The Mollified (Discrete) Uniform Distribution and its Applications

TL;DR

The paper introduces and explores the mollified uniform distribution, a soft version of the uniform distribution, highlighting its properties and diverse applications in modeling, generalized linear models, and education.

Contribution

It rediscoveres the mollified uniform distribution, details its properties, and discusses its potential applications and a discrete variant, expanding understanding and utility.

Findings

Models platykurtic, mesokurtic, and leptokurtic shapes

Serves as a soft-clipping response function in generalized linear models

Provides an example for convolution of random variables

Abstract

The mollified uniform distribution is rediscovered, which constitutes a ``soft'' version of the continuous uniform distribution. Important stochastic properties are presented and used to demonstrate potential fields of applications. For example, it constitutes a model covering platykurtic, mesokurtic and leptokurtic shapes. Its cumulative distribution function may also serve as the soft-clipping response function for defining generalized linear models with approximately linear dependence. Furthermore, it might be considered for teaching, as an appealing example for the convolution of random variables. Finally, a discrete type of mollified uniform distribution is briefly discussed as well.