Properties and application of the form $A\cdot exp(-(x-c)^2/(a(x-c)+2b^2))$ for investigation of ultra high energy cascades

TL;DR

This paper introduces a new asymmetric distribution form for analyzing ultra high energy cascades, exploring its properties, approximation methods, and practical applications in modeling atmospheric shower distributions.

Contribution

It presents a novel distribution form, details its properties, and demonstrates its effectiveness in modeling ultra high energy atmospheric shower data.

Findings

The distribution effectively models ultra high energy shower data.

Approximation methods improve data fitting accuracy.

Relationship with known distributions enhances practical usability.

Abstract

The form $A\cdot exp(-(x-c)^2 /(a(x-c)+2b^2))$ is an asymmetric distribution intermediate between the normal and exponential distributions. Some specific properties of the form are presented and methods of approximation are offered. Appropriate formulae and table are presented. The practical problems of approximation by the form are discussed with connection to the quality of original data. Application of the methods is illustrated by using in the problem of calculating and studying distribution function of maximum ultra high energy atmospheric showers. Relationship with exponential and normal distributions makes usage of the form to be effective in practice.