Probability Theory

Probability is an elusive concept, much like time in Augustine’s Confessions—we understand it until asked to define it. From ancient games to modern quantum mechanics, probability has puzzled and fascinated great minds. Even today, its true meaning is debated: is it about counting, measuring uncertainty, or something inherent in the universe? Regardless, we need probability to solve real-world problems. Modern probability theory, built on set theory, boolean algebra, and measure theory, often feels disconnected from practical applications....

Motivating the definition of measurability for random variables

1. What Is a Measure? Imagine you’re trying to figure out how to assign a size, or “measure,” to different kinds of sets—some simple, like intervals on the real line, and some more complex. A measure is just a mathematical way to do this, and it needs to follow a few basic rules: Non-negativity: The measure of any set should be non-negative (you can’t have negative length or size). The empty set gets zero: The set with nothing in it has a measure of zero....

A fairly rigorous proof of the Central Limit Theorem (CLT) using characteristic functions.

An elementary derivation of the PDF of a normal distribution.

A motivated introduction to measure theory.

An algebraic derivation of the biased sample variance estimator using expected value.

Posts concerning stochastic calculus