An integral over $(0,\pi)$ for the distribution function of a sum of independent gamma random variables and for quadratic forms of Gaussian variables

TL;DR

This paper derives an integral representation over (0,π) for the distribution function of sums of independent gamma variables with different parameters, and applies it to quadratic forms of Gaussian variables with specific shape parameters.

Contribution

It provides a novel integral formula for the distribution of sums of gamma variables and quadratic forms of Gaussian variables, extending existing methods.

Findings

Integral formula for gamma sum distribution

Distribution of quadratic forms with shape parameter 1/2 derived

Applicable to diverse statistical models

Abstract

An integral over the interval $(0,\pi)$ is given for the cumulative distribution function of a sum of independent gamma random variables with different scale and shape parameters. The cumulative distribution function of a positive definite quadratic form is obtained as a special case with identical shape parameters $\alpha = 1/2$.