Probability & Statistics

This course provides a comprehensive foundation in probability and statistics, essential for understanding randomness and applying statistical models to real-world phenomena. The course integrates theoretical concepts, practical applications, and computational techniques.

Course Resources

• Lecture Notes
  • Lecture 1: Introduction to Probability and Random Number Generation
  • Lecture 2: Foundations of Randomness (Part 2)
  • Lecture 3: Static Network Models
  • Lecture 4-6: Probability Relationships and Markov Chains
  • Lecture 7: Continuous Random Variables
  • Lecture 8: Fundamentals of Statistics
  • Lecture 9: Estimation
  • Lecture 10: Prediction
  • Lecture 11: Non-Linear Prediction
  • Lecture 12: Hypothesis Testing

Course Content

1. Introduction to Probability
   • Overview of discrete probability, random variables, and their applications in cryptography and simulations.
2. Foundations of Randomness
   • Independence, expectation, the Law of Large Numbers, and their implications in probability theory.
3. Random Network Models
   • Exploration of Erdos-Renyi and preferential attachment models to understand network phenomena.
4. Markov Chains and Probability Relationships
   • Conditional probability, Bayes’ theorem, and modeling time-evolving systems with Markov Chains.
5. Continuous Random Variables
   • Transitioning from discrete to continuous variables, focusing on distributions like Exponential and Normal.
6. Statistics Fundamentals
   • Exploring datasets, descriptive statistics (mean, median, variance), and basic visualization techniques.
7. Estimation
   • Statistical modeling, parameter estimation, and bootstrap methods for uncertainty estimation.
8. Prediction
   • Developing predictive models, from linear regression to logistic regression, focusing on minimizing prediction error.
9. Non-Linear Prediction
   • Neural networks, Generalized Linear Models (GLMs), and Generalized Additive Models (GAMs) for complex data relationships.
10. Hypothesis Testing
    • Principles of statistical inference, p-values, significance levels, and examples like the Clairvoyance Test.