These notes are based on lectures by Perla Sousi in the academic 2025-2026. Mistakes are almost surely mine and this is not official nor a recreation of what was in the lectures.
Summary of the course [Expand]
Lectures 1-3 are about the definition and basic properties of probability spaces and probability measures, and some elementary combinatorics that we need for probability purposes.
Lectures 4-5 are about conditional probability and some counterintuitive results that arise from this.
Lectures 6-8 are about properties of discrete random variables.
Lectures 9-10 are about some useful inequalities: Markov's, Chebyshev's, AM-GM and Jensen's.
Lectures 11-14 are about some processes such as random walks and branching processes. We review properties of probability generating functions and recurrence relations as a useful tool to analyze these.
Lectures 14-18 are about continuous random varibles (when they have an integrable density). We also introduce the change of variables formula and the ideal of joint densities of more than one random variable.
Lectures 18-21 are about moment generating functions and multivariate normals. It is here that we heavily rely on the results from complex analysis.
Lectures 22-24 are about different types of convergence for random variables, and we use this to formalize the weak and strong laws of large numbers. We also talk about applications of the central limit theorem which we proved in level 6.