Random Variables aren't Random

TL;DR

This paper redefines random variables as measurable functions within measure-theoretic probability, enabling logical inference and information-based assessment without relying on repeated sampling or hypothetical scenarios.

Contribution

It introduces a measure-theoretic framework that characterizes random variables without infinite sampling, facilitating logical inference and information-based evaluation of statistical procedures.

Findings

Random variables are fully characterized as measurable functions.

Logical inference can be extended beyond probabilistic reasoning.

Information-based methods improve assessment of statistical procedures.

Abstract

This paper examines the foundational concept of random variables in probability theory and statistical inference, demonstrating that their mathematical definition requires no reference to randomization or hypothetical repeated sampling. We show how measure-theoretic probability provides a framework for modeling populations through distributions, leading to three key contributions. First, we establish that random variables, properly understood as measurable functions, can be fully characterized without appealing to infinite hypothetical samples. Second, we demonstrate how this perspective enables statistical inference through logical rather than probabilistic reasoning, extending the reductio ad absurdum argument from deductive to inductive inference. Third, we show how this framework naturally leads to information-based assessment of statistical procedures, replacing traditional inference metrics that emphasize bias and variance with information-based approaches that better describe the families of distributions used in parametric inference. This reformulation addresses long-standing debates in statistical inference while providing a more coherent theoretical foundation. Our approach offers an alternative to traditional frequentist inference that maintains mathematical rigor while avoiding the philosophical complications inherent in repeated sampling interpretations.