On Generating Distributions with the Memoryless Property

Abstract

The exponential and geometric distribution are well-known continuous and discrete family of distributions with the memoryless property, respectively. The memoryless property is emphasized in introductory probability and statistics textbooks even though no distribution beyond these two families of distributions has been explored in detail. By examining the relationship between these two families of distributions, we propose a general algorithm for generating distributions with the memoryless property. Then, we show that the general algorithm uniquely determines the distribution with the memoryless property given the parameter value, and nonnegative support which contains zero and is closed under addition. Furthermore, we present a few nontrivial examples and their applications to demonstrate the richness of such distributions.

KEYWORDS:

• Exponential distribution
• Geometric distribution
• Memorylessness

Acknowledgments

The authors are grateful to the Editor, Associate Editor, and two anonymous reviewers for their helpful comments. The authors are also thankful to Amites Sarkar, Alexander Kuhn, Patrick Carroll, Samuel Burnett, and Fiona Cleary for carefully proofreading this article.