A refinement of the multinomial distribution with application

TL;DR

This paper introduces a refined multinomial distribution that incorporates the number of inversions in outcome sequences, providing a new way to assess homogeneity in experiments.

Contribution

It presents a joint distribution of outcomes and inversions, offering a novel statistical tool for analyzing sequence homogeneity.

Findings

The refined distribution models the relationship between outcomes and inversions.

Inversions serve as a proxy for sequence homogeneity.

Application demonstrates improved assessment of outcome sequences.

Abstract

A refinement of the multinomial distribution is presented where the number of inversions in the sequence of outcomes is tallied. This refinement of the multinomial distribution is its joint distribution with the number of inversions in the accompanying experiment. An application of this additional information is described in which the number of inversions acts as a proxy measure of homogeneity (or lack thereof) in the sequence of outcomes.