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

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.