Saddlepoint Approximation to Cumulative Distribution Function for Poisson–Exponential Distribution

The random sum distribution is a key role in probability theory and its applications as well, these applications could be used in different sciences such as insurance system, biotechnology, allied health science, etc. The statistical significance in random sum distribution initiates when using the applications of probability theory in the real life, where the total quantity X can be only observed, which is included of an unknown random number X of random contributions. Saddlepoint approximation techniques overcome this problem. Saddlepoint approximations are effective tools in getting exact expressions for distribution functions that are not known in closed form. Saddlepoint approximations usually better than the other methods in which calculation costs, but not necessarily about accuracy. This paper introduces the saddlepoint approximations to the cumulative distribution function for random sum Poisson- Exponential distributions in continuous settings. We discuss approximations to random sum random variable with dependent components assuming existence of the moment generating function. A numerical example of continuous distributions from the Poisson- Exponential distribution is presented.