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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.

There’s no real reason for anyone other than me to care about this, but if you’ve ever heard the term pgf or characteristic function and you’re curious what it’s about, hop on for the ride!

Imagine you are holding five regular playing cards in your hand. Maybe your hand is QQA97, i.e. a pair of queens, an ace, a nine, and a seven. We’re playing some sort of weird poker variant where I get to blindly draw one of your cards. We’re curious about the probability distribution of the outcome of that draw.

In words, most cards (e.g. 2, 4, 8, J and others) have a probability of zero of being drawn from your hand (because they are not in your hand.) Some cards (ace, seven, nine) have a 20 % probability of being drawn, and then there’s a 40 % probability that a queen is drawn, since you have two of them.

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