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The Distribution of Two Independent Random Variables from Chi-Square and Exponential Distributions Mr. Mohammad Y. Alshoqran Department of Mathematical Sciences, College of Arts, Science and Education Abstract The distribution of two independent random variables is great of interest in Mathematical Statistics specially and in different areas of sciences, This investigation provides an analytical study of the distributions of the sum, the ratio and the product of two independent random variables, one belongs to the Chi-square distribution with (r) parameter and the second variable belongs to the Exponential distribution with (λ) parameter, the applied method is based on the Change of variables and Distribution Function methods. This paper concluded that the distribution of sum of these two independent random variables is an Exponential Distribution approximately. Some specific cases of interest were discussed and some results were derived as special cases. Keywords: Random Variable, Chi-Square, Exponential, Change of variables, Distribution Function. Introduction Methodology Conclusions The distributions of functions of random variables are playing an important role in the mathematical statistics and its applications, like its applications in the biological, physical and different areas of sciences. For example, D. L. Evans and L. M. Leemis (2004) presented an algorithm of computing the probability density function of the sum of two independent discrete random variables, with an implementation of the algorithm in a computer algebra system. Also they gave some examples to illustrate the utility of that algorithm. The distribution of the ratios of two random variables is the stress-strength model in the context of reliability .This is done by Nadarajah and Kotz (2007). Distributions of Two Independent Random Variables : In this investigation, we applied the distribution function and the change of variables techniques to find the distribution of the sum, ratio and the multiplication of two random variables. We concluded that the distribution of sum of two independent random variables, from chi-square distribution and exponential distribution is an Exponential Distribution approximately. The distribution of ratio of two independent random variables, from chi-square distribution and exponential distribution is special distribution very near to F-Distribution. The main goal of this paper is to find the probability density function of two independent random variables: Y1 x y , Y2 y / x , Y3 x. y . References T. Kadri, K.Smaili(2014), The exact distribution of the ratio of two independent hypoexponential random variables, British Journal of Mathematics & Computer Science,Vol.4: pp.2665-2675. S. Nadarajah, and S.Kotz,(2007) ,On the product and ratio of t and Bessel random variables. Bulletin of the Institute of Mathematics, Academia Sinica, Vol. 2 No.1: pp. 55-56. P.E. Oguntunde; O.A.Odetunmibi; and A. O. Adejumo (2013), On the sum of exponentially distributed random variables: a convolution approach, European Journal of Statistics and Probability Vol.1, No.2:pp.1-8. A. Seijas-Macías;. Oliveira (2012),an approach to distribution of the product of two normal variables, Discussiones Mathematicae Probability and Statistics, Vol.32: pp.87-99. K. Teerapabolarn (2014),Binomial approximation for a sum of independent Beta Binomial random variables, Applied Mathematical Sciences, Vol. 8, no. 179: pp.8929 – 8932. Distribution of Y1= x+y : Also we found the mean and the variance for the ratio function. Also, we concluded that some approximations needed to find the distribution of multiplication of two independent random variables, from chi-square distribution and exponential distribution.