# Creating new distributions using integration and summation by parts

## TL;DR

This paper introduces a novel methodology for creating new probability distributions by applying integration and summation by parts, offering a systematic way to modify and analyze distributions through integral transformations.

## Contribution

It presents a general framework for generating new distributions from existing ones using integration and summation by parts, with practical examples and data fitting applications.

## Key findings

- New distributions can be derived via integration by parts.
- Summation by parts extends the method to discrete distributions.
- Examples demonstrate the method's applicability to real data.

## Abstract

Methods for generating new distributions from old can be thought of as techniques for simplifying integrals used in reverse. Hence integrating a probability density function (pdf) by parts provides a new way of modifying distributions; the resulting pdfs are integrals that sometimes require computation as special functions. Summation by parts can be used similarly for discrete distributions. The general methodology is given, with some examples of distribution classes and of specific distributions, and fits to data.