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

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.

💡 Research Summary

The paper introduces and thoroughly investigates a novel asymmetric probability density function (PDF) of the form

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