Browsed byCategory: Probability Theory

Extensions of the Single Server Efficiency Model

Extensions of the Single Server Efficiency Model

For the full paper, which includes all proofs, click here. Abstract Editor's note: this paper comprises the third chapter of the PhD dissertation by Rachel Traylor. Visit here and here to see chapters one and two, respectively. Herein we further generalize the single server model presented in . In particular, we consider a multichannel server…

Generalizing the Negative Binomial Distribution via First-Kind Dependence

Generalizing the Negative Binomial Distribution via First-Kind Dependence

This paper generalizes the negative binomial random variable by generating it from a sequence of first-kind dependent Bernoulli trials under the identity permutation. The PMF, MGF, and various moments are provided, and it is proven that the distribution is indeed an extension of the standard negative binomial random variable. We examine the effect of complete…

On Permuted First-Kind Dependence of Categorical Random Variables

On Permuted First-Kind Dependence of Categorical Random Variables

This paper discusses the notion of horizontal dependency in sequences of first-kind dependent categorical random variables. We examine the necessary and sufficient conditions for a sequence of first-kind dependent categorical random variables to be identically distributed when the conditional probability distribution of subsequent variables after the first are permuted from the identity permutation used…

On Server Efficiency

On Server Efficiency

For the full text of the paper, including all proofs and supplementary lemmata, click to download  Abstract Editor's note: This paper comprises the second chapter of the PhD dissertation by Rachel Traylor. Cha and Lee defined a mathematical notion of server performance by measuring efficiency defined as the long run average number of jobs completed per…

A Generalized Geometric Distribution from Vertically Dependent Bernoulli Random Variables

A Generalized Geometric Distribution from Vertically Dependent Bernoulli Random Variables

For full proofs and derivations, read here. Abstract This paper generalizes the notion of the geometric distribution to allow for dependent Bernoulli trials generated from dependency generators as defined in Traylor and Hathcock's previous work. The generalized geometric distribution describes a random variable that counts the number of dependent Bernoulli trials until the first success. The…

Stochastic Reliability of a Server under a Random Workload

Stochastic Reliability of a Server under a Random Workload

For the full paper, which includes all proofs, click here if you dare. Abstract Editor's note: This paper is the first chapter from a PhD thesis published in 2016 by Rachel Traylor. We generalize a 2011 model from Cha and Lee that gave a closed form for the survival function of a server under a random…

Vertical Dependency in Sequences of Categorical Random Variables

Vertical Dependency in Sequences of Categorical Random Variables

For the full text, including proofs, download the pdf here. Abstract This paper develops a more general theory of sequences of dependent categorical random variables, extending the works of Korzeniowski (2013) and Traylor (2017) that studied first-kind dependency in sequences of Bernoulli and categorical random variables, respectively. A more natural form of dependency, sequential dependency, is…

A Generalized Multinomial Distribution from Dependent Categorical Random Variables

A Generalized Multinomial Distribution from Dependent Categorical Random Variables

For the full paper, which includes all proofs, download the pdf  here. Abstract Categorical random variables are a common staple in machine learning methods and other applications across disciplines. Many times, correlation within categorical predictors exists, and has been noted to have an effect on various algorithm effectiveness, such as feature ranking and random forests.…