Difference Equations of Moment-Generating Functions for some Probability Distributions

Nature ◽  
1950 ◽  
Vol 165 (4192) ◽  
pp. 370-370
Author(s):  
P. V. KRISHNA IYER
Keyword(s):  
Difference Equations ◽  
Generating Functions ◽  
Probability Distributions ◽  
Moment Generating Functions

scholarly journals A Finite-Interval Uniqueness Theorem for Bilateral Laplace Transforms

2007 ◽  
Vol 2007 ◽  
pp. 1-6 ◽  
Author(s):  
Patrick Chareka
Keyword(s):  
Laplace Transform ◽  
Uniqueness Theorem ◽  
Generating Functions ◽  
Finite Interval ◽  
Probability Distributions ◽  
Laplace Transforms ◽  
Complex Argument ◽  
Moment Generating Functions

Two or more bilateral Laplace transforms with a complex argument “s” may be equal in a finite vertical interval when, in fact, the transforms correspond to different functions. In this article, we prove that the existence of a bilateral Laplace transform in any finite horizontal interval uniquely determines the corresponding function. The result appears to be new as we could not find it in the literature. The novelty of the result is that the interval need not contain zero, the function need not be nonnegative and need not be integrable. The result has a potential to be useful in the context of fitting probability distributions to data using Laplace transforms or moment generating functions.

scholarly journals Lattice Paths, Sampling Without Replacement, and Limiting Distributions

10.37236/156 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
M. Kuba ◽  
A. Panholzer ◽  
H. Prodinger
Keyword(s):  
Generating Functions ◽  
Probability Distributions ◽  
Random Variable ◽  
Phase Changes ◽  
Lattice Paths ◽  
Sampling Without Replacement ◽  
Sample Paths ◽  
Quarter Plane ◽  
Explicit Formulae ◽  
Weighted Lattice Paths

In this work we consider weighted lattice paths in the quarter plane ${\Bbb N}_0\times{\Bbb N}_0$. The steps are given by $(m,n)\to(m-1,n)$, $(m,n)\to(m,n-1)$ and are weighted as follows: $(m,n)\to(m-1,n)$ by $m/(m+n)$ and step $(m,n)\to(m,n-1)$ by $n/(m+n)$. The considered lattice paths are absorbed at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$. We provide explicit formulæ for the sum of the weights of paths, starting at $(m,n)$, which are absorbed at a certain height $k$ at lines $y=x/t -s/t$ with $t\in{\Bbb N}$ and $s\in{\Bbb N}_0$, using a generating functions approach. Furthermore these weighted lattice paths can be interpreted as probability distributions arising in the context of Pólya-Eggenberger urn models, more precisely, the lattice paths are sample paths of the well known sampling without replacement urn. We provide limiting distribution results for the underlying random variable, obtaining a total of five phase changes.

Bounds of moment generating functions of some life distributions

2005 ◽  
Vol 46 (4) ◽  
pp. 575-585 ◽  
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Generating Functions ◽  
Moment Generating Functions ◽  
Life Distributions
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