# Tagged Questions

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### How to obtain probability distribution from the generating function $G(s) = e^{a(s-1)^2}$?

I was trying to get the probability distribution $p(n)$ from a generating function $G(s)$ like this: $G(s) = e^{a(s-1)^2}=\sum s^np(n)$ I need first to do Maclaurin expansion of the exponential and ...
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### Generating function for picking j balls without replacement from an urn

In an urn, each balls is labeled with one of $\{0,1,2,...,k\}$. For each $i\in{0,1,2,...,k}$, there are exactly $n_i$ balls labeled $i$. Let $f(x)=\sum\limits_{i=0}^k n_ix^i$. Let ...
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### Probability Generating Functions with Three Dice

Three identical dice are thrown. The dice are fair, that is, for all three dice the probability of turning up face $j$ is $1/6$, $1 \le j \le 6$. Let $X_1,\ X_2,\ X_3$ be the independent random ...
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### Probability of getting SUCCESS AND FAILURE at number n-1 and n trial

In a sequence of Bernoulli trials let $u_n$ be the probability that the combination SF occurs for the first time at the trials number n-1 and n. To find the generating function I wrote the following ...
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### Proving that a moment generating function converges pointwise

I have found a moment generating function $M_n$ given by $\cfrac{(1-e^t)e^{\frac tn}}{n(1-e^{\frac tn})}$ if $t\ne 0$ and 1 if $t =0$ How do I prove that $M_n$ converges point-wise to the moment ...
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### Dummy variable in the probability generating function

I'm struggling to understand what the purpose of the dummy variable $t$ in the probability generating function is? I know it takes a value between 0 and 1, and have heard it described as a 'relative ...
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### Finding generating functions - how was this jump made?

I'm going through examples of probability-generating functions in a book and am confused by the following example: $$1+2s+4s^2+...=\sum_{n=0}^\infty (2s)^n=(1-2s)^{-1}$$ I understand the summation but ...
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### How to recover the probability mass function from probability generating function?

Would someone please provide me an example of where we take a p.g.f and use it to derive the p.m.f. ? I understand that you were have to take the derivatives of the pmf, which is understandable ...
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### find expectation of non-negative integer valued RV from generating function

How can we find $E\left(X\right)$ and $E\left(X^{2}\right)$ if all we have is that $G\left(s\right)$ is the generating function for X, which takes non-negative integer values. I know ...
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### generating function and binomial distribution - counting

I am trying to understand generating function. I have the following problem: There are 50 students in the International Mathematical Olympiad (IMO) training programme. 6 of them are to be selected to ...
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### Card game probability

Suppose the following solitaire with a standard deck. I turn four cards visible on the board and on each turn, I remove those suits that appears more than once in the board. Then I fill the board such ...
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### Utility of Probability Generating Function .

The utility of Probability Generating Function , how far known to me , is basically to generate PMF uniquely (what all the popular books of probability have written ) . Now , PGF is constructed with ...
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### Probability question with trees and fruit using probability generating functions

Each year a tree of a particular type flowers once and the probability that it has n flowers is $(1-p)p^n$, $n=0,1,2...,$ where $0<p<1$. Each flower has probability $1/2$ of producing a ripe ...
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### Finding a probability distribution given the moment generating function

The $n$-th moment ($n \geq 1$) of a random variable $X$ is given by: $m_n = \frac{2^n}{n+1}$. Find the probability distribution of $X$. Here's my attempt at a solution: I expand the moment generating ...
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### A classical problem in combinatorics/probability

I read this problem in Cognition and Chance by Raymond Nickerson (the problem is stated not discussed) ...
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### Calculating coefficient of generating function with Coin

The problem I'm currently looking over requires use of generating functions to solve the following: If a coin is flipped $25$ times with eight tails occurring, what is the probability that no run of ...
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### Computing the probability of rolling a sum of 18 on 4 six-sided dice

The following PDF gives an explanation on page 11. Unfortunately I do not know how to reproduce it here. http://web.mit.edu/~qchu/Public/TopicsInGF.pdf In short, I am not sure how the symmetry ...
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### Proving a Probability Generating Function satisfies a partial differential Equation

We have N animals grazing in a field. The animals graze independently, and periods of grazing and resting alternate for the animals. If an animal is resting at time t, the probability it begins ...
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### Poisson distribution with exponential parameter

I don't know how to solve Exercise 8, Section 5.2 from Geoffrey G. Grimmett, David R. Stirzaker, Probability and Random Processes, Oxford University Press 2001. For those who don't have this book: ...
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### Geometric probability splitting

So, this question has two parts... It's not homework, just something I wanted to calculate, but don't know how to for sure. So, given an item with a value of X, it has a 50% chance to split/double, ...
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### Moment Generating Function of $X$

The moment generating function of $X$ is given by $M_X(t)=e^{2e^t-2}$ and that of $Y$ by $M_Y(t)=\left(\frac34e^t+\frac14\right)^{10}$. If $X$ and $Y$ are independent, what are $P(XY=0)$?
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### To obtain the closed-form expression of CDF and PDF from the recurrence relation

Now I have a question, in which I need to find the probability mass function and the cumulative distribution function. But now I only have the recurrence relation. Here is the details: Assume ...
I am trying to understand the solution of a problem. $X_1,X_2,....$ a sequence of independents randoms variables and same probability distribution. $N$ rv. taking its values in $\mathbf{N}$ ...