0
votes
1answer
14 views

Find the probability generating function of a GW process

Consider a Galton-Watson process with offspring distribution $\mathrm{Poisson}(1)$. That is, $\textbf{p}(k) = \frac{e^{-1}}{k!}$. Given this information, and that $P(z) = ...
2
votes
1answer
59 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
37 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
32 views

How is this step completed?

User Did, did this step in his answer to my previous question: $$\sum_{k=0}^n{n\choose k}(zp)^kq^{n-k}=(q+pz)^n.$$ How is it done? Is it simply an identity, or something more?
0
votes
2answers
50 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
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 ...
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 ...
3
votes
1answer
20 views

prove tail probabilities equation

Let's say $p(z) = E[z^X]$, which is a probability generating function of a random variable X. Could we prove following equation? $\sum_{k \geq 0} Pr[X \geq k] z^k = \frac{1-zp(z)}{1-z}$ in which ...
1
vote
1answer
114 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
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 ...
0
votes
0answers
176 views

Probability generating function for logarithmic series distribution, support $k\geq1$

I'm trying to derive the probability generating function (pgf) for the logarithmic series distribution, and not getting the expected form $\frac{\log{(1-qs)}}{\log{(1-q)}}$. It seems that pgfs are ...
2
votes
1answer
77 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 $0$. ...
0
votes
1answer
111 views

Probability density function for the normalised sum of N random variables

I was wondering what the PDF looks like for Z= (1/N)*SUM(z_1+...+ z_n), where each z_i is computationally represented by RAND(). What is the behaviour of the PDF as N -> infinity?
2
votes
1answer
53 views

Finding $G_{cX}(t)$ given $G_{X}(t)$

I have recently been studying probability generating functions, and have seen the proof that the sum of two independent Poisson random variables has a Poisson distribution also. This used the fact ...
4
votes
2answers
748 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: ...
2
votes
2answers
351 views

Obtaining cumulants using the characteristic function

If a random variable $x$ has a characteristic function $\phi(\omega)$, then the $n^{\mathrm{th}}$ moment of the distribution of $x$, $\mu_n$ can be calculated as: $$\mu_n = ...
1
vote
0answers
172 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 ...
5
votes
1answer
125 views

Relationship between moments of a random variable

Let $X$ be a random variable with continuous density $\rho(x)$. Assume that $X$ is symmetric and $\vert X\vert<L$. Since it has a bounded support, all moments of $X$ are well-defined. Let $m_i$ ...
1
vote
1answer
147 views

Partial fractions for geometric probability-generating function wrong

Let $X\sim \text{Geo}(1/4), Y\sim \text{Geo}(1/2)$ be given. First I have to compute $\mathbb{E}[z^{X+Y}]$: ...
3
votes
2answers
399 views

Generating function for Banach's matchbox problem

Here's the description for Banach's matchbox problem from Concrete Mathematics EXERCISE 8.46 (edited) Stefan Banach used to carry two boxes of matches, one containing $m$ matches and the other one ...
3
votes
1answer
73 views

How to solve a functional equation of the form $1-g(f(s))=m(1-g(s))$?

I have arrived to this equation in several contexts within branching processes. It arises from textbook exercises, so it must be solvable somehow. Here $f$ is a probability generating function which ...
1
vote
2answers
215 views

Showing that a random sum of logarithmic mass functions has negative binomial distribution

Specific questions are bolded below. I've been unsuccessful in solving the following problem., which is exercise 5.2.3 from Probability and Random Processes by Grimmett and Stirzaker. Let $X_1, ...