DIRICHLET-RELATED PROBABILITY PROBLEMS AND NEW TOOLS FOR SOLVING THEM

ABSTRACT

In this paper we primarily consider waiting time problems under three different sampling rules. SR₁ is the usual sampling with replacement, SR₂ is without replacement, and SR₃ is also with replacement, but uses no repetitions. We develop a new methodology for solving a wide variety of waiting time problems under each of the three sampling rules. A connection between waiting time problems under SR₂ and SR₃ is established which enables one to simultaneously solve waiting time problems under both of these sampling rules. The methods are illustrated with a large number of examples.

Keywords:

• Dirichlet I, J, C, and D functions
• Waiting time problems
• Generating functions

ACKNOWLEDGMENTS

The second author would like to thank the Natural Sciences and Engineering Research Council of Canada for funding this research. We would also like to thank the referee for comments that led to an improvement in the presentation of the paper.