Introduction to Probability with Applications

For these notes, I will be referencing Statistical Inference, 2nd ed. (2002) by Casella & Berger, Introduction to Probability Models, 10th ed. (2009) by Sheldon Ross, and Introduction to Probability and its Applications 1st ed. (1950) by William Feller.

1 Probability Theory
1.1 Set Theory
1.2 The Sample Space and Events
1.3 Basics of Probability Theory
1.4 Counting
1.5 Conditional Probability and Independence
1.6 Random Variables
1.7 Distribution Functions
1.8 Density and Mass Functions
2 Transformations and Expectations
2.1 Distributions of Functions of a Random Variable
2.2 Expected Values
2.3 Moments and Moment Generating Functions
2.4 Differentiating Under an Integral Sign
3 Common Families of Distributions
3.1 Discrete Distributions
3.2 Continuous Distributions
3.3 Exponential Families
3.4 Location and Scale Families
3.5 Inequalities and Identities
4 Multiple Random Variables
4.1 Joint and Marginal Distributions
4.2 Conditional Distributions and Independence
4.3 Bivariate Transformations
4.4 Hierarchical Models and Mixture Distributions
4.5 Covariance and Correlation
4.6 Multivariate Distribution
4.7 Numerical Inequalities
4.8 Functional Inequalities
5 Properties of a Random Sample
5.1 Basic Concepts of Random Samples
5.2 Sums of Random Variables from a Random Sample
5.3 Sampling from the Normal Distribution
5.4 Order Statistics
5.5 Convergence in Probability
5.6 Almost Sure Convergence
5.7 Convergence in Distribution
5.8 The Delta Method
5.9 Generating a Random Sample