Part III — Random Variables

Random variables turn outcomes into numbers. This part builds the distributional toolkit: CDFs, PMFs, and PDFs, joint, marginal, and conditional distributions, and what happens to distributions under transformations.

Topics

Topic	Focus
Random Variables	Random variables as mappings from outcomes to numbers
Cumulative Distribution Functions	The CDF and its defining properties
Probability Mass Functions	PMFs for discrete random variables, with examples
Probability Density Functions	Densities, non-negativity, and normalization
Joint Distributions	Joint PMFs and PDFs for multiple random variables
Marginal Distributions	Recovering single-variable distributions from a joint distribution
Conditional Distributions	Conditioning on events and on other random variables
Functions of Random Variables	Distributions of transformed random variables
Change of Variables	The change-of-variables formula for densities under monotone transformations