Probability & Statistics

This course provides a comprehensive foundation in probability and statistics, essential for understanding randomness and applying statistical models to real-world phenomena.

Course Overview

The 2023 course covered the following major topics:

• Lecture 1: Random Number Generation
  • Introduction to randomness, pseudo-random number generation, and the importance of random number generators (RNGs) in simulations and cryptography.
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• Lecture 2: Static Network Models
  • Exploration of network models, with focus on Erdos-Renyi graphs, small-world phenomena, and preferential attachment models.
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• Lecture 3: Bayes’ Theorem & Conditional Probability
  • Introduction to Bayes’ theorem, conditional probability, and their applications in decision-making and inferential statistics.
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• Lecture 4: Introduction to Markov Chains
  • Covering Markov chains, transition matrices, and stationary distributions. Modeling time-evolving systems using probabilistic transitions.
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• Lecture 5: Queuing Theory
  • Introduction to queuing models and their applications in analyzing systems handling tasks over time, with real-world applications in networks.
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• Lecture 6: Poisson Processes
  • Poisson processes and their applications in modeling random events occurring over time, such as traffic flow and customer arrivals.
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• Lecture 7: Quasi-Reaction Systems
  • Introduction to quasi-reaction systems, stochastic processes for modeling interactions in systems like gene expression and ecological dynamics.
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• Lecture 8: Combinatorics in Reaction Systems
  • Application of combinatorics in reaction systems and stoichiometry in chemical and ecological models, with examples of predator-prey dynamics.
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• Lecture 9: Predictive Modeling & Logistic Regression
  • Exploration of predictive models, error analysis, and logistic regression for binary classification, including estimation methods.
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• Lecture 10: Neural Networks
  • Introduction to neural networks, from logistic regression as a foundation to deeper networks, including forward and backward propagation.
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• Lecture 11: Hypothesis Testing
  • Hypothesis testing, including Type I and Type II errors, p-values, significance levels, and applications in real-world decision making.
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• Lecture 12: Permutation Tests
  • Permutation testing as a non-parametric method for hypothesis testing and how to apply this in various research contexts.
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Learning Objectives (2023)

1. Understand and apply fundamental probability concepts and stochastic models.
2. Analyze real-world problems using queuing theory, Poisson processes, and Markov chains.
3. Develop predictive models using tools like logistic regression and neural networks.
4. Perform hypothesis testing with a solid grasp of statistical inference principles.
5. Apply advanced techniques like permutation tests in non-parametric hypothesis testing.