This course provides the theoretical basis for studying the properties of modern statistical and machine learning methods. Students will learn the definitions and properties of probability spaces, random variables, distributions, expectations and limit theorems. Students will work with density functions, conditional expectations, and convergence of random variables. After successful completion of this course, students will be able to determine the probability distribution function and density of random variables/vectors, use the properties of expectations and higher-order moments in computations, and examine the appropriate convergence of random variables given specific situations. Students will also be able to apply results from probability theory to study the properties of sample statistics such as estimators.