UAH > Math > Colloquia > 9/9/2005
Let S denote the collection of all finite subsets of N+. We will define an operation on S that makes S into a positive semigroup with set inclusion as the associated partial order. Positive semigroups are the natural mathematical home for probability distributions with "exponential" properties (such as the memoryless, constant rate, and maximum entropy properties). We will show that there are no exponential distributions on S, but that S can be partitioned into a countable collection of sub-semigroups, each of which supports a one-parameter family of exponential distributions. We then turn to the problem of finding the distribution on S that is as close to exponential as possible. This work may have some practical value in terms of the most random way to pick a finite subset of a countably infinite population.