Muth introduced a probability distribution with application in reliability theory. We propose a new model from the Muth law. This paper studies its statistical properties, such as the computation of the moments, computer generation of pseudo-random data and the behavior of the failure rate function, among others. The estimation of parameters is carried out by the method of maximum likelihood and a Monte Carlo simulation study assesses the performance of this method. The practical usefulness of the new model is illustrated by means of two real data sets, showing that it may provide a better fit than other probability distributions.

A two-parameter probability distribution with bounded support is derived from the shifted Gompertz distribution. It is shown that this model corresponds to the distribution of the minimum of a random number with shifted Poisson distribution of independent random variables having a common power function distribution. Some statistical properties are written in closed form, such as the moments and the quantile function. To this end, the incomplete gamma function and the Lambert W function play a central role. The shape of the failure rate function and the mean residual life are studied. Analytical expressions are also provided for the moments of the order statistics and the limit behavior of the extreme order statistics is established. Moreover, the members of the new family of distributions can be ordered in terms of the hazard rate order. The parameter estimation is carried out by the methods of maximum likelihood, least squares, weighted least squares and quantile least squares. The performance of these methods is assessed by means of a Monte Carlo simulation study. Two real data sets are used to illustrate the usefulness of the proposed distribution.

The shifted Gompertz distribution was introduced by Bemmaor ( 1994 Bemmaor , A. C. ( 1994 ). Modelling the diffusion of new durable goods: Word-of-mouth effect versus consumer heterogeneity . In: Laurent , G. , Lilien , G. L. , Pras , B. , eds. Research Traditions in Marketing . Boston : Kluwer , pp. 201 – 229 . [Google Scholar] ) as a random model of adoption timing of innovations in a market. The purpose of this article is threefold. We provide explicit expressions for the expectation and variance of this model, which are functions of the Euler constant. In addition, for simulation purposes, we derive a closed-form expression for the quantile function of the shifted Gompertz distribution in terms of the Lambert W function. Finally, the limit distributions and computer generation of extreme order statistics are considered. In particular, we show that the domains of attraction for maxima and minima are the Gumbel and Weibull distributions, respectively.

In this paper, we consider the sequence <!-- MATH $(\theta_n)_{n\ge 0}$ --> solving the Ramanujan equation <!-- MATH \begin{displaymath} \frac{e^n}{2}=\sum_{k=0}^{n}\frac{n^k}{k!}+\frac{n^n}{n!}\,(\theta_n-1),\qquad n=0,1,\dots. \end{displaymath} -->

The purpose of this paper is to establish a connection between the polylogarithm function and a continuous probability distribution. We provide closed-form expressions in terms of the polylogarithm function for all the moments of a continuous random variable related to the Bass diffusion model, which was introduced by Bass and is widely used in marketing science. In addition, a new integral representation of the polylogarithm of order n is achieved from a probabilistic formulation.