Probability Theory
Lectures
11/08 The need for σ-algebra structure in the domain of definition of a measure, Lebesgue measure, and Lebesgue σ-algebra. An example of a real subset not in Lebesgue σ-algebra. Definition of an abstract measure, The definition of probability measure.
16/08 How to define a random variable mathematically? Explanation of Kolmogorov's model of random variables, Definition of measurable functions, Notion of "almost sure". Construction of Bernoulli and Normal random variable. Note
18/08 Various notions of convergence of a sequence of random variables.
22/08 Measure theoretic interpretation of expectation. Integration of simple functions. Definition of integrable functions and their integration, convergence of a sequence of integrals, Notion of absolute continuity of measures, Radon-Nikodym Theorem, R-N derivative. Note. Statement of all major theorems on convergence of integrals. Note
23/08 Revisit definition of algebra, σ-algebra and monotone class. Definition of σ-algebra generated by a collection of subsets. A monotone algebra is a σ-algebra. A monotone class generated by an algebra is a σ-algebra generated by that algebra (Monotone Class Theorem for sets). Note
25/08 Definition of π-class, λ-class generated by a collection of subsets. A collection that is a π-class, as well as a λ-class is a σ-algebra. The λ-class generated from a π-class is the σ-algebra generated by that π-class. Note
29/08 Definition of Borel σ-algebra, σ-algebra generated by a random variable. Independence of random numbers. The distribution measure of a random variable. Note
30/08 Definition of Cumulative Distribution Function(CDF) and example; Properties of CDF. Definition of semi-algebra, extension of measure from a semi-algebra, The Lebesgue Stieltjes measure induced by a distribution function. Existence of a distribution measure for a given CDF; Existence of a random variable for a given CDF; Jump of CDF, countability of the set of jump points; Decomposition of a CDF as a convex combination of discrete and continuous CDFs. Note
01/09 Example of the conditional expectation of the outcome of a rolling dice given a dependent random variable. Definition of the conditional expectation of a random variable(having finite expectation) given a σ-algebra, or given a measurable set. Definition of the conditional probability of an event given a measurable set. Proof of P(A|B)P(B)=P(A&B). View conditional expectation as a Radon-Nikodym derivative. Note
05/09 Properties of the conditional expectation: linearity, monotonicity, MCT, FL, DCT. If X is G-measurable, E(XY|G)= XE(Y|G), and hence E(X|G)=X, with proofs. Note
06/09 Revisiting the definition of independence of random variables. Proof of the fact that the expectation of product of independent numbers (having finite expectation) is the product of expectations. If X is independent of G, then E(X|G)= E(X). Note
08/09 Conditional Expectation as the projection of L^2 random variables in the subspace of random variables measurable w.r.t. a sub σ-algebra. Proof of V(X)>=V(E[X|G]), Note. Association inequality Note.
12/09 Independence of X & Z does not imply "E[X|Y, Z]=E[X|Y]". An example to illustrate this. Statement: If σ(X) and D1 are independent to D2, then E[X|D1V D2]=E[X|D1 ]. Properties of conditional probability, Definition of Regular conditional probability. Note
13/09 Class Cancelled.15/09 Conditional expectation as integration wrt regular conditional probability measure. Definition of Conditional Distribution of X given a sub σ-algebra G. Recollection of Carathéodory extension theorem. Note
19/09 Quiz 1 out of 15 [Max=15.0, Min=1.0, Mean=7.4, Median=6.5, STD=4.4]20/09 Definition of the n-dimensional distribution function. Proof of Doob's theorem on the existence of regular conditional distribution measure of X given a sub σ-algebra G. Note
22/09 Revision of previously taught concepts and tutorial.
28/09 10.00 am to 12.00 Midsem Exam out of 30 [Max=30.0, Min=2.0, Mean=15.4, Median=12.5, STD=9.5]
Mid Semester Break06/10 Midsem correction checking and Solution keys.
10/10 Given a sequence of events, the notion of occurrence of infinitely many often, Kolmogorov's 0-1 Law, Statement of Borel Cantelli Theorem
11/10 Proof of Borel Cantelli Theorem. Definition of Characteristic Function, Statement, and proof of the Lévy Inversion Formula. Note
13/10 Levy's Continuity Theorem on the sequence of CDFs. Motivation for the Central Limit Theorem (CLT). Lindeberg condition on a sequence of independent random variables, Statement, and proof of CLT of Lindeberg and Feller. Note
17/10 Definition of filtration, martingale, the example of SSRW. Stopping time relative to a filtration. [Notes]
18/10 Definition and few properties of the stopping time σ-algebra. Closed martingales [Notes]
20/10 Class Cancelled
24/10 Closed Holiday
25/10 Class Cancelled27/10 Decomposition of sub- and super-martingale, an example of a bounded martingale, a convex function of a martingale is a submartingale, Kolmogorov's inequality, and its extension. (Extra half an hour) [Notes]
31/10 Almost sure and L2 convergence of martingale with bounded variance [Note] Supplementary:- 1a, 1b and 2
01/11 Proof of Doob's up-crossing inequality assuming a lemma, Proof of submartingale convergence theorem [Notes]
03/11 Quiz 2 out of 15 [Max= 15.0, Min= 5.0, Mean= 10.8, Median= 11.0, STD= 3.2], Proof of the lemma (Extra half an hour) [Notes]07/11 Tutorial and solution key to Quiz 2 questions.
08/11 Closed Holiday10/11 Doob's Optional Stopping Theorem, Wald's Equation. Note
14/11 Strong and Weak law of large number part 1
15/11 Strong and Weak law of large number part 2 Combined Note
17/11 Not from the syllabus: Example of a martingale that is not Markov. Open discussions based on the doubts of the participants. Note
21/11 Quiz 3 out of 15, Max=13.0, Min=2.0, Mean=8.8, Median=10.0, STD=3.722/11 Tutorial M-Z WLLN Note Recorded Lecture
Supplementary results:- 3, 4-5, 6-7
28/11 Endsem Test [Max=35.5, Min=0.0, Mean=22.4, Median=22.3, STD=9.6 ]