Questions tagged [probability-generating-fn]

Question 1

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 ...

Question 2

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 ...

Question 3

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 ...

Question 4

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 $S$...

Question 5

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 ...

Question 6

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 ...

Question 7

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. ...

Question 8

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 ...

Question 9

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 ...

Question 10

For a discrete random variable $X$ with the PGF P(z). I have been given $Y = 2X$ and I need to compute the PGF, G(Z) of this new random variable. We know that the probability generating function is $E(...

Question 11

Suppose $\{X_{n}|n\geq 1\}$ is independent, identically distributed distribuited. Define $S_{0}=X_{0}=1$ and for $n\geq 1$ $$S_{n}=X_{0}+X_{1}+\cdots+X_{n}.$$ For $n\geq 1$ the distribution of $X_{n}$ ...

Question 12

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 ...

Question 13

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)=\langle e^{itx}\...

Question 14

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? ...

Question 15

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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Questions tagged [probability-generating-fn]

Lognormal distribution and Galton Watson process

Probability generating function and binomial coefficients

Difficulties with the PGF of X+Y with Y~Poisson(1) and X~Poisson(Y)

Probability Generating Function for The Difference of Two Binomially Distributed Random Variables?

Average cost of getting a specific card from deck of 9?

PGF of sum of RVs is a composition of PGFs

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Generating function of a random walk

Likelihood loss function for finite support probability distribution in Neural Networks

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Branching process Galton Watson

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