The article dives into the concept of probability-generating functions, explaining how they are used to encode sequences of probabilities into a single object. The author goes into detail about the historical context, explaining how de Moivre used polynomial functions as a way to work with sequences before vectors were invented. The article also explores the properties of probability-generating functions, including how they can be used to calculate expectations and variances. The author touches upon the transformation from probability-generating functions to characteristic functions when certain values for the parameter are more sensible. The characteristic function is visualized as a spiral on the complex plane. Overall, the article provides a comprehensive and unique exploration of this mathematical concept.
https://entropicthoughts.com/probability-generating-functions