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

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 $\alpha = 1/2$.