#### 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. De Morgan’s Law, Union Bounds

###### Distributions : [Weeks 2]

Random variables, expectation, and variance.

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.

OPTIONAL (if time permits): Function of One Random Variable: Change of Variables.

###### Conditional Probability : [Weeks 3]

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

###### Independence: [Weeks 4]

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

###### Tail Bounds: [Week 5]

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

###### Applications: [Week 6 & 7]

Cover one or more applications: For example,

- Statistics: Hypothesis Testing
- Parameter Estimation: MLE, Least Squares
- Introduction to the probabilistic method. Applications to random graphs and number theory. Lovasz Local Lemma and applications
- Information theory
- Markov chains
- Randomized algorithms/ Streaming

Books / References:

- William Feller, An introduction to probability theory and its applications.
- Sheldon Ross, A first course in probability.
- David Stirzaker, Elementary probability.
- Kai Lai Chung, A course in probability theory.
- Athanasios Papoulis and S Pillai, Probability - Random Variables and Stochastic Processes, Chapter 5, for Change of Variable concept