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Zbl 1153.60400
Coelho, Carlos A.
The joint characterization of discrete and continuous `waiting times' through their reciprocal relationships. (English)
[J] J. Interdiscip. Math. 9, No. 2, 297-318 (2006). ISSN 0972-0502

Summary: Let us suppose there is an event $C$ (occurrence of a given defect or disease) that is part of a group of events we are interested in, and whose probability of occurrence in that group is known. The distribution of the waiting times for the $r_1$th event in that group, given that we expect $r$ events $C$, and the distribution of the number of events in the group that are not events $C$, given that we waited for a length $y$ waiting for the $r_1$th event in the group (given that we expected $r$ events $C$) are derived based on very mild assumptions. Relations of the distributions obtained with known distributions, their expression as mixtures and a limiting case are also studied. \par Cases where $r_1<r$ and $r_1>r$ are studied in detail since they correspond to two different situations of interest, the one in which the event $C$ is one of the rarest ones in its group, is not easy to identify or its occurrence $r$ times kills or disables the observation unit, or the case in which it may be rather common in the group and easy to identify. \par Examples of application in epidemiology, industry, transportation and agriculture are used for illustration.

MSC 2000:: *60K99 Special processes
60E05 General theory of probability distributions

Keywords: waiting times discrete and continuous; mixtures; negative binomial; generalized gamma distribution; non-central generalized gamma distribution

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Zentralblatt MATH Berlin [Germany]