Characteristic Functions: Properties and Applications

This lecture introduces characteristic functions, a powerful tool in probability theory, defined as the expectation of e to the i tx. The instructor explains how characteristic functions are always well defined and provides examples for discrete and continuous random variables. The lecture covers key properties of characteristic functions, including symmetry and positive semi-definiteness. The instructor also discusses the inversion formula, which allows for the unique identification of a distribution from its characteristic function. Furthermore, the lecture explores the relationship between characteristic functions and moments of random variables, highlighting the moment generating function property. The factorization property of characteristic functions is presented as a crucial tool for determining independence between random variables.