Exact distribution of the sum of two correlated chi-square variables and its application

The exact distribution of the sum of two chi-square random variables is known if the variables are independent. We derive the exact distribution of the sum of two correlated chi-square variables when they are correlated through a bivariate chi-square distribution. The distribution is important in estimating the common variance of a bivariate normal population. Some properties of the distribution, namely, the characteristic function, cumulative distribution function, raw moments, mean centered moments, coefficient of skewness and kurtosis are derived. The graph of the density function is also presented.