On Extended Exponential Distribution: Properties and Applications In Tracking the Pandemic Covid-19

This article reviews and discusses an extended form of exponential model distributions as well as some important cases of these distributions. A new four parameters extended exponential model is proposed. The family of the exponential distributions and their derivate models finds many applications. The author proposes a new four parameter extended exponential model distribution, which generalizes the model of Weibull-exponential distributions. The applicability of the new models is well confirmed using two real data sets. Various properties of this new model are investigated and then exploited to derive several related results, especially characterizations in probability. As a motivation, the statistical applications of the results based on health related data are included to investigate the reliability properties of a flexible extended exponential model of distributions. These findings will be useful for the practitioners in various fields of theoretical and applied sciences. We use the general formula to track the spread of the Corona virus in the world for a period of six months in all countries of the world, guess the values of the parameters that reflect the values of indicators that express the expectation of the epidemic and its acceleration in the world, and find margins that can help lead the crisis and manage the pandemic.

The exponential distribution model was first defined in applied cases as a special case of distribution is one of the widely used continuous distributions. It is a special case of type III Pearson distribution. It is also used for products with constant failure or arrival rates. After a period of 70 years the Exponential Distribution becomes just a special case of the Weibull gamma distribution to appear on its own [1]. Karl Pearson discussed the Weibull and Gamma distributions in 1895. History of exponential density function has been obtained. Generalized exponential density function where it is assumed that independent events occur at a constant rate. Exponential distribution refers to a statistical distribution used to model the time between independent events that happen at the average with constant rate. It is often concerned with the amount of time until some specific event occurs. The Exponential distribution is a continuous distribution bounded on the lower side. Its shape is always the same, starting at a finite value at the minimum and continuously decreasing at larger x. The exponential distribution decreases rapidly for increasing x. The Exponentiated exponential or generalized exponential distribution is introduced as a special case of the exponentiated Weibull with three parameters. Exponentiated exponential distribution has many mathematical properties which have not been known completely or have not been known in simple and general forms. This distribution presented a comprehensive survey of the mathematical properties and most attractive generalization of the exponential distribution, and has received widespread attention [2]. The generalized exponential (GE) distribution with three-parameter (scale, shape, and location) proved its suitability in probabilistic models related to Rayleigh distribution or recurrence modelling [3]. The GE distribution shares many physical properties of the gamma and Weibull distributions, unlike the two parameter exponential distribution preserves its memory property strength [2-6].

Assume X be a random variable, the probability density function (pdf) with a parameter can be defined using an alternative parameterization. It describes the arrival time of a randomly recurring independent event sequence. If is the mean waiting time for the next event recurrence, its pdf is given by [7]:

Where represent the scale parameter and is called the rate of the distribution is characterized by the single parameter, or alternatively the most commonly used form of the Exponential distribution. If be a positive real number we write X~ exponential and say that X is an exponential random variable with parameter if the pdf is:

The basic parameter expresses the waiting time in random processes with exponential distribution, X, before an event occurs will have an exponential distribution if the probability of the event to occur during a certain time interval is directly proportional to the length of that time interval. Exponential distribution is a special case of the two parameter gamma distribution. The representation of the shape of the pdf for exponential distribution with single parameter is shown in figures (Figure 1).

Figure 1: Graph of pdf for various values of lamina.

Figures 2: Graphs of pdf and CDF for various values.

The probability distribution function (CDF) can be formulated when integration takes, and it becomes CDF of the exponential random variable with parameter is given by:

In order to clarify the basic conditions of this function we show some special shapes for exponential its probability density function and cumulative distribution function for various values of the parameter is shown in figures (Figure 2).