# 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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### 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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### 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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### 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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### 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 ...