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Wyatt's Notes — University

Probability Theory

Probability Theory

Contents

  1. Probability Spaces
  2. Random Variables
  3. Joint Distributions and Independence
  4. Limit Theorems
  5. Transformations and Convolutions

Overview

University-level probability theory notes covering probability spaces, random variables, and limit theorems.

Topics Covered

  • Probability Spaces: Sigma-algebras, measures, conditional probability
  • Random Variables: Discrete and continuous distributions, expectation, variance
  • Joint Distributions: Independence, covariance, correlation
  • Limit Theorems: Law of large numbers, central limit theorem

Prerequisites

  • Real analysis (sequences, series, integration)
  • Basic set theory and logic
  • Mathematical proofs and logic

How to Use These Notes

Start with probability spaces to build foundational knowledge, then progress to random variables and limit theorems. Each section includes worked examples and practice problems.

Use the sidebar to browse topics, or start with the introductory pages linked from the sidebar.

Additional Resources

Each section includes:

  • Detailed explanations of key concepts
  • Worked examples with step-by-step solutions
  • Practice problems with answers
  • Common pitfalls and how to avoid them
  • Connections to other areas of mathematics

Study Tips

  1. Master the definitions: Probability theory requires precise understanding of measures and distributions
  2. Practise proofs: Learn to write clear, rigorous proofs
  3. Draw diagrams: Visualise distributions and random variables
  4. Learn standard examples: Know the properties of common distributions (normal, binomial, Poisson)
  5. Connect to applications: Relate probability theory to statistics, finance, and physics