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Manjunath Krishnapur

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

Probability and statistics (Fall 2013)

Mon, Tue, Fri : 11-12, 9:30-10:30, 11-12, respectively.

Teaching assistants: Aadirupa Saha, Chandramouli Kamanchi, Somnath Pradhan (Tutorials : Wed 11:00-12:00)
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 (20%), two mid terms (30% together) and the final exam (50%). Homeworks are due on Mondays in class (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: My lecture notes will be updated often. 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.

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.

Homeworks:
Old papers: Have a look at the papers I set in previous years. Final (2013), Final (2012). And also 2nd mid-term (2013) and 2nd mid-term (2012). There could be a couple of problems on topics that I did not cover in class this year. Hopefully you can make out which!