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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Bibliographic Details
Published in:Statistics & probability letters 2007, Vol.77 (1), p.32-39
Main Authors: Antzoulakos, Demetrios L., Boutsikas, Michael V.
Format: Article
Language:English
Subjects:
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
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Summary: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 ) .
ISSN:0167-7152
1879-2103
DOI:10.1016/j.spl.2006.05.009