1
vote
1answer
29 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
28 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
30 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
20 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
42 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
75 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
42 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
78 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
64 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
33 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
65 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
100 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
53 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
103 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
464 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
62 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
67 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
65 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
37 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
202 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
74 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
204 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
53 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
53 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
777 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
145 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
93 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
77 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
152 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
590 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
50 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
149 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 $P(XY=0)$?
0
votes
0answers
161 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
222 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
61 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 $\mathbf{N}$ ...
3
votes
0answers
335 views

Asymptotics for the expected length of the longest streak of heads.

As Introduction to Algorithms (CLRS) describes, the problem is Suppose you flip a fair coin $n$ times. What is the longest streak of consecutive heads that you expect to see? The book claims ...
3
votes
2answers
379 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
2answers
797 views

“infinite moments” or “moments don't exist”?

what is the difference between "infinite moment" and "moments don't exist"? moreover, if I find out the moment generating function for some distribution, and take first derivative set s=1, and if I ...
7
votes
3answers
828 views

Probability that sum of rolling a 6-sided die 10 times is divisible by 10?

Here's a question I've been considering: Suppose you roll a usual 6-sided die 10 times and sum up the results of your rolls. What's the probability that it's divisible by 10? I've managed to solve it ...
4
votes
4answers
378 views

Find the ordinary generating function $h(z)$ for a Gambler's Ruin variation.

Assume we have a random walk starting at 1 with probability of moving left one space $q$, moving right one space $p$, and staying in the same place $r=1-p-q$. Let $T$ be the number of steps to reach ...
1
vote
2answers
196 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, ...