Density Function of Weighted Sum of Chi-Square Variables with Trigonometric Weights

Abstract

We have investigated a weighted chi-square distribution of the variable $\hat{I}¾$ which is a weighted sum of squared normally distributed independent variables whose weights are cosines of angles $\ddot{I}{•}_k=2\ddot{I}€k/N$, where $k\in \{0,1,...,N-1\}$ and $N$ is the number of the freedom degrees. We have found the exact expression for the density function of this distribution and its approximation for large $N$. The distribution is compared with the Gaussian distribution.

8 pages, 2 figures