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CS 203: Mathematics for Computer Science - III

Units: 3-0-0-9 (modular first half)
Pre-requisites: None
 
Basics of Probability Theory : [Weeks 1]

Sets, The concept of a discrete sample space in probability theory. The definition of an event. The definition of a probability distribution.  Random variables, expectation and variance.

 

Distributions : [Weeks 2 & 3]

Discrete distributions: Bernoulli trials, Geometric, Binomial and Hypergeometric and Negative Binomial distributions, Poisson distribution.  Continuous distributions: normal and other continuous distributions Exercises based on the analysis of applications to computer science.

Linearity of expectation. Higher moments of a random variable, moment generating function. Computing the moments of geometric, binomial, normal and Poisson distributions.

Conditional Probability, Conditional expectation of a random variable with respect to an event. Bayes' Theorem and examples of applications in computer science.

 

[Weeks 4]

The concept of k-wise and mutual independence of random variables. Applications of independence and k-wise independence in computer science.

 

[Week 5]

Tail bounds: Markov inequality, Chebyshev's Inequality, Chernoff bound, and Examples of applications to the analysis of randomized algorithms.

 

[Week 6 & 7]

Introduction to the probabilistic method. Applications to random graphs and number theory. Lovasz Local Lemma and applications.

If time permits, one of the following topics can be covered: Information theory/ Markov chains/ Randomized algorithms/ Streaming.


Books:
  1. William Feller, An introduction to probability theory and its applications.
  2. Sheldon Ross, A first course in probability.
  3. David Stirzaker, Elementary probability.
  4. Kai Lai Chung, A course in probability theory.