On the distribution of the sum of independent exponential-geometric random variables


Abstract


In this article, we derive exact expressions for the probability density function and cumulative distribution function of the sum of independent and non-identical exponential-geometric random variables. Then we discuss the corresponding result for independent and identically distributed exponential-geometric random variables. A saddlepoint approximation is also utilized to approximate the derived distribution. Finally, numerical simulations are used to investigate the precision of the saddlepoint approximation.

DOI Code: 10.1285/i20705948v16n3p694

Keywords: Exponential-geometric distribution; Divided differences; Independent and non-identically distributed random variables; Saddlepoint approximation.

References


Adamidis, K., and Loukas, S. (1998). A lifetime distribution with decreasing failure rate. Statistics and Probability Letters, 39: 35--42.

Akkouchi, M. (2008). On the convolution of exponential distributions. Journal of the chungcheong mathematical society, 21(4): 501--501.

Atkinson, K. E. (1989). An introduction to numerical analysis, 2nd edition. Wiley, New York.

Balakrishnan, N., Zhu, X., and AL-Zahrani, B. (2015). A recursive algorithm for the single and product moments of order statistics from the exponential-geometric distribution and some estimation methods. Communications in Statistics - Theory and Methods, 44: 3576--3598.

Daniels, H. E. (1954). Saddlepoint approximations in statistics. The Annals of Mathematical Statistics, pages 631--650.

Daniels, H. E. (1987). Tail probability approximations. International Statistical Review/Revue Internationale de Statistique, pages 37--48.

Gradshteyn, I. S. and Ryzhik, I. M. (2007). Table of integrals, series, and products, 7th ed. Academic Press.

Khuong, H. V., and Kong, H. Y. (2006). General expression for pdf of a sum of independent exponential random variables. IEEE Communications Letters, 10(3): 159--161.

Kitani, M., and Murakami, H. (2020). On the distribution of the sum of independent and non-identically extended exponential random variables. Japanese Journal of Statistics and Data Science: 3(1), 23--37.

Kitani, M., Murakami, H., and Hashiguchi, H. (2021). The distribution of the sum of independent and non identically generalized Lindley random variables. Communications in Statistics-Theory and Methods, pages 1-13.

Levy, E. (2022). On the density for sums of independent exponential, Erlang and gamma variates. Statistical Papers, 63(3):

--721.

Louzada--Neto, F. (1999). Polyhazard models for lifetime data. Biometrics, 55(4): 1281--1285.

Lugannani, R., and Rice, S. (1980). Saddle point approximation for the distribution of the sum of independent random variables. Advances in applied probability, 12(2): 475--490.

Mathai, A. M. (1982). Storage capacity of a dam with gamma type inputs. Annals of the Institute of Statistical Mathematics, 34(3): 591--597.

Moschopoulos, P. G. (1985). The distribution of the sum of independent gamma random variables. Annals of the Institute of Statistical Mathematics, 37(1): 541--544.

Murakami, H. (2014). A saddlepoint approximation to the distribution of the sum of independent non-identically uniform random variables. Statistica Neerlandica, 68(4): 267--275.

Murakami, H. (2015). Approximations to the distribution of sum of independent non--identically gamma random variables. Mathematical Sciences, 9(4): 205--213.

Nadarajah, S., Jiang, X., and Chu, J. (2015). A saddlepoint approximation to the distribution of the sum of independent non--identically beta random variables. Statistica Neerlandica, 69(2): 102--114.

Ross, S. M. (2014). Introduction to probability models. Academic press, New York.

Sadooghi-Alvandi, S. M., Nematollahi, A. R., and Habibi, R. (2009). On the distribution of the sum of independent uniform random variables. Statistical papers, 50(1): 171--175.

Smaili, K., Kadri, T., and Kadry, S. (2013). Hypoexponential distribution with different parameters. Applied Mathematics, 4: 624--631.


Full Text: pdf
Bookmark and Share


Creative Commons License
This work is licensed under a Creative Commons Attribuzione - Non commerciale - Non opere derivate 3.0 Italia License.