# Tagged Questions

1answer
56 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 ...
2answers
36 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 ...
1answer
30 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?
2answers
48 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 ...
1answer
27 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 ...
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 ...
1answer
36 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 ...
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 ...
1answer
105 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 ...
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 ...
0answers
164 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 ...
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 $0$. ...
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?
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 ...
2answers
724 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: ...
2answers
338 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 = ...
0answers
170 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 ...
1answer
123 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 $m_i$ ...
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 $\mathbb{E}[z^{X+Y}]$: ...
2answers
394 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 ...
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 ...
2answers
211 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, ...