Probability and Statistics | Xingyu Yang

Reading guide

Start with counting and conditional probability, then use random variables to describe uncertain quantities. Distributions, expectation, and variance provide the language for comparing them. Continue into joint distributions, continuous models, limit theorems, inference, and stochastic processes as those chapters are developed.

Basic set notation and finite sums are useful prerequisites. Calculus becomes necessary for continuous distributions and limits. The reading path distinguishes developed notes from outlines.

Reading path

Counting and Probability Axioms

The product, sum, inclusion-exclusion, division, and pigeonhole principles, followed by a precise finite probability model.

Conditional Probability and Independence

Conditional probability, Bayes' theorem, the law of total probability, and the distinction between independence and conditional independence.

Random Variable

A foundational probability note defining random variables as measurable mappings from outcomes to numbers, with discrete, continuous, CDF, PMF, and PDF examples.

Probability Distributions

A probability note on distributions, support, PMF, PDF, CDF, common discrete and continuous distributions, and how distributions connect to moments.

Expectation and Variance

A compact note on expected value, variance, standard deviation, LOTUS, linearity, and the basic proof patterns used throughout probability theory.

Moment Generating Functions

A note on moment generating functions, moment extraction by differentiation, uniqueness, sums of independent variables, and common distribution examples.

Weak Law of Large Numbers

A note on the weak law of large numbers, convergence in probability, sample means, and the Chebyshev-style proof pattern.

Planned

Continuous Distributions

An explicit migration scaffold for Continuous Distributions, preserved from the archived Math-for-Computer-Science outline until the mathematical exposition is written and reviewed.

Joint Cumulative Distributions

An explicit migration scaffold for Joint Cumulative Distributions, preserved from the archived Math-for-Computer-Science outline until the mathematical exposition is written and reviewed.

Limit Theory in Probability

An explicit migration scaffold for Limit Theory in Probability, preserved from the archived Math-for-Computer-Science outline until the mathematical exposition is written and reviewed.

Stochastic Processes

An explicit migration scaffold for Stochastic Processes, preserved from the archived Math-for-Computer-Science outline until the mathematical exposition is written and reviewed.

Statistics: Foundations and Inference

An explicit migration scaffold for Statistics: Foundations and Inference, preserved from the archived Math-for-Computer-Science outline until the mathematical exposition is written and reviewed.