Chapter 9: Generating Functions and Transforms

Learning Objectives

• Define the moment generating function, characteristic function, and probability generating function, and explain their domains of existence
• Compute transforms for common distributions and verify the uniqueness and inversion theorems
• Use the product-of-transforms property to derive distributions of sums of independent random variables
• Apply the Levy continuity theorem to prove the law of large numbers and central limit theorem via characteristic functions
• Define the cumulant generating function and relate cumulants to moments
• Derive the Chernoff bound and state Cramer's theorem for large deviations
• Apply the probability generating function to compound sums, branching processes, and extinction probability