Applied Probability and Uncertainty Analysis

Introduction to uncertainty concepts through toy examples, exploring methods to control and exploit uncertainty in real-world applications.

Mathematical formulation of probability, introduction to random variables, and concept of independence.

Study of expectation, variance, standard deviation, and their applications in uncertainty analysis.

Introduction to universality, exploration of the law of large numbers with proofs and applications.

Study of the central limit theorem and its applications in statistical analysis.

Analysis of heavy-tailed random variables and large deviation principles.

Introduction to random processes, simple random walk on a line, and applications to gambling.

Study of universality in random walks and Galton-Watson trees.

Introduction to Markov processes and chains with examples.

Study of stationary distribution and universal principles for Markov chains.

Exploration of Markov chain Monte Carlo, mixing theory, and cutoff phenomenon.

Study of stochastic optimizations, gradient descent methods, and applications to deep learning.