Overview

In this section, we move from the static description of probability spaces to the dynamic behavior of random experiments. We will explore how to describe outcomes numerically using Random Variables, and how to analyze the relationships between different events.

Key Topics

1. Limits: What does it mean for a sequence of events to happen “infinitely often”?
2. Independence: What does it fundamentally mean for two events or $\sigma$-fields to be “independent”?
3. Random Variables: How to rigorously define variables that depend on random outcomes, and when they are independent.
4. Distribution: How a random variable induces a probability measure on the real line.
5. Types of Distributions: Absolutely continuous, discrete, and singular distributions.
6. Borel-Cantelli Lemmas: First major tools to decide if an event happens infinitely often or not.
7. Kolmogorov’s 0-1 Law: A surprising result showing that “tail events” (events happening in the asymptotic future) are deterministic for independent sequences.

These tools form the bedrock for proving the Laws of Large Numbers and the Central Limit Theorem.