Tagged Questions

0
votes
1answer
36 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
41 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 ...
0
votes
0answers
34 views

is it right to subs m=-m into ordinary generating function to be z-transform?

i success convert exponential generating function into ordinary generating function ordinary generating function = summation(p(x)*z^(-m),m=0..infinity) (Maple code) substitute m=-m into this ...
4
votes
2answers
145 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
101 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 = ...
0
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0answers
94 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
88 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
87 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
261 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
63 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
128 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, ...