Probabilistic Solution of Rational Difference Equations System with Random Parameters

Stochastic systems of difference equations usually appear in the investigation of systems with discrete time or in the numerical solution of systems with continuous time. A lot of difference systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment changes. Recently, there has been great interest in studying difference equation systems. One of the reasons for this is the necessity for some techniques that can be used in investigating equations arising in mathematical models describing real-life situations in population biology, economics, probability theory, genetics, psychology, and so forth. There are many papers related to the difference equations system; for example, Çinar 1 studied the solutions of the system of difference equations:


Probabilistic Solution of Rational Difference Equations System with Random Parameters
Seifedine Kadry and Chibli Joumaa American University of the Middle East, Kuwait Abstract: In this article, we study the periodicity of the solutions of the of rational difference equations system of type (p $ 1), then we propose new "exact" procedure to find the probability density

INTRODUCTION
Stochastic systems of difference equations usually appear in the investigation of systems with discrete time or in the numerical solution of systems with continuous time.A lot of difference systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, etc.Recently, there has been great interest in studying difference equation systems.One of the reasons for this is the necessity for some techniques that can be used in investigating equations arising in mathematical models describing real life situations in population biology, economics, probability theory, genetics, psychology etc.There are many papers related to the difference equations system for example: Cinar (2004) studied the solutions of the system of difference equations: Papaschinopoulos and Schinas (1998) studied the oscillatory behavior, the boundedness of the solutions, and the global asymptotic stability of the positive equilibrium of the system of nonlinear difference equations: Ozban (2007) studied the positive solutions of the system of rational difference equations: , Yang and Xu (2007) propose a method to deal with the mean square exponential stability of impulsive stochastic difference equations.In this paper, we investigate the analytic solution of the stochastic difference equations system: (1) where a, b, x 0 = N and y 0 = M are independent random variables.

PERIODICITY OF THE SOLUTIONS
In this section, we study the periodicity of the solutions of system (I).
Theorem 1: All solutions of (I) are periodic with period 2. Proof:

ANALYTIC STOCHASTIC SOLUTIONS
In this section, we develop an analytic technique to find the stochastic solutions of (cf.theorem 2): The solution of a stochastic system of difference equations is obtained when evaluating the statistical characteristic of the solution process like the mean, standard deviation, high order moments and the most important characteristic "the probability density function" (p.d.f.).Our proposed technique is based on the transformation of random variables to get the p.d. f of x n and y n .

Transformation of random variables technique (TRVT):
Definition: Let X be a continuous random variable with generic probability density function f(x) defined over the supportc 1 <x<c 2 .And, let Y = u(X) be an invertible function of X with inverse function X = v(Y).Then, using the Transformation of Random Variable Technique (TRVT) defined by Kadry and Younes (2005), Kadry (2007), El-Tawil et al. (2009) and Kadry and Smaili (2007), the probability density function of Y is: defined over the supportu (c 1 ) <y<u (c 2 ).
To simplify our technique, let us consider the following stochastic equation: , where, x and y are two independent random variables.To find the p.d.f of z, firstly we linearize the denominator then we find the p.d.f of the product of two random variables: We suppose * = 1/ x " then we apply the TRVT technique to get the p.d.f of *: Once the linearization step has been done, it's required to find the p.d.f of the product of two random variables.To do that, we developed the following theorem: Theorem 3: Let X be a continuous random variable with distribution function f(X) which is defined on the interval By induction, suppose the result holds for n -1: proof is completed by induction.
[a, b], where, 0 < b < 4. Similarly, let Y be a random variable of the continuous type with distribution function g(Y) which is defined on the interval[c, d]  , where, 0 < c < d < 4. The p.d.f of z = XY, h(z), is: