#### Description

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.

#### Prerequisites

MATH 240 and MATH 241; or MATH 340 and MATH 341.

#### Topics

**Text of Ross, chapters 1-8 including:**

Axioms of Probability and basic properties

Combinatorial problems

Conditional probability

Random variables and distributions in one and several variables, including change-of-variable techniques

Expectation and conditional expectation

Moments

Moment generating functions

Law of Large Numbers and Central Limit Theorem

**Optional Topics from among:**

Characteristic functions

Fourier transforms

Borel-Cantelli Lemma

Meaning of convergence with probability 1

Filling in missing steps of the book's proof of the Central Limit Theorem