1. Introduction

Here, we proposed a distribution with the hope it will attract wider applicability in other fields. The generalization which is motivated by the work of Eugene et al. will be our guide. Eugene et al. (2002) [2] defined the beta G distribution from a quite arbitrary cumulative distribution function (cdf), G(x) by

(1)

where a > 0 and b > 0 are two additional parameters whose role is to introduce skewness and to vary tail weight and is the beta function. The class of distributions (1) has an increased attention after the works by Eugene et al. (2002) [2] and Jones (2004) [5]. Application of to the random variable V following a beta distribution with parameters a and b, V ∼ B(a, b) say, yields X with cdf (1). Eugene et al. (2002) [2] defined the beta normal (BN) distribution by taking G(x) to be the cdf of the normal distribution and derived some of its first moments. General expressions for the moments of the BN distribution were derived (Gupta and Nadarajah, 2004 [4]). An extensive review of scientific literature on this subject is available in Abid and Hassan (2015) [1]. We can write (1) as,

(2)

Where, denotes the incomplete beta function ratio, i.e., the cdf of the beta distribution with parameters a and b. For general a and b, we can express (2) in terms of the well-known hypergeometric function defined by,

Where denotes the ascending factorial. We obtain,

The properties of the cdf, F(x) for any beta G distribution defined from a parent in (1), could, in principle, follow from the properties of the hypergeometric function which are well established in the literature; see, for example, Section 9.1 of Gradshteyn and Ryzhik (2000) [3]. The probability density function (pdf) corresponding to (1) can be written in the form,

(3)

where is the pdf of the parent distribution.

Now, since the pdf and cdf of [0,1] truncated Fréchet distribution are respectively,

(4)

(5)

Graphs for some arbitrary parameters values of pdf and cdf are shown in figure (1) and figure (2) respectively,

Figure 1 and 2

Now, Given two absolutely continuous cdfs, H and G, so that h and g are their corresponding pdfs. We suggest a new distribution F by composing H with G, so that is a CDF,

(6)

With pdf,

(7)

With being a baseline distribution, we define in (6) and (7) above, a generalized class of distributions. We will name it the [0,1] truncated Fréchet -G distribution.

In the following section, we will assume that G is Generalized Gamma distribution.

2. [0,1] truncated Fréchet-generalized Gamma Distribution

Assume that and are pdf and cdf of Generalized Gamma random variable [6, 8] respectively, then, by applying (6) and (7) above, we get the cdf pdf of [0,1] TFGG random variable as follows,

(8)

(9)

So, the reliability and hazard rate function are respectively

The rth raw moment can be derived as follows,

Since, then,

Since,

By using, and

(10)

We get,

and then,

let then,

By using [6],

(11)

Where, is the Lauricella function of type A, then,

(12)

And then, the characteristic function is

So, the mean and variance of the of [0,1] TFGG random variable are,

(13)

(14)

Since, then By solving the nonlinear equation

we obtain the median of X.

The skewness of [0,1] TFGG random variable will be, so

(15)

Also, the kurtosis is,

(16)

The quantile function of [0,1] TFGG random variable can be obtained by solving the following nonlinear equation as,

since

So by using inverse transform method we can generate [0,1] TFGG random variable as follows,

Where U is uniformly distributed random number in the unit interval [0,1].

2.1. Shannon and Relative Entropies

An entropy of a random variable X is a measure of variation of the uncertainty. The Shannon entropy of random variable X can be found as follows,

Let,

Since, then

By using equation (10) we get,

and then,

by using expansion series incomplete Gamma function

(17)

We get,

let then,

Now, since the Lauricella function of type A, can be defined as

where is the ascending factorial defined by (with the convention that ).

by using equation (10) we get,

let then,

By using equation (11) we get,

and,

By using equation (10) we get,

By using expansion series of incomplete Gamma function, we get

Since,

By using (10) we get,

let then,

By using (11) we get,

then

(18)

The relative entropy (or the Kullback–Leibler divergence) is a measure of the difference between two probability distributions and . It is not symmetric in and . In applications, typically represents the "true" distribution of data, observations, or a precisely calculated theoretical distribution, while typically represents a theory, model, description, or approximation of . