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### Manjunath Krishnapur

Department of Mathematics, Indian Institute of Science, Bangalore 560 012

### Probability models (Fall 2011)

Tue, Thu, Fri 2:00 - 3:00, in LH III

Teaching assistants: Indrajit Jana and Kartick Adhikari (Tutorials: Tue 5:30 PM)
Some generalities: This is the first course in probability theory. Probability is an intuitive concept, and we often use the words such as probably, by chance, luckily, an unbelievable coincidence etc., which indicate uncertainty. Yet, probability theory is a part of mathematics. The aim in this course is to learn the basic concepts of this mathematical theory of probability, as well as develop an intuitive understanding of what these concepts mean and how they are applied in "real life situations". In this course we will stick to discrete probability spaces, where the mathematical sophistication needed is little. A natural continuation is the Probability theory course offered in the next semester, where you will learn the measure theoretical foundations of probability.

Grading: The final grade will be based on weekly homeworks (25%), two class tests (25% together) and the final exam (50%). Homeworks are due on Mondays (delayed submissions not accepted). Solving problems (preferably many more than given in the homeworks) is absolutely crucial to develop an understanding of the subject.

Texts and other resources: Sheldon Ross' Introduction to probability models is the primary text for the course. We will try to cover most of chapters 1-6 and chapter 11, although I will not strictly follow the book at all times. Another excellent book is William Feller's classic treatise An introduction to probability theory and its applications - vol. 1. Both these book have lots of examples and problems (and available at Tata book house). In addition to books, there are various resources on the web (for example) regarding basic probability.

Tentative list of topics: Probability space, events. Basic rules for calculating probabilities. Inclusion exclusion. Combinatorial examples. Independence and conditioning. Bayes formula. Random variables. Distribution function. Simulation. Examples: Binomial, Geometric, Poisson, Hypergeometric etc. Expectation, variance and covariance, generating functions. Independence and conditioning of random variables. Joint disribution, Distribution of the sum. The conceptual difficulty of picking a point at random from [0,1] or tossing a coin infinitely many times. Working rules for continuous distributions and densities. Simulation. Examples: Normal, exponential and gamma, uniform and beta, etc. Useful inequalities: Markov, Chebyshev, Cauchy-Schwarz, Bonferroni. IID random variables (existential issues overlooked). WLLN, SLLN (?), Demoivre-Laplace CLT, General CLT. Interesting processes: (a) Random walks (b) Branching processes (c) Polya's urn scheme. Markov chains: Basic theory. Many examples. Irreducible aperiodic MC. Stationary distribution. Reversibility. Recurrence and positive recurrence. Convergence theorem. Strong Markov property. CLT for statistics. Mixing time idea introduced. Continuous time markov chains: Infinitesimal description, generator. Poisson process. Examples.

Progress of lectures:
 Lec 1 04 Aug Introductory lecture Lec 2 05 Aug Probability space, events. Lec 3 09 Aug Rules for calculating probabilities. Homework 1 Due on 16th Lec 4 11 Aug More examples. Inclusion-Exclusion. Lec 5 16 Aug Bonferroni's inequalities. Applications. Lec 6 18 Aug Infinite sums, absolute convergence. Lec 7 19 Aug Random variables. Examples. Lec 8 23 Aug Distributions. Homework 2 Due 2nd Sep Lec 9 25 Aug Countable additivity revisited. CDF. Lec 10 26 Aug Continuous CDFs and a conceptual difficulty. Lec 11 30 Aug Working rules for continuous distributions and densities. Lec 12 02 Sep Examples of continuous distributons Lec 13 06 Sep Simulating random variables Homework 3 Due 16th Sep Lec 14 08 Sep Change of variable in one dimension Lec 15 09 Sep Examples of change of variables Lec 16 13 Sep Joint distributions, CDFs, densities Lec 17 15 Sep Examples of joint distributions Lec 18 16 Sep Change of variables in higher dimensions Some notes Lec 19 20 Sep Change of variables - examples Lec 20 22 Sep Independence and conditioning Homework 4 Due 07 Oct Lec 21 23 Sep Independence and conditioning 01 Oct FIRST CLASS TEST - 10:30 AM Lec 22 04 Oct CLASS CANCELLED Lec 23 07 Oct Independence Lec 24 11 Oct Conditioning on events Lec 25 13 Oct Conditioning on random variables Lec 26 14 Oct Conditioning on random variables Lec 27 18 Oct Expectation when a pmf or pdf exists Lec 28 20 Oct Expectation - general properties. Cauchy-Schwarz inequality. Homework 5 Due 03 Nov Lec 29 21 Oct Variance, covariance, correlation, conditional expectation Lec 30 25 Oct Markov and Chebyshev ineuqalities. WLLN. Lec 31 27 Oct Applications: Monte-Carlo integration. Wierstrass' theorem. Lec 32 28 Oct CLT for Bernoulli random variables. Lec 33 03 Nov Markov chains - definition and examples. Lec 34 04 Nov Lec 35 08 Nov Lec 34 11 Nov Homework 6 Due 24 Nov Lec 33 15 Nov Lec 34 17 Nov Lec 33 18 Nov Lec 34 22 Nov Notes on Markov chains Lec 35 23 Nov Problems