Manjunath Krishnapur

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

Probability and statistics (Aug-Dec 2018)

Mon, Wed, Fri : 9:30-10:30

Teaching assistants: Anwoy Maitra, Sanjay Jhawar, Lakshmi Kanta Mahato, Raghavendra Tripathi, G V K Teja (Tutorials : Tue 9:30-10:30)
Some generalities: This is a first course in probability theory and statistics. About the first 20 lectures will be spent on basic probability and then another 20 lectures on methods in statistics. Grading: The final grade will be based on weekly homeworks/quizzes (30% together), one mid term (20%) and the final exam (50%). Homeworks are due in tutorials (delayed submissions not accepted). Solving problems (preferably many more than given in the homeworks) is absolutely crucial to develop an understanding of the subject. We shall try to put up problem sets every week. Some of them will be homework problems. Problems in the quizzes will be similar to problems in problem sets. Roughly, homework submission and quizzes will be be on alternate weeks. Homeworks must be submitted on the due date, in tutorials (unless otherwise indicated). Late homeworks will not be accepted. One quiz and one homework will be omitted when marking, to allow for unforeseeable emergencies.

Texts and other resources: My lecture notes on probability and statistics will be updated often. The same content, but with many additional interactive experiments and better format for reading, were created by Siddhartha Gadgil and may be found here. For more explanations on various concepts, you may consult the following books.
  1. Sheldon Ross' Introduction to probability and statistics for scientists and engineers
  2. William Feller's classic treatise An introduction to probability theory and its applications - vol. 1
  3. Freedman, Pisani, Purves Statistics
  4. An introductory statistics course on Udacity: http://www.udacity.com/overview/Course/st101/CourseRev/1
  5. Many resources can be found online. A fascinating one is David Aldous' homepage.
I will not follow any book verbatim. However most of the material is contained in the first book which is very well-written. The second book is for those who want more mathematics and probability. The third book is for those who prefer to get the ideas of statistics with minimal use of sophisticated mathematics. All these three books are beautifully written.

Old content: Have a look at the homeworks/papers I set in previous years. The course page from 2016 with all homeworks etc. And from even earlier time: Final (2013), Final (2012). And also 2nd mid-term (2013) and 2nd mid-term (2012).

Tentative list of topics:
Probability: 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). Weak law of large numbers, Demoivre-Laplace CLT, General CLT.
Statistics: Summarizing the data. Mean, median, quantiles, standard deviation etc. Histograms, scatter plots etc. Linear regression. Estimation of parameters (Least squares, maximum likelihood, minimax). Testing of hypotheses. Kolmogorov-Smirnov and Chi-squared tests. Many examples.

Quizzes: 28/Aug, 04/Sep, 18/Sep, 23/Oct, 06/Nov

Homeworks: For practise problems look at the previous years, especially my course page from 2016 or this compiled list of probability and statistics problems. And of course problems from the books cited above. Here only the problems to submit as homework and some new problems are put up.