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Probability-generating function

About: Probability-generating function is a research topic. Over the lifetime, 752 publications have been published within this topic receiving 9361 citations.


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Journal ArticleDOI
24 Mar 2020
TL;DR: The main reliability characteristics of the system—the reliability function and the steady state probabilities—have been found in analytical form and can be used in the studies of various applications of systems with dependent failures between their elements.
Abstract: In this paper a two component redundant renewable system operating under the Marshall–Olkin failure model is considered. The purpose of the study is to find analytical expressions for the time dependent and the steady state characteristics of the system. The system cycle process characteristics are analyzed by the use of probability interpretation of the Laplace–Stieltjes transformations (LSTs), and of probability generating functions (PGFs). In this way the long mathematical analytic derivations are avoid. As results of the investigations, the main reliability characteristics of the system—the reliability function and the steady state probabilities—have been found in analytical form. Our approach can be used in the studies of various applications of systems with dependent failures between their elements.

1 citations

Posted Content
TL;DR: The probability generating functions of the number of turns required to end the games of various probabilistic games with piles for one player or two players are investigated and interesting recurrence relations for the sequences of such functions in n are derived.
Abstract: We consider various probabilistic games with piles for one player or two players. In each round of the game, a player randomly chooses to add $a$ or $b$ chips to his pile under the condition that $a$ and $b$ are not necessarily positive. If a player has a negative number of chips after making his play, then the number of chips he collects will stay at $0$ and the game will continue. All the games we considered satisfy these rules. The game ends when one collects $n$ chips for the first time. Each player is allowed to start with $s$ chips where $s\geq 0$. We consider various cases of $(a,b)$ including the pairs $(1,-1)$ and $(2,-1)$ in particular. We investigate the probability generating functions of the number of turns required to end the games. We derive interesting recurrence relations for the sequences of such functions in $n$ and write these generating functions as rational functions. As an application, we derive other statistics for the games which include the average number of turns required to end the game and other higher moments.

1 citations

01 Jan 2003
TL;DR: In this article, a class of probability generating functions for N, the sample size, and a NaS condition that implies the convergence to an ID (MID) law by convergence to a ϕ-ID (ϕ)-MID law and vise versa are discussed.
Abstract: Infinitely divisible (ID) and max-infinitely divisible (MID) laws are studied when the sample size is random. ϕ-ID and ϕ-MID laws introduced and studied here approximate random sums and random maximums. The main contributions in this study are: (i) in discussing a class of probability generating functions for N, the sample size, (ii) a NaS condition that implies the convergence to an ID (MID) law by the convergence to a ϕ-ID (ϕ-MID) law and vise versa and thus a discussion of attraction and partial attraction for ϕ-ID and ϕ-MID laws.
Journal ArticleDOI
TL;DR: This paper presents some tips for generating multiple sequences of two- and three-valued random variables and explains the importance of knowing the values of these variables.
Abstract: We present some tips for generating multiple sequences of two- and three-valued random variables.

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20236
202211
20217
202014
201912
20188