The course is a solid introduction to the formulation and manipulation of probability models, leading up to a rigorous proof of the law of large numbers and the central limit theorem. The emphasis is on concepts: sets and combinatorics allow a precise mathematical formulation of probability models, multivariable calculus supplies machinery for changing variables and calculating probabilities and average values relating to vectors of real-valued random variables, and limit theorems allow event-occurrences which are individually unpredictable to become predictable in the aggregate.
MATH 240 and MATH 241; or MATH 340 and MATH 341.
Text of Ross, chapters 1-8 including:
Axioms of Probability and basic properties
Random variables and distributions in one and several variables, including change-of-variable techniques
Expectation and conditional expectation
Moment generating functions
Law of Large Numbers and Central Limit Theorem
Optional Topics from among:
Meaning of convergence with probability 1
Filling in missing steps of the book's proof of the Central Limit Theorem