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Author: Rachel Traylor

The Red-Headed Step-Distributions

The Red-Headed Step-Distributions

Almost every textbook in probability or statistics will speak of classifying distributions into two different camps: discrete (singular in some older textbooks) and continuous. Discrete distributions have either a finite or a countable sample space (also known as a set of Lebesgue measure 0), such as the Poisson or binomial distribution, or simply rolling a…

(Commentary) Spectre and Meltdown: Spokes on a Wheel

(Commentary) Spectre and Meltdown: Spokes on a Wheel

There has been a flurry of articles and discussions related to Intel’s Spectre and Meltdown vulnerabilities. Many good writings discuss the technical nature and implications to hardware, and you can find a selection here, here, and here. As of this writing, many software developers and security experts are frantically trying to create patches to protect…

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…

Welcome to GF(4)

Welcome to GF(4)

Everyone has solved some version of a linear system in either high school or college mathematics. If you've been keeping up with some of my other posts on algebra, you know that I'm about to either take something familiar away, or twist it into a different form. This time is no different; we're going to…

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…

A Partition by any Other Name

A Partition by any Other Name

I promise I'm actually a probability theorist, despite many of my posts being algebraic in nature. Algebra, as we've seen in several other posts, elegantly generalizes many things in basic arithmetic, leading to highly lucrative applications in coding theory and data protection.  Some definitions in mathematics may not have obvious "practical use", but turn out to yield theorems and results so…

Time Series Analysis Part 1: Regression with a Twist

Time Series Analysis Part 1: Regression with a Twist

We're surrounded by time series. It's one of the more common plots we see in day-to-day life. Finance and economics are full of them - stock prices, GDP over time, and 401K value over time to name a few. The plot looks deceptively simple; just a nice univariate squiggle. No crazy vectors, no surfaces, just…

Commentary: Infrastructure Considerations for Machine Learning

Commentary: Infrastructure Considerations for Machine Learning

Welcome to another brief commentary and departure from the heavier mathematics. I have been endeavoring to expand the breadth of my knowledge on the tech side of things, and chronicling some things I've learned and observed from speaking with different companies, both as an independent and as a Tech Field Day delegate. Many of these…

Poisson Processes and Data Loss

Poisson Processes and Data Loss

There are many applications for counting arrivals over time. Perhaps I want to count the arrivals into a store, or shipments into a postal distribution center, or node failures in a cloud cluster, or hard drive failures in a traditional storage array. It's rare that these events come neatly, one after the other, with a…