Constant Rate Distributions on Partially Ordered Sets

We consider probability distributions with constant rate on partially ordered sets, generalizing distributions in the usual reliability setting that have constant failure rate. In spite of the minimal algebraic structure, there is a surprisingly rich theory, including moment results and results concerning ladder variables and point processes. We concentrate mostly on discrete posets, particularly posets whose graphs are rooted trees. We pose some questions on the existence of constant rate distributions for general discrete posets.Comment: 21 page

Upward and Downward Runs on Partially Ordered Sets

We consider Markov chains on partially ordered sets that generalize the success-runs and remaining life chains in reliability theory. We find conditions for recurrence and transience and give simple expressions for the invariant distributions. We study a number of special cases, including rooted trees, uniform posets, and posets associated with positive semigroups.Comment: 13 page

Broadcasting in Partially Ordered Sets

Let # be a partial order on a countable set V such that there exists a minimum element a # V and {u # V : u # v} is finite for each v # V . An item of information is to be broadcast to the elements of V according to the policy # in the following sense: a is initially informed and broadcasting to v #= a begins as soon as all elements u # v are informed; the time required to broadcast the information is X(v) where {X(v) : v #= a} is a collection of independent, identically distributed variables with mean 1. The natural random variables associated with this model are the time required to broadcast the information to a given set of vertices and the set of informed vertices at a given time. These variables are studied in terms of the structure of the directed graph associated with #. A number of stochastic comparison results are obtained. The common model for deterministic broadcasting in trees is a special case. 1 Introduction Suppose that # is a partial order on a..

Random Samples

Created by Kyle Siegrist of the University of Alabama-Huntsville, this is an online, interactive lesson on random samples. The author provides examples, exercises, and applets concerning many different topics. Some of these include: sample mean, law of large numbers, sample variance, partial sums, central limit theorem, special properties of normal samples, order statistics, and sample covariance and correlation. Additionally, the author provides links to external resources for students wanting to engage further with the topic. This is simply one of seventeen lessons. They are all easily accessible as the author has formatted his site much like an online textbook

Point Estimation

Created by Kyle Siegrist of the University of Alabama-Huntsville, this is an online, interactive lesson on point estimation. The author provides examples, exercises, and applets about the topic. More specifically, they concern estimators, method of moments, maximum likelihood, Bayes' estimators, best unbiased estimators, and sufficient, complete and ancillary statistics. Additionally, the author provides links to external resources for students looking to engage in a more in-depth study of the topic. This is simply one lesson in a series of seventeen. They are easily accessible as the author has created the site in an online textbook format

The Poisson Process

Created by Kyle Siegrist of the University of Alabama-Huntsville, this online, interactive lesson on the Poisson process provides examples, exercises, and applets. Specific topics include the exponential distribution, gamma distribution, Poisson distribution, splitting a Poisson process, analogy with Bernoulli trials, and higher dimensional Poisson processes. Additionally, the author offers external resources for those interested in further study of this statistical concept. Overall, this is a nice resource as it provides students with definitions and then allows them to apply these theories in the form of interactive applets

Markov Chains

Created by Kyle Siegrist of the University of Alabama-Huntsville, this is an online, interactive lesson on Markov chains. The author provides examples, exercises, and applets to introduce the subject. More specifically, the lesson covers: recurrence, transience, periodicity, time reversal, as well as invariant and limiting distributions. In addition, the author has provided links to external resources for further research. Overall, this is a great resource for those interested in this statistical process. It provides basic examples to introduce the topic, but also provides a more in-depth study to further challenge students

Expected Value

This online, interactive lesson on expected value provides examples, exercises, and applets in which students will explore relationships between the expected value of real-valued random variables and the center of the distribution. Students will also examine how expected values can be used to measure spread and correlation

Partially Ordered Sets

We consider probability distributions with constant rate on partially ordered sets, generalizing distributions in the usual reliability setting 0, ∞ , ≤ that have constant failure rate. In spite of the minimal algebraic structure, there is a surprisingly rich theory, including moment results and results concerning ladder variables and point processes. We concentrate mostly on discrete posets, particularly posets whose graphs are rooted trees. We pose some questions on the existence of constant rate distributions for general discrete posets