Probability Distributions on Partially Ordered Sets

Dr. Kyle Siegrist

Department of Mathematical Sciences
University of Alabama in Huntsville

April 18, 2008
219 Shelby Center
3:00 PM (Refreshments at 2:30)

Abstract

The geometric distribution on N, and the exponential distribution on [0, ∞) both have the constant rate property: the upper probability function F and the probability density function f (with respect to counting measure in the first case and Lebesgue measure in the second) are related by f = α F for some α > 0. Moreover, these distributions are the only ones (on N and [0, ∞), respectively) with this property. The two distributions are the building blocks of other important distributions (negative binomial in the first case, gamma in the second), which in turn lead to fundamentally important random processes (Bernoulli trials in the first case, the Poisson process in the second).

In this talk I will discuss probability distributions on general partially ordered sets that have the constant rate property. In spite the generality, and the lack of any other algebraic structure, a surprising amount of the theory stills goes through--constant rate distributions have nice moment properties and lead to "gamma" distributions and a "Poisson" process. We will see that, in many respects, constant rate distributions describe the "most random way" to put points in the poset.

Finally, I will discuss the characterization problem: when does a poset support constant rate distributions, and how many "free parameters" will the distribution have? The answer is fairly easy, but still interesting, in the case of rooted trees, but to the best of my knowledge the problem is open for more general classes of posets.