Inequalities for sums of independent geometrical random variables
type: Item
instance of: scholarly article
title: Inequalities for sums of independent geometrical random variables
zbMATH Open document ID: 0893.60006
DOI: 10.1007/BF02755453
published in: Aequationes Mathematicae
publication date: 1998-08-10T00:00:00Z
full work available at URL: https://eudml.org/doc/137701
review text: The concepts of majorization and Schur convexity are examined in relation to the probability distribution function of the sum of independent geometric and negative binomial variables. It is shown in particular, that the function \(P(X_1+ X_2+\cdots+ X_n\leq t)\), where \(X_1, X_2,\dots, X_n\) are independent geometric random variables with parameters \(p_1,p_2,\dots, p_n\) is Schur-concave with respect to \((p_1,p_2,\dots, p_n)\) for every real \(t\). The result is extended to the case of the negative binomial distribution, where each of the random variables \(X_i\), \(i= 1,2,\dots, n\), represents the number of trials until the \(r\)th success in a sequence of Bernoulli trials. The case where the random variables \(X_i\), \(i=1,2,\dots, n\), are distributed independently according to the negative binomial distribution with unequal parameters \(r_i\), \(i= 1,2,\dots,n\), is also considered.
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 62E99
zbMATH DE Number: 1053606
zbMATH Keywords: majorization
zbMATH Keywords: Schur convexity
zbMATH Keywords: geometric and negative binomial variables
author: Milan Merkle
author: Ljiljana Petrović
reviewed by: Evdokia Xekalaki
MaRDI profile type: Publication
cites work: Inequalities for linear combinations of gamma random variables
cites work: On the distribution of the number of successes in independent trials
cites work: On the Distribution of the Number of Successes in Independent Trials
cites work: Probability inequalities for convex sets and multidimensional concentration functions
cites work: Peakedness of Distributions of Convex Combinations
OpenAlex ID: W1994912998
Recommended article: Inequalities for Sums of Independent Random Variables
Recommended article: Q3471277
Recommended article: Q5748680
Recommended article: Q3987001
Recommended article: Probability inequalities for sums of independent unbounded random variables
Recommended article: On the geometric sum of iid random variables
Recommended article: Q3973424
description: scientific article; zbMATH DE number 1053606