This course covers the concepts and applications of Characteristic, Moment Generating, and Factorial Generating Functions in statistics. By the end of the course, students will be able to understand and apply properties of these functions, inversion formulas, and generating functions for various distributions. The course teaches skills such as deriving moment generating functions, using generating functions for different distributions, and applying characteristic functions in statistical analysis. The teaching method includes theoretical explanations, derivations, and examples to illustrate the concepts. This course is intended for students and professionals in the field of statistics who want to deepen their understanding of probability distributions and statistical functions.
Characteristic, Moment Generating, Factorial Generating Functions
Overview
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.
Taught by
statisticsmatt
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