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Probability-generating functions
I have long struggled with understanding what probability-generating functions are and how to intuit them. There were two pieces of the puzzle missing for me, and we’ll go through both in this article.
Today, the inverse operation is probably more common in that we take the coefficients of a long polynomial and plug them into a vector to e.g. find roots with linear algebra. I believe the latter was one of the motivating cases for probability-generating functions, because this lets us solve the problem of the points, i.e. how to distribute the betting pot if we have to quit a game of dice early. If we focus on the case where \(u=1\), then each of the \(e^{iuX}\) represent a distance \(X\) traveled counter-clockwise (for \(X\) positive) or clockwise (for \(X\) negative) around the unit circle in the complex plane.
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