Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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.

Wikipedia has an article https://en.wikipedia.org/wiki/Probability-generating_function with further references.

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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? Computing $E[X]$ is fairly straight forward: $E[X] =…
Brian
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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 answered without referring to characteristic…
StubbornAtom
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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 values. I hope someone could assist me. Thanks
Heisenberg
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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 Focusing on only discrete random variable, we have…
hephaes
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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
Shreya Bhandari
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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 $$ X \sim \text{Poisson} (\lambda), \quad \text{where }\lambda \sim Bin(n,p) $$ where $n$ is very large. Detail and…
Ian Sudbery
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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 distribution of $X$? We consider a partition of…
StubbornAtom
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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…
Cherryl
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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 $D$, but I know the generating function $G(s) =…
bpeter
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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(t) = \sum_{i \geq 0} { e^{- \lambda} \lambda ^i…
rannoudanames
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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 sides, and two dice tossed) the multiplication of the…
Antoni Parellada
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Extreme birthday problem

I have an extreme version of the birthday problem. I want to know: The probability that $m$ individuals will share a birthday The expected $m$ given the number of individuals The slight complication is that: An individual can have multiple…
Anonymous Scientist
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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 however I’m having difficulty with G2 and I am not able…
lauren
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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 that the $n^{th}$ derivative at unity…
user1936752
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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 generalize a generating function for when more than one die…
John P.
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