The Two Ignored Components of Random Variation

TL;DR

This paper introduces a new paradigm for understanding random variation, emphasizing the dual sources of identity instability and external factors, and develops a bivariate distribution model that encompasses many traditional univariate distributions as special cases.

Contribution

It proposes a comprehensive bivariate distribution framework for random variation, unifying many existing distributions and providing a new perspective on their origins.

Findings

Exponential and normal are special cases of a single distribution model.

The new bivariate distribution captures a wide range of observed variations.

Empirical data supports the validity of the new paradigm.

Abstract

A random phenomenon may have two sources of random variation: an unstable identity and a set of external variation-generating factors. When only a single source is active, two mutually exclusive extreme scenarios may ensue that result in the exponential or the normal, the only truly univariate distributions. All other supposedly univariate random variation observed in nature is truly bivariate. In this article, we elaborate on this new paradigm for random variation and develop a general bivariate distribution to reflect it. It is shown that numerous current univariate distributions are special cases of an approximation to the new bivariate distribution. We first show that the exponential and the normal are special cases of a single distribution represented by a Response Modeling Methodology model. We then develop a general bivariate distribution commensurate with the new paradigm, its properties are discussed and its moments developed. An approximating assumption results in a univariate general distribution that is shown to include as exact special cases widely used distributions like generalized gamma, log-normal, F, t, and Cauchy. Compound distributions and their relationship to the new paradigm are addressed. Empirical observations that comply with predictions derived from the new paradigm corroborate its scientific validity.