Random Sum of Mixtures of Sum of Bivariate Exponential Distributions

Abstract: Problem statement: The distributions of R X Y = + and /( ) W X X Y = + , where X and Y follow Lawrance and Lewis`s bivariate exponential distribution, is generalized. Approach: In this research we found mixtures of sum of bivariate expo nential random variables. Results: Also we calculated the probability density function (pdf) o the random sum of mixtures of sum of bivariate exponential random variables. Conclusion/Recommendations: In this study we investigated the pdf of random sum of mixtures of sum of bivariate expon ential random variables.


INTRODUCTION
One of the standard stochastic models used in various areas of applied probability, such as insurance risk theory and queuing theory, is the random sum model. Random sum is defined as a sum of n random variables and the number n is also random variable. Random sum play an important role in insurance application, for example, let i X , 1, 2, ..., i N = be independent and identically distributed random variables determine the losses of a unit of production caused by fire, explosion, destruction, and so on. The random sum of i X , 1, 2, ..., i N = is the total losses of this unit, and then one can get tail probability, i.e., ( ) t N a Pr S t = > . Compound geometric distributions play an important role in reliability theory, queuing theory, and regenerative processes. For example, the equilibrium waiting time in G\G\1 queue has a compound geometric distribution and so does the maximal aggregate loss of surplus process under the classical and renewal risk models. For more details, Gertsbakh [4][5][6]10] . Mixture of exponential distribution is the only one of the most important distributions in reliability theory because of their properties to distribution function which given in Feller [3] . Brawn [1] discussed moment based bounds on the tial of compound geometric distributions. Drozdenko [2] studied some examples of generalizations of mixtures of exponential distributions. Ma and King [8] discussed the calculation of the n-fold convolution of generalized exponential-sum distribution functions. But here we discuss the random sum of mixtures of sum of bivariate exponential random variables.
In this study, we consider the distributions of R X Y = + and /( ) W X X Y = + when X and Y are correlated exponential random variables with the joint pdf given by: for , , 0 x y λ > and 0 If λ = 1 in relation (1) we obtain the distribution of Lawrance and Lewis`s bivariate exponential distribution [9] .
The research is organized as follow. First, we derive the exact expressions for the pdfs of R X Y = + and /( ) W X X Y = + . The pdf of mixture of sum of bivariate exponential distribution is obtained. Results give the pdf of the random sum of mixture of sum of bivariate exponential random variables. Finally, conclusion is given.

MATERIALS AND METHODS
The distributions of R and W: Theorems 1 and 2 derive the pdfs of R X Y = + and /( ) W X X Y = + when X and Y are distributed according to (1).
Theorem 1: If X and Y are jointly distributed according to (1) then: Proof: The result immediately follows by elementary integration [9] .
Theorem 2: If X and Y are jointly distributed according to (1) then: for 0<w<1.

Mixture of sum of bivariate exponential distributions:
We derive the mixture of the distribution of sums R X Y = + .
A random variable X has a mixture of distributions if one or more parameters of the distribution of X is also random variable. Let g (x, y) be probability function (for simplicity pf) of X, where y is a parameter of distribution of X. If y is a value of a random variable Y, then X has a mixture of distributions. The random variable Y may take continuous distribution with probability density function (pdf) h (y), then the pf of X: The random variable Y may take discrete distribution with probability h(y), thus the pf of X becomes: By using Eq. 2 and the definition of mixture of distribution we have: Remark 1: The relation (6) is a general case of a mixture of sum of bivariate exponential distribution (i.e., we are not substituting the value of the distribution of the parameter λ). When we use a certain probability distribution as the probability distribution of the parameter λ, we find that the values taken by the random variable in such distribution must be in the domain of the possible values for the parameter λ (i.e., all the value belong to the parameter space for the main distribution).

RESULTS AND DISCUSSION
Random sum of mixtures of sum of bivariate exponential distributions: We derive the pdf of random sum of mixtures of sum of bivariate exponential distributions. Let i X , 1, 2, ..., i N = be independent and identically distributed random variables.
has a mixture of sum of bivariate exponential distribution with pdf as given by using Eq. 4 as follow: The Laplace transform of this pdf given as follow: after evaluate above integral in Eq. 8 we have: Then, the Laplace transform is: where, P N (θ) is the probability generating function (pgf) of the random variable N . From relations (9) and (10) we get: where, Let: Substituting from relations (13) and (14) into relation (12) we get: By using complete multinomial expansions theorem in relation (16), see Ma [7] , we find: Consequently from relation (17) we have: