KindiakMath

Here are my write-ups on the theory of probability and some non-trivial exercises therein. While there are many levels that we can introduce probability, we will introduce it at the undergraduate level, which will involve measure-theoretic machinery to rigorously underpin any mathematically meaningful application of probability into STEM. Enjoy!

Theory

Basic Probability

1. Probability Nomenclature
2. How to Count
3. Probabilistic Events
4. Bayes’ Theorem

Discrete Probability

1. Neither Random Nor Variable
2. Adding Random Variables
3. The Mathematical Average
4. The Real Standard Deviation

Measure Theory

1. Extending the Reals
2. The Infinite Coin Toss
3. Measuring the Reals
4. The Lebesgue Integral
5. The Measure-Theoretic Trifecta

Continuous Probability

1. Integrable Sanity Checks
2. Common Continuous Probabilities
3. Legitimate Integral Swapping
4. Density Functions on Steroids
5. Adding Random Variables…Revisited
6. A Student’s Nightmare
7. The Central Limit Theorem

Bonus: Multi-Armed Bandits

1. Return to Bandits
2. A Zeroth-Order Solution
3. Automating Confidence
4. The Thompson Sampling Template
5. The Ultimate Winner
6. Risky Bernoulli Bandits

Exercises

Basic Probability

1. Counting Arguments
2. Binomial Theorem Corollaries
3. The Union Bound

Discrete Probability

1. The Poisson Distribution
2. The Geometric Distribution
3. Several Probability Puzzles

Measure Theory

1. Formalising the Dirac Delta
2. Proving Feynman’s Trick

Continuous Probability

1. The Exponential Family
2. Multivariate Normal Bookkeeping
3. Jordan Decomposition
4. The Gaussian Integral
5. Real-Life Hypothesis Tests
6. The Friendship Formula