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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.