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The main purpose of this paper is to present

The gamma family distributions was discussed by Karl Pearson in 1895 as pointed out in Balakrishnan and Basu [

An extension of Exponential Distribution was proposed by Weibull (1951). The Exponential Distribution is a special case wherein the shape parameter equals one. The Weibull distribution has many applications in survival analysis and reliability engineering; for reference see Lai et al. (2006). Some other applications in industrial quality control are discussed in Berrettoni (1964).

We begin with some definitions which provide a base for the definition of

The Euler gamma function

The beta function of two variables

In (

The beta distribution has an application to model a random phenomenon whose set of possible values is a finite interval

A continuous random variable

For

We have

One has the following:

Put

Putting the value of

A random variable

Weibull distribution is widely used in engineering practice due to its versatility. It was originally proposed for the interpretation of fatigue data but now it is also used for many other problems in engineering. In particular, in the field of life phenomenon, it is used as the distribution of lifetime of some object, particularly when the “weakest link” model is appropriate for the model; that is, consider an object consisting of many parts and suppose that the object experiences death (failure) when any of its parts fail; it has been shown [

The Gamma and Weibull distributions are commonly used for analyzing any lifetime data or skewed data. Both distributions have nice physical interpretation and several desirable properties. Unfortunately both distributions have drawbacks, one major disadvantage of the gamma distribution is that the distribution function or the survival function can not be computed easily if the shape parameter is not an integer. By using mathematical tables or computer software one obtains the distribution function, the survival function, or hazard function. This makes the gamma distribution unpopular as compared to Weibull distribution whose distribution function, hazard function, or survival function is easy to compute. It is well known that even though the Weibull distribution has convenient representation of distribution function, the distribution of the sum of independent and identically distributed (i.i.d) Weibull random variables is not simple to obtain. Therefore, the distribution of the mean of random sample from Weibull distribution is not easy to compute whereas the distribution of sum of independent and identically distributed (i.i.d) gamma random variables is well known. For more details see Mudholkar, Srivastava and Freimer (1995), Mudholkar and Srivastava [

Recently R. D. Gupta and D. Kundu have introduced three-parameter Exponential Distribution (location, scale, and shape) and studied the theoretical properties of this family and compared them with respective good studies properties of Gamma and Weibull distributions. The increasing and decreasing hazard rate of the Generalized Exponential Distribution (GED) depends on the shape parameter. Generalized Exponential Distribution (GED) has several properties that are quite similar to gamma distribution but it has distribution function similar to that of the Weibull distribution which can be computed easily. Since the Generalized Exponential family has the likelihood ratio ordering on the shape parameter, one can construct a uniformly most powerful test for testing one sided hypothesis on the shape parameter when the scale and location parameters are known.

A continuous random variable

The Generalized Exponential Distribution (GED) introduced by Mubeen et al. [

Replacing

The main aim of this paper is to present interesting extensions of Generalized Exponential Distribution (GED) in various ways and to study their moment generating functions (MGFs). We shall first define Generalized Exponential Distribution (GED) in terms of a new parameter

Let

If we take

Clearly

In this section, we derive MGF of the random variable

Let

For

Clearly

To see this, we take

In this section, we derive MGF of the random variable

We also present the following three-parameter extension of Generalized Exponential Distribution (GED). In fact, we prove the following.

Let

Clearly

Now

If we replace

For

Taking

Put

For

If in

then

Letting

Finally we present the following more general interesting result which among other things includes Weibull distribution as a limiting case.

Let

Clearly

Weibull Distribution is the limiting case of Theorem

Replacing

The applications of Exponential Distribution have been widespread, which include models to determine bout criteria for analysis of animal behaviour [

The Exponential Distribution is often used to model the reliability of electronic systems, which do not typically experience wear-out type failures. The distribution is called “memoryless,” meaning that the calculated reliability for, say, a 10-hour mission is the same for a subsequent 10-hour mission, given that the system is working properly at the start of each mission. Given a hazard (failure) rate,

In this paper the authors conclude the following.

(i) Weibull distribution is a special case of

(ii) Also if

(iii) The moment generating functions (MGFs) obtained in this paper generalize the classical moment generating functions (MGFs) of the given distributions.

The authors declare that they have no conflicts of interest.

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