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The definition of a Probability Generating Functions makes sense. However, I want to learn the applications of PGF's and why they are useful.

For example:

  • Probability Mass Functions (PMFs) and Probability Distribution Functions (PDFs) are useful for describing the probability of a random variable - this is useful in scientific applications for predicting the behavior of some scientific phenomena that is believed to be follow one of these functions

  • Moment Generating Functions (MGFs) are useful for estimating the parameters of a Probability Distribution Function when the Likelihood Function is difficult to deal with (e.g. Gamma Distribution - involves differentiating the Gamma Function)

Thus, this naturally leads to the question: what are the main purposes of Probability Generating Functions and what kind of situations do they help us in?

Thanks!

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    $\begingroup$ Grimmett and Stirzaker's Probability and Random Processes Section 5.1 may be of interest to you. $\endgroup$
    – Andrew
    Commented Nov 20, 2022 at 5:35
  • $\begingroup$ Thank you! Will try to check it out! $\endgroup$
    – stats_noob
    Commented Nov 20, 2022 at 5:43
  • $\begingroup$ There are many applications of PGFs in Feller, "An Intoduction to Probability Theory And Its Applications", 3rd edition, starting in Chapter XI. $\endgroup$
    – awkward
    Commented Nov 20, 2022 at 14:16
  • $\begingroup$ You can also consider that PGF are just GF. And GF is basically the same as the Z-transform which in turn is a discrete analog of the Laplace transform, which is connected to the Fourier transform... You can guess that the applications and connections are endless. $\endgroup$
    – leonbloy
    Commented Jan 4, 2023 at 1:54

1 Answer 1

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There are many applications of Probability Generating Functions (PGFs), however, one natural example occurs in the study of discrete time Markov Chains and Stochastic Processes.

Let us start with the setup:

A Branching Process is the sequences $(X_n)_{n \in \mathbb{N}} \in \mathbb{N}_0$ where $X_n$ denotes the number of individuals in a population at time $n$. Each individual $i$ in this process gives birth to a ($Z_i^n$) offspring (independently and identically distributed) at time $n$ which forms the next generation at the time $n+1$. Therefore, $X_{n+1}= \sum_{i=1}^{n}Z_i^n$.

Some quick observations we make here is firstly that $(X_n)$ is a Markov Chain and that once the chain is at $0$, it will stay at $0$ forever. Therefore, this is useful as a simple modelling technique for population growth. We don't impose a constraint on the offspring distribution $Z_i^n$ other than them being independent and identically distributed.

Now we can turn to our application of PGFs.

The Extinction Probability is the smallest non-negative root of the equation $\alpha = G(\alpha)$ where G is the PGF of our offspring distribution

Therefore, by solving this relatively simple equation above, we can determine the probability that the population will die out under our Branching Process model.

The Extinction Mean Criterion tells us that for when our PGF $G(0)>0$, then:

  1. $\mathbb{E}(Z_i^n) \leq 1 \space \implies \space$ Certain Extinction

  2. $\mathbb{E}(Z_i^n) > 1 \space \implies \space$ Uncertain Extinction

And so as we can see, the PGF does turn out to be quite a useful tool here - helping with this class of population models.

As an additional comment, Branching Processes may seem like one of many population modelling choices available - however, many of the most common modelling techniques can fall under the heading of a Branching Process including SIR modelling, SIS modelling, and (with a specific choice $\gamma$ of the recovery rate), the Reed-Frost model too.

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  • $\begingroup$ Here's a related exercise in MGFs, PGFs and CGFs denoted $M,\,G,\,K$. Given $S:=\sum_{i=1}^NX_i$ for random $N$ and IIDs $X_i$ whose distribution is that of $X$,$$M_S(t)=\sum_nP(N=n)M_X^n(t)=G_N(M_X(t))=M_N(K_X(t))\implies K_S(t)=K_N(K_X(t)).$$ $\endgroup$
    – J.G.
    Commented Jan 24, 2023 at 13:01

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