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### Manjunath Krishnapur

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

### Probability and statistics (Fall 2013)

Mon, Wed, Fri 9:30-10:00

Teaching assistants: Kartick Adhikari, Nanda Kishore Reddy, Nidhin Koshy Vaidhiyan, Rajeev Gupta, Pranav Haridas (Tutorials (batchwise): Mon-Thu 2:00-3:00)
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 (25%), two mid terms (25% 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:
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. An introductory statistics course on Udacity: http://www.udacity.com/overview/Course/st101/CourseRev/1
4. 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.

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.

Notes and homeworks: