1. Probability

Probability theory is the mathematical framework for reasoning about uncertainty before observing data. It provides the deductive foundation for modeling random phenomena, deriving expected outcomes, and quantifying the likelihood of events.

1.1 Core Philosophy

Probability answers the question: "Given a known model or process, what outcomes should we expect?"

Starting from axioms and probability distributions, we build a rigorous mathematical structure that lets us: - Quantify uncertainty in future events - Derive properties of random variables - Establish limit theorems and convergence results - Model stochastic processes and information flow over time

1.2 Topics Covered

• Expected Value —
• Filtration —
• Tower Property —
• Gaussian Integral —
• Jensen's Inequality —
• Moment Generating Functions —

Deductive reasoning from models to outcomes