Characteristic, Moment Generating, Factorial Generating Functions

Overview

Explore the mathematical foundations of probability theory through an in-depth video series on characteristic, moment generating, and factorial generating functions. Delve into key concepts such as inequalities for absolute value of expected value, properties of characteristic functions, inversion formulas, and applications to various probability distributions. Learn how to derive moment generating functions for common distributions like Gamma, Poisson, Normal, Binomial, and Multinomial. Discover advanced topics including joint characteristic functions, applications of characteristic functions, and derivations using univariate and multivariate LOTUS. Gain practical skills in analyzing quadratic forms, Gumbel distributions, and sums of negative binomial random variables using moment generating functions.

Syllabus

Inequality for Absolute Value of Expected Value.
Derivatives of a Characteristic Function are Bounded.
Properties of the Characteristic Function (part 1).
Properties of the Characteristic Function (part 2).
Inversion Formula for a Characteristic Function (part 1).
Inversion Formula (part 2).
Inversion Formula Example (part 3).
Properties of the Moment Generating Function (part 1).
Properties of the Moment Generating Function (part 2).
Factorial Moment Generating Function. Probability Generating Function..
Joint Characteristic Function.
Generating Functions for Gamma Distribution.
Generating Functions for Poisson Distribution.
Generating Functions for Normal Distribution.
Generating Functions for Binomial Distribution.
Generating Functions for Multinomial Distribution.
Generating Functions for Cauchy Distribution.
Some Applications of Characteristic Functions.
Illustration using univariate LOTUS: Derive the MGF for a 1 df noncentral Chi square Distribution.
Illustration using multivariate LOTUS: Derive the MGF or a k df noncentral Chi square Distribution.
Distribution of quadratic form n(xbar-mu)Sigma(xbar-mu), where x~MVN(mu,sigma).
Mean, Variance, MGF, & CDF of a Gumbel Distribution.
Distribution for the Sum of Negative Binomial Random Variables Using the MGF.
Derive the MGF of a Logistic Distribution and use it to derive the Mean and Variance.