A new discrete exponential distribution: properties and applications

In this work, we propose a novel discrete counterpart to the continuous exponential random variable. It is defined on N0=0,1,2,⋯ and is constructed to have a step-wise cumulative distribution function that minimizes the Cramér distance to the continuous cumulative distribution function of the exponential random variable. We show that its distribution is a particular case of the zero-modified geometric distribution. The probability mass function is analyzed in detail, and the characteristic function is derived, from which the moments of the distribution can be readily obtained. The failure rate function, the zero-modification index, Shannon’s entropy, and the stress-strength reliability parameter are also derived and discussed. Parameter estimation is examined, by considering the maximum likelihood method, the method of moments, and the least-squares method. A two-parameter generalization is also introduced and investigated. A real data analysis is provided, where the proposed distribution is fitted to a data set and compared to a well-known counting distribution. Finally, an application of the proposed discrete model is presented, focusing on the determination of the distribution of a compound sum of i.i.d. continuous random variables, with a specific application to the insurance field.