# An integral formula for the powered sum of the independent, identically and normally distributed random variables

## TL;DR

This paper derives an integral formula for the distribution of the sum of powered standard normal variables, generalizing chi-squared, which aids in evaluating their density functions.

## Contribution

It introduces a new integral representation for the density of the powered sum of i.i.d. normal variables, based on the inversion formula and hyperfunction analysis.

## Key findings

- Provides a convergent integral formula for the density function.
- Facilitates evaluation of the distribution of powered sums of normal variables.
- Connects the formula with hyperfunction theory.

## Abstract

The distribution of the sum of r-th power of standard normal random variables is a generalization of the chi-squared distribution. In this paper, we represent the probability density function of the random variable by an one-dimensional absolutely convergent integral with the characteristic function. Our integral formula is expected to be applied for evaluation of the density function. Our integral formula is based on the inversion formula, and we utilize a summation method. We also discuss on our formula in the view point of hyperfunctions.