Course Archives Theoretical Statistics and Mathematics Unit
Course: Probability III
Level: Undergraduate
Time: Currently not offered
Past Exams


- 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

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