# Questions tagged [probability-generating-fn]

A probability generating function is a function defined as a power series which contain all the probability mass function values of a discrete probability distribution. It is related to the moment generating function, and also known as a z-transform.

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### Can the pdf of the difference of two independent random variables be found out using their Probability Generation Functions?

I recently learned of probability generation functions and that the sum of two independent random variables can be found out by multiplying the PGFs. I wanted to know if anything similar can be done ...
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### Reference request: Table of probability mass function / probability generating function pairs?

I have a probability generating function $G(z) = \sum_{k=0}^\infty z^n p(n)$ for a discrete random variable which is somewhat complicated. I would like to "invert" it to obtain the pmf $p(n)$, which ...
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### Probability generating function of mixture of discrete random variables

Consider a discrete distribution $X$ that is a mixture of two discrete distributions $A$ and $B$. Explicitly, $X=A$ with probability $p$ and $X=B$ with probability $1-p$. Denote the pgfs of $A$ ...
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### How is $s^k$ in p.g.f. likened to Dirac measure $\delta_k$?

How is $s^k$ (sometimes $z^k$, $\theta^k$) in probability generating function likened to Dirac measure $\delta_k$? I don't see how these are similar. Yet I see them equated. Clearly the Dirac ...
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### Probability Generating Functions: How to use them?

For a discrete variable $X$ that takes on nonnegative integer values $\{0,1,2,\ldots\}$, the probability generating function is defined as $$G(s) = \sum_{k=0}^\infty P(X=k) s^k$$ It is easy to show ...
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### In general, how should we find the pmf given only the moment generating function without comparing its form to that of famous pmf?

Background It is known that moment generating function generates moments, but does it hold information about the probability of the random variable being realised at a particular value? Example ...
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### Fit data based on generating function

Suppose I have iid data generated from a discrete random variable $X_i \sim D(\lambda)$, and I would like to infer the parameter $\lambda$. Unfortunately, I do not know the likelihood function for $D$,...
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### Generating Function for sum of N dice [or other multinomial distribution] where lowest N values are “dropped” or removed

Background I found this interesting question Formula for dropping dice (non-brute force) and excellent answer https://stats.stackexchange.com/a/242857/221422, but couldn't figure out how to ...
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### Stopping condition for MCTS

I'm trying to come up with a stopping condition for MCTS with exploration. Say we have $m$ actions at the root state, and after $N$ playouts they have $n_1\geq ...\geq n_m$ visits. I want to test the ...
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### Stochastic Processes: Randomly Stopped Sums *vs* the sum of I.I.D. Random Variables? (For Finding PGF's)

So I'm studying Stochastic Processes (specifically PGF's). I see two definitions but I don't seem to know how to differentiate between them properly. Theorem 1 Suppose $X_1, ... , X_n$ are ...
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### Proof of Algebraic Formula for the Sum of Two-Dice Toss as a Convolution

To figure out exactly the expected frequency of a given sum in a dice toss (given a certain number of dice and sides/dice), the following formula is posted here by @Glen_b (adapted to dice of six ...