Course Archives Theoretical Statistics and Mathematics Unit

Course: Probability III Level: Undergraduate Time: Currently not 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 Top of the page Past Exams

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

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