1
vote
2answers
93 views

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 ...
2
votes
1answer
60 views

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 ...
1
vote
2answers
41 views

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 ...
1
vote
1answer
76 views

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 ...
1
vote
1answer
37 views

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 ...
0
votes
2answers
48 views

Generating function(really simple????)

With $X_n$ as a discrete random variable that takes the values $1,2,\dots,n$ with equal probability $\frac1n$. I want to: Evaluate the probability generating function of $X_n$!!! $G_{X_n}= ...
0
votes
2answers
53 views

Joint probability generating functions, help please!

With a sequence of $N$ independent Bernoulli trials performed, where $N \in \mathbb{Z}^+$ and the probability of success on any trial is $p$, and $S$ and $F$ being total number of success and fails ...
0
votes
1answer
30 views

Probability generating function question

The probability generating function of a non-negative, integer valued random variable $A$ is given by: $G(b) = \cfrac{e^{2(b-1)}}{2-b}, (|b| \lt 2)$ To determine ...
1
vote
3answers
52 views

Probability of even number of successes in a series of independent trials

Consider a series of independent trials at each of which there is a success of a failure with probabilities $p$ and $1-p$ respectively. I am finding it difficult to derive the probability of even ...
1
vote
2answers
34 views

Probability of obtaining equal number of each outcome of a fair die at the nth trial

Suppose a fair die is tossed repeatedly. I am concerned in deriving the probability of the occurrence of obtaining equal number of each possible outcome at the nth trial. Clearly this is only possible ...
0
votes
0answers
29 views

Multivariate generating function - am I to assume bivariate normal?

I've run into a problem that I can't quite seem to get started on, and I think it might be because of an assumption that may (or may not) be made. My textbook frequently references to bivariate ...
2
votes
1answer
67 views

Generating functions and the blue eyed daughters

There is a famous problem, given that a man has a number of daughters and if you were to meet two of them at random there is a 50% chance that both have blue eyes. How many daughters does the man ...
0
votes
0answers
86 views

Expectation and Variance of a Discrete Uniform Distribution using the Probability Generating Function and Cumulant Generating Function

Hi I just derived the MGF of a discrete uniform distribution and found it to be: [e^t - e^t(m+1)]/(1 - e^t)m and the pgf is ...
0
votes
1answer
28 views

how to find f(21) of the following probability generating function

I have a pgf that seems to me would take more than 5 minutes to find f(21) of. Does anyone know how to compute f(21) of this pgf within the specified time? ...
1
vote
1answer
34 views

relation between multivariate probability generating function and univariate ones

Suppose I have two independent integer random variables $X_1$, $X_2$ (with constraint that $X_1+X_2\le N,0\le X_1\le N,0\le X_2\le N$), with probability generating functions $g_1(z)$, $g_2(z)$. Now I ...
2
votes
1answer
33 views

Show that a function is a probability generating function

I'm doing past papers in order to revise for exams in June and my university irritatingly doesn't provide any mark schemes and I'm very stuck on a question. The question says: Let ...
0
votes
1answer
37 views

multivariate probability generating function

Suppose I have three random variables $X_1$, $X_2$ and $X_3$, with probability generating functions $g_1(z)$, $g_2(z)$ and $g_3(z)$. Now I have a joint-distribution $P(X_1-X_2,X_1-X_3)$, whose ...
0
votes
0answers
24 views

probability in graphs - degree distribution

I am reading this paper on networks which employs probability in analyzing graphs. Suppose that a graph has $n$ vertices. Furthermore, if each vertex has a probability $p_k$ of having $k$ neighbors, ...
1
vote
1answer
55 views

Find the probability generating function

I have an exercise of this type that I just can not solve "Are $x$ and $y$ be independent random variables, $X$-Poisson($a$), $Y$-Poisson($b$). Find the probability generating function of the random ...
2
votes
1answer
84 views

Differentiate $P_{x_n}(z) = \prod_{i=1}^n\frac{1+z+z^2+…+z^{i-1}}{i}$ twice to calculate the variance of involutions.

Use the Probability Generating Function for Involutions: $P_{x_n}(z) = \prod_{i=1}^n\frac{1+z+z^2+...+z^{i-1}}{i}$ To Calculate the Variance of Involutions where: $Variance \space X_n = ...
0
votes
1answer
59 views

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 ...
3
votes
4answers
81 views

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 ...
1
vote
1answer
125 views

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 ...
0
votes
2answers
45 views

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 ...
0
votes
2answers
72 views

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 ...
4
votes
0answers
116 views

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 ...
0
votes
0answers
58 views

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 ...
3
votes
1answer
144 views

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 ...
1
vote
1answer
51 views

Prove that $P_{S_N}(t) = P_N(P_X(t))$ for $S_N = X_1 + \cdots + X_N$.

