Probability-generating functions
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Probability-generating functions
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Re: Probability-generating functions
#2Probably worth mentioning the moment-generating function as well, since it's a bit more elementary than the characteristic function and shares many of the properties. It's also more simply related to the probability generating function, you can go from one to the other with a basic change of coordinates (t -> log(x)). I also estimate calculating the moment generating function to be easier in most cases.
In fact most properties of the PGF come from the monent-generating/characteristic function. Including why the second derivative is related to the variance. The second derivative of the moment generating function is the second moment E[X^2]. The second derivative of the logarithm of the MGF is the variance by definition.
The one property that's somewhat unique to the PGF is how composition relates to drawing a randomly-sized sample, which I can see could be useful.
Re: Probability-generating functions
#3Herbert S. Wilf (1990): Generatingfunctionology. https://www2.math.upenn.edu/~wilf/gfology2.pdf
Re: Probability-generating functions
#4Generating functions are also a great tool in combinatorics (see for example the book Analytic Combinatorics by Flajolet and Sedgewick).
Re: Probability-generating functions
#5Probably worth mentioning the moment-generating function as well, since it's a bit more elementary than the characteristic function and shares many of the properties. It's also more simply related to the probability generating function, you can go from one to the other with a basic change of coordinates (t -> log(x)). I also estimate calculating the moment generating function to be easier in most cases. In fact most…
One can also think of probability generating functions as (flipped) Z transforms, moment generating functions as (flipped Laplace transforms), and characteristic functions as Fourier transforms of the respective PMF/PDF. Lot of their properties then follow from simple properties of Signals and Systems.
Re: Probability-generating functions
#6Tangentially related: http://bactra.org/notebooks/cumulants.html
Re: Probability-generating functions
#7Lost the plot at "... and for sensible v, this is equivalent to ..." :(
Re: Probability-generating functions
#8"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."
Great article. For more, I really recommend Analytic Combinatorics:
Re: Probability-generating functions
#9For those interested in looking slightly more into the characteristic function, it may be worth pointing out that the characteristic function is equal to the Fourier-transform (with the sign of the argument being reversed) of the probability distribution in question.
In my own experience teaching teaching probability theory to physicists and engineers, establishing this connection is often a good way of helping people build intuition for why characteristic functions are so useful, why they crop up everywhere in probability theory, and why we can extract so much useful information about a distribution by looking at the characteristic function (since this group of students tends to already be rather familiar with Fourier-transforms).
Re: Probability-generating functions
#10For those interested in looking slightly more into the characteristic function, it may be worth pointing out that the characteristic function is equal to the Fourier-transform (with the sign of the argument being reversed) of the probability distribution in question. In my own experience teaching teaching probability theory to physicists and engineers, establishing this connection is often a good way of helping peopl…
I had not made that connection and find that incredibly useful. Thank you for pointing that out.