I and II.. (c) J. Pitman: Probability. (d) Sheldon Ross: Probability Models. (e) Santosh S. Venkatesh: Theory of Probability - Explorations and Applications. (f) R. Meester: A Natural Introduction to Probability Theory. (g) S. R. Athreya and V. S. Sunder: Measure and Probability Evaluation: Mid-term 30 marks Assignment 20 marks Final Exam 50 marks Total 100 marks Top of the page Past Exams

Midterm 24.pdf 25.pdf Semestral 24.pdf Supplementary and Back Paper 25.pdf Top of the page

[ Semester Schedule ][ Statmath Unit ] [Indian Statistical Institute]