Let $N$ and $(X_i)_{(i \ge 1)}$ be independent random variables ($X_i$ have the same density). Let $S_N = X_1 + \cdots + X_N$. Prove that $P_{S_N}(t) = P_N(P_X(t))$ where $$P_X(t) = E(t^X) = ...
2
votes
1answer
570 views

finding probability generating function and the sum of two independent random variables

Let $X$ be a discrete random variable with probability mass function $$P_X(x) = p(1-p)^x,\qquad x=0,1,2,3,\ldots$$ (a) Find the probability generating function for $X$ and hence ...
1
vote
1answer
83 views

Change of variable in an infinite sum

I'm currently trying to understand a derivation from WolframMathWorlds. I got to step 6 where a change of variable happens. You can see the equation here. I understand everything except how they get ...
1
vote
1answer
108 views

CDF from generating function

Is there a way to obtain the CDF of a discrete random variable directly from one of its generating functions?
1
vote
1answer
68 views

moment generating function of gambling [duplicate]

Suppose a gambler starts with one dollar and plays a game in which he or she wins one dollar with probability $p$ and loses one dollar with probability $1-p$. Let $f_n$ be the probability that he or ...
0
votes
2answers
40 views

algebraic manipulation question

$M_{z_n}(t)$ is a particular moment generating function, and it is given that $\lambda_n$ approaches $\infty$ as $n$ approaches $\infty$: Could someone help me see how the above was derived?
2
votes
1answer
234 views

Proof that $\frac{(\bar X-\mu)}{\sigma}$ and $\sum_{i=1}^n\frac{(X_i-\bar X)^2}{\sigma^2}$ are independent

Let $X_i\sim N(\mu,\sigma^2)$ ; where$[i=1,2,\ldots,n]$ $Z_i\sim N(0,1)$ ; where$[i=1,2,\ldots,n]$ Proof that $\bar Z=\frac{(\bar X-\mu)}{\sigma}$ and $\sum_{i=1}^{n}(Z_i-\bar ...
2
votes
1answer
79 views

How the generating function $P(s)=\mathbb E[s^X]$ uniquely determines probabilities $p_n$, $n=1,2,\ldots$

for determining the probabilities, it has been written on the book that: $$p_n=\frac{\frac{d^n}{ds^n}P(s)|_{s=0}}{n!};\ldots(A)$$ But if i set $s=0$ then $p_n$ becomes $0$. ...
1
vote
1answer
206 views

How many compositions of n are there with same parity

Let n be a non-negative integer. How many compositions of n are there where the i-th part has the same parity as i? For example, compositions of 7 that satisfy this condition are (7), (5,2), (3,4), ...
1
vote
1answer
55 views

What kind of functions can be moment-generating functions for a random variable?

Given an infinitely differentiable function $ g: \mathbb{R} \rightarrow \mathbb{R}$, can we always find a distribution function $f_X$ of some random variable $X$ so that $g(t) = \int_{-\infty}^\infty ...
0
votes
1answer
59 views

Moment generating function, change of variable

If $X=aY$ where $X, Y$ are random variables and $a$ is a constant. Do I have $$M_X(t)=M_Y(t/a)$$ where $M$ is the moment generating function. How can I prove it?
2
votes
1answer
900 views

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 ...
4
votes
2answers
159 views

A classical problem in combinatorics/probability

I read this problem in Cognition and Chance by Raymond Nickerson (the problem is stated not discussed) ...
0
votes
1answer
162 views

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 ...
1
vote
1answer
102 views

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 ...
0
votes
1answer
165 views

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 ...
4
votes
2answers
793 views

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: ...
0
votes
1answer
51 views

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, ...
0
votes
3answers
166 views

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 $P(XY=0)$?
1
vote
0answers
177 views

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 ...
0
votes
2answers
243 views

Calculating coefficients of generating function

Fist I'll explain the problem I had to solve (and which I solved), and then ask a related question. We have a bin with 2 balls: black and white. We take one from the bin and put back. Than we add a ...
0
votes
1answer
63 views

Generating functions of discrete random variable

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 $\mathbf{N}$ ...