A direct method to obtain the joint distribution of successes, failures and patterns in enumeration problems

Let E be a pattern (single or composite) defined over a sequence of Bernoulli trials. Let X n , S n and F n be the number of occurrences of E , the number of successes and the number of failures in n trials, respectively. Also let T r , S T r and F T r be the waiting time for the rth occurrence of E...

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Vydané v:	Statistics & probability letters Ročník 77; číslo 1; s. 32 - 39
Hlavní autori:	Antzoulakos, Demetrios L., Boutsikas, Michael V.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Amsterdam Elsevier B.V 2007 Elsevier
Edícia:	Statistics & Probability Letters
Predmet:	Applications Combinatorial probability Distribution theory Double generating function Exact sciences and technology Mathematics Patterns Patterns Success runs Scans Probability generating function Double generating function Waiting time distribution Probability and statistics Probability generating function Probability theory and stochastic processes Reliability, life testing, quality control Scans Sciences and techniques of general use Statistics Success runs Waiting time distribution Success runs Waiting time distribution Probability generating function Double generating function Patterns Scans Scanning Bernoulli trial Generating function Probability theory Probability distribution Joint distribution Waiting time Enumeration(counting) Direct method Statistical method Time distribution Distribution function Failure time
ISSN:	0167-7152, 1879-2103
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Abstract	Let E be a pattern (single or composite) defined over a sequence of Bernoulli trials. Let X n , S n and F n be the number of occurrences of E , the number of successes and the number of failures in n trials, respectively. Also let T r , S T r and F T r be the waiting time for the rth occurrence of E in an infinite sequence of trials, the number of successes and the number of failures observed at that time, respectively. In this article we present some simple results revealing that, if we know either the distribution of X n or the distribution of T r we can instantly derive the joint distribution of ( X n , S n , F n ) and ( T r , S T r , F T r ) .
AbstractList	Let be a pattern (single or composite) defined over a sequence of Bernoulli trials. Let Xn, Sn and Fn be the number of occurrences of , the number of successes and the number of failures in n trials, respectively. Also let Tr, STr and FTr be the waiting time for the rth occurrence of in an infinite sequence of trials, the number of successes and the number of failures observed at that time, respectively. In this article we present some simple results revealing that, if we know either the distribution of Xn or the distribution of Tr we can instantly derive the joint distribution of (Xn,Sn,Fn) and (Tr,STr,FTr). Let E be a pattern (single or composite) defined over a sequence of Bernoulli trials. Let X n , S n and F n be the number of occurrences of E , the number of successes and the number of failures in n trials, respectively. Also let T r , S T r and F T r be the waiting time for the rth occurrence of E in an infinite sequence of trials, the number of successes and the number of failures observed at that time, respectively. In this article we present some simple results revealing that, if we know either the distribution of X n or the distribution of T r we can instantly derive the joint distribution of ( X n , S n , F n ) and ( T r , S T r , F T r ) .
Author	Boutsikas, Michael V. Antzoulakos, Demetrios L.
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Keywords	Success runs Waiting time distribution Probability generating function Double generating function Patterns Scans Scanning Bernoulli trial Generating function Probability theory Probability distribution Joint distribution Waiting time Enumeration(counting) Direct method Statistical method Time distribution Distribution function Failure time
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Snippet	Let E be a pattern (single or composite) defined over a sequence of Bernoulli trials. Let X n , S n and F n be the number of occurrences of E , the number of... Let be a pattern (single or composite) defined over a sequence of Bernoulli trials. Let Xn, Sn and Fn be the number of occurrences of , the number of successes...
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SubjectTerms	Applications Combinatorial probability Distribution theory Double generating function Exact sciences and technology Mathematics Patterns Patterns Success runs Scans Probability generating function Double generating function Waiting time distribution Probability and statistics Probability generating function Probability theory and stochastic processes Reliability, life testing, quality control Scans Sciences and techniques of general use Statistics Success runs Waiting time distribution
Title	A direct method to obtain the joint distribution of successes, failures and patterns in enumeration problems
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