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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Why do we use exponent in characteristic function?

A student who is attending probability 101, learned about normal distribution and generating functions recently. We are given a "generating function" as follows: $$G(t)=<e^{itx}>=\int_{-\infty}^...
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How do I analytically calculate variance of a recursive random variable?

Suppose I have a chest. When you open the chest, there is a 60% chance of getting a prize and a 40% chance of getting 2 more chests. Let $X$ be the number of prizes you get. What is its variance? ...
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Branching process Galton Watson

Using the Galton Watson branching process Assume that a fox had 0,1,2,3 offspring with probabilities p0,p1,p2,p3 respectively. find the probability distribution for G1 and G2. I worked out G1 ...
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Extending normalized probability generating function (pgf) in branching process

My question is in the context of branching processes and simply how to extend a normalized negative binomial probability generating function (pgf) from $\frac{1}{y}[G(s)]^y$ to $\frac{n}{y}[G(s)]^y$ ...
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Generating function of relaxed Bernoulli

In [1,2], an approximate continuous relaxation of the Bernoulli distribution is introduced as follows: $$X = \frac{1}{1 + e^{- (\theta - L) / \tau}}$$ where $L$ is a Logistic random variable, $\tau&...
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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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112 views

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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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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Using Chebyshev's inequality to obtain lower bounds

Let $X_1$ and $X_2$ be i.i.d. continuous random variables with pdf $f(x) = 6x(1-x), 0<x<1$ and $0$, otherwise. Using Chebyshev's inequality, find the lower bound of $P\left(|X_1 + X_2-1| \le\...
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Can a $\require{enclose}\enclose{horizontalstrike}{z-transform}$ Probability Generating Function be used to solve this recurrence relation?

Imagine a simple random walk where $x_n=\sum_{i=0}^ny_i$ and $y_i = \begin{cases} 1, & \text{with prob 1/2} \\ -1, & \text{with prob 1/2} \end{cases}$ Then say we want to find the ...
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What is the distribution of a Poisson variable, where the Poisson rate is Normal (or Binomial)?

What is the distribution of $X$ if $$ X \sim \text{Poisson}(\lambda), \quad \text{where }\lambda \sim N(\mu,\sigma^2)$$ or $$ X \sim \text{Poisson} (\lambda), \quad \text{where }\lambda \sim ...
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Probability Generating Functions and Discrete Time Queuing Chains

I've been learning about Markov Chains for a reading course I'm doing and I've become stuck on some of the notes I was given. It defines a discrete time queuing chain for a server by $$X_{t+1}=\begin{...
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587 views

Compound Poisson Process, probability generating function

If we have a iid random variables $X_i$ with probability generating function $\xi(t) = E[t^{x_i}]$ and $N$ is Poisson with mean $\lambda$ with Probability generating function : $$ \begin{aligned} \phi(...
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PMF of the number of trials required for two successive heads

A coin with probability $p$ of landing a head is tossed repeatedly till the occurrence of two consecutive heads. Let $X$ be the random variable denoting the the number of trials needed. What is the ...
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For independent RVs $X_1,X_2,X_3$, does $X_1+X_2\stackrel{d}{=}X_1+X_3$ imply $X_2\stackrel{d}{=}X_3$?

Let $X_1,X_2$, and $X_3$ be independent random variables such that $X_1+X_2$ and $X_1+X_3$ have the same distribution. Does it follow that $X_2$ and $X_3$ have the same distribution? Can this be ...
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Formula for dropping 2 dice (non-brute force)

This is a follow up from my previous question: Formula for dropping dice (non-brute force) which asked how to determine the statistics for dropping the lowest die but now I'm asking how to determine ...
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Probability Generating Function for Geometric using conditioning

I was wondering if someone could look over my derivation of the probability generating function for geometric RV. I think I arrived at the right conclusions. I wanted to take the alternate route over ...
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198 views

What is the meaning of a probability generating function given domain T?

What does it mean when I plug in $0, 1, ..., n$ into a probability generating function? What does a probability generating function tell me about the number I plugged in?
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Poisson binomial distribution-like problem

Given n trials, where, on each trial, you have a given probability of either winning or losing a set amount of money (with both the amount of money and the probability changing for each trial)- what ...
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25 views

What would we obtain if we set any value other than 0 or 1 to $s$ in $G_X(s)$?

Let $X$ be a discrete RV with PGF $G_X(s) = \frac{s \cdot (2 + 3s^2)}{5}$ . Find the distribution of $X$. Here, if we continue to differentiate and set $s=0$ in $G_X$, we obtain $P(X=0,1,2,… … …)$. ...
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If $G_X$ is a Probability Generating Function of $X$, what does it mean by $G_X(0)=\mathbb P(X=0)$ and $G_X(1)=1$?

Let $G_X$ is a Probability Generating Function of $X$. What does it mean by $G_X(0)=\mathbb P(X=0)$ ? $G_X(1)=1$ ? Why are these identities important?
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Generating from an unknown discrete posterior distribution

My question is that if I have an informative prior with density $$f(N) \propto \frac{1}{N^2}$$ and posterior for N is $$ P(N|T) \propto \big\{{n^A \choose T} {N-n^A \choose n^B-T}/{N \choose n^B}\big\...
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When is a generating function a probability generating function?

Given a generating function, say $G_1(z) = \frac{1}{1-z}$ it can be verified it is not a probability generating function by inspection that the sum of the probabilities (if it were a pgf) is not 100% ...
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MGF of Poisson Z=X+2Y

If $X\sim P(2)$ and $Y \sim P(3)$ using the moment generating function, what kind of distribution has random variable $Z=X+2Y$. So far as I know : $$ M_X(t)=e^{-\lambda(1-e^t)}=e^{-2(1-e^t)} $$ $$ ...
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Negative probabilities - what are the two ordinary pgfs that correspond to the gf of a half-coin?

In Half of a Coin: Negative Probabilities, author considers pgf of a fair coin represented by random variable, $X = 1_H$: $$G_X(z) = E[z^X] = \sum_{x=0,1} z^xP(X=x) = (z^0)(1/2) + (z^1)(1/2) = \frac{...
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How to define distribution, given probability generating function?

Given some probability generating function $G_X:(-1,1) \rightarrow \mathbb{R}$, Since $$P_X(k)=\frac{G^{(k)}(0)}{k!}$$ then how does one define the distribution $P_X(k)$ in practice? The difficulty ...
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380 views

Whats is the probability generating function of Poisson varied with parameter lambda

I have a question which says: "Show that the probability generating function of Poisson varied with parameter $\lambda$ is $e^{-\lambda(1-t)}$. I am not quite sure about the question because the ...
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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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Probability generating function for negative values of random variables?

What if we have negative integral values for a random variable?Then is it possible to write a probability generating function for it? All definitions I have seen so far is for non negative integer ...