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

## Keywords:

Bivariate chi-square distribution, characteristic function, correlated chi-square variables, cumulative distribution function, sum of correlated chi-square variables.## Abstract

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.

## References

Ahmed, S. E. 1992. Large sample pooling procedure for correlation. The Statistician, 41: 415- 428.

Gerkmann, T. Martin, R. 2009. On the statistics of spectral amplitudes after variance reduction by temporal cepstrum smoothing and cepstral nulling. IEEE Transactions on Signal Processing, 57(11): 4165-4174.

Gordon, N. H. Ramig, P. F. 1983. Cumulative distribution function of the sum of correlated chi-squared random variables. Journal of Statistical Computation and Simulation, 17(1): 1-9.

Gradshteyn, I.S. Ryzhik, I.M. 1994. Table of Integrals, Series and Products, Academic Press.

Gunst, R. F. Webster, J. T. 1973. Density functions of the bivariate chi-square distribution. Journal of Statistical Computation and Simulation, 2: 275-288.

Joarder, A. H. 2009. Moments of the product and ratio of two correlated chi-square random variables. Statistical Papers, 50(3): 581-592.

Johnson, N. L., Kotz, S. Balakrishnan, N. 1994. Continuous Univariate Distributions. John Wiley, New York.

Krishnaiah, P. R.; Hagis, P. Steinberg, L. 1963. A note on the bivariate chi distribution, SIAM Review, 5, 140-144.

Lawson, J. L. Uhlenbeck, G. E. 1950. Threshold Signals, McGraw-Hill Book Co, New York.

Provost, S.B. 1988. The exact density of a general linear combination of gamma variables. Metron, 46: 61-69.