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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Lognormal distribution and Galton Watson process

I have been studying a Galton Watson process that creates random binary trees with probability of survival $p_{s}$ and offspring distribution $p(k)=p_{s}\delta(k-2)+(1-p_{s})\delta(k)$. I'm ...
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Probability generating function and binomial coefficients

I'm reading an article where the authors derive the mass function of a compound distribution by considering the generating function. The generating function of interest for a random variable $N$ is a ...
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Difficulties with the PGF of X+Y with Y~Poisson(1) and X~Poisson(Y)

The pair of random variables $(X, Y )$ is distributed as follows. $Y$ has probability mass function $\text{Poisson}(1).$ Given $Y , X$ has probability mass function $\text{Poisson}(Y ).$ Show that the ...
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Probability Generating Function for The Difference of Two Binomially Distributed Random Variables?

Suppose I have 2 random variables: $X\sim \textrm{Bin}(m,p_1)$ and $Y\sim \textrm{Bin}(n,p_2).$ I want to find the distribution of $S=X-Y$ using the probability generating function ($PGF$) treating $S$...
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Average cost of getting a specific card from deck of 9?

In a game, I am looking to draw the hero card out of 9 possible cards. The first card (full deck) costs 300 gems. All subsequent draws cost 600 gems. I can either keep drawing one card at a time till ...
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PGF of sum of RVs is a composition of PGFs

Let $\\\{X_n\\\}$ be a sequence of i.i.d. random variables whose values are non-negative integers. Let $N$ be a random variable that is independent of $\\\{X_n\\\}$. $N$'s values are also non-negative ...
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Find probability generating function(p.g.f) of a compound distribution

I have a discrete compound distribution for a random variable: $$S_N = X_1 + X_2 +\dots + X_N,$$ where $X_1, X_2, \dots, X_N$ are IID count random variables, and $N$ is a count random variable too. ...
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Generating function of a random walk

Consider a random walk with $S_n=\sum^n_{i=1}X_i$, where the random i.i.d. steps $X_i$ take values $-1,0,2$ with probabilities $1/9,1/9,7/9$ respectively. Set $S_0=1$. I would like to calculate the ...
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Likelihood loss function for finite support probability distribution in Neural Networks

I have managed to reproduce solution from this article and made it work for my dataset. Instead of making a Neural Network output a scalar (regression), we make it output two parameters of a ...
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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 ...
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Generating function from the infinitesimal generator of a continuous-time markov chain?

Summary: The basic goal is to find the time evolution of the probability generating function (or the moment generating function or the characteristic function if you prefer) for a continuous-time ...
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