The Moments Generating Function of a Poisson Exponential Joint Distribution
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1 Journal of Basrah Researches ((Sciences)) Volume 37.Number 4.D ((2011)) Available online at: ISSN The Moments Generating Function of a Poisson Exponential Joint Distribution Kareema Abul -Kadhum Mukhrib Khafaji and Ali Hussein Mahmood Al-Obaidi Kareema_kadim@yahoo.com Abstract In this paper, we derive the moment generating function of this joint p.d.f. random vector where is defined as and is a random sample of size from exponential distribution with be a Poisson random variable, and use it in deriving some moments of this distribution. Keywords: Random vector, Poisson random variable, exponential distribution, and moments. 1.1 Introduction From Poisson Process, the Poisson distribution is closely related to the exponential distribution because the waiting time between observations is exponentially distributed, while the number of observations is Poisson distribution( see Ross (2007)). The researchers can consider the queuing theory another application of this research, which has many applications to businesses concerned with customer wait times but the maximum customer wait times is studies in this research( see Ross (2007)). Park (1970) found properties of the joint distribution of the total time for the emission of particles emitted from a radioactive substance that is and the number of these particles, where the waiting times of the emissions are exponentially distributed with parameter, the number of particles has a Poisson distribution with parameter and the conditional distribution of the total length of time required for the emission, i.e., the sum, for a given number of particles is a Gamma variate: 6
2 Khafaji & Al-Obaidi: The Moments Generating Function of a Poisson Exponential Joint (see Park (1970)). Sarabia and Guill en (2008) emphasize the value of studying a joint distribution, rather than Park's joint distribution with constant. They use joint distribution for the random vector, where is the sum of random variables, and is the number. Their application of interest is from the insurance industry where represents the number of claims and represents the total claim amount ( Sarabia, and Guill en (2008)). Kozubowski and Panorska(2008) studied the joint distribution of, where has a geometric distribution and is the maximum of i.i.d. exponential variables, independent of. While Al-Obaidi, and Al-Khafaji(2010) studied some properties of the joint probability distribution of random vector where is defined as and is a random sample of size from exponential distribution. In this work, we derive the moment generating function of this joint p.d.f. where the probability density function of each and cumulative distribution function c.d.f. Also, be a Poisson random variable with parameter and probability mass function (p.m.f.) such that be independent of. We find the following: 1. The joint p.d.f. of a random vector is as follow 7
3 Journal of Basrah Researches ((Sciences)) Volume 37.Number 4.D ((2011)) and c.d.f. of a random vector as follow 2. The can be represented as a mixed distribution, which with probability is a point mass at or with probability is a random vector given by the p.d.f. The following proposition formalizes this idea. Proposition 1 If has joint distribution, then Where is a random vector with p.d.f., and 8
4 Khafaji & Al-Obaidi: The Moments Generating Function of a Poisson Exponential Joint is an indicator random variable, independent of taking on the values of and with probabilities and, respectively. 2.1 The Moment Generating Function We know that the moment generating function is as follows: 3.1 The Moments Now, since 9
5 Journal of Basrah Researches ((Sciences)) Volume 37.Number 4.D ((2011)) Then we get and We can write the mth partial derivative of the moment generating function with respect to is as follows Where And since 0
6 Khafaji & Al-Obaidi: The Moments Generating Function of a Poisson Exponential Joint Then we get that And the mth partial derivative of the moment generating function with respect to is as follows Where Now we use the equation to derive, then 1
7 Journal of Basrah Researches ((Sciences)) Volume 37.Number 4.D ((2011)) Then we get 2
8 Khafaji & Al-Obaidi: The Moments Generating Function of a Poisson Exponential Joint Where 4.1 Conclusions We note that the moment generating function provides us with easy tool in deriving the moments of our Poisson Exponential joint probability density function. REFERENCES Al-Obaidi, A. H., and Al-Khafaji, K. A. (2010), "Some Probabilistic Properties of a Poisson Exponential Joint Distribution",Unpublised thesis, Department of Mathematics, College of Education, University of Babylon. Kozubowski,T.J., and Panorska, A.K.(2008), "A Mixed Bivariate Distribution Connected with Geometric Maxima of Exponential Variables", Commmunications in Statistics-Theory and Methods,37: Park, S. (1970), "Regression and Correlation in a bivariate distribution with different marginal densities", Technometrics, Vol. 12, No. 3, Ross, S. M. (2007),"Introduction to Probability Models", Ninth Edition, Academic Press. Sarabia, J.M. and Guill en, M. (2008), "Joint Modeling of the Total Amount and the Number of Claims by Conditionals", Insurance: Mathematics and Economics. Vol. 43,
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