# SI417: Introduction to Probability Theory

Prof. Koushik Saha

Course Description:

This is the 1st course in probability theory. The aim in this course is to learn the
basic concepts of this mathematical theory of probability, as well as to develop an intuitive understanding of what these concepts mean and how they are applied in “real life situations”. A prerequisite course to do any other minor course in statistics.

Course Contents:

1. Sample spaces, events, sigma algebra, probability space
2. Properties of probabilities, conditional probability, independence, Bayes formula
3. Poly’s urn model, some combinatorial problems
4. Discrete random variables, probability mass function, independent random variable, sum of random variables, random vector
5. Expectation of discrete random variable, properties of expectation and variance,
6. Continuous random variable, distribution function, density of a continuous random variable, expectation, change of variable formula, random vector
7. Joint distribution of random variables, joint density, distribution of sums and products of random variables, conditional density, conditional expectation
8. Order statistics, moment generating function, characteristic function
9. Inequalities: Markov, Chebyshev, one sided chebyshev, Schwarz and Chernoff bound
10. Almost sure convergence, strong law large number (SLLN), convergence in probability, weak law of large number (WLLN), convergence in distribution, central limit theorem(CLT). Relation between three modes of convergence

No compulsory attendance.

Classes + tutorials for 4 hours per week. Classes and tutorials are enough for the course.

Policy: Quizzes (10% each), midsem(30%) , endsem(50%)

References:

Main : Introduction to probability models, by Sheldon Ross

Additional : Introduction to probability theory, by Hoel, Port and Stone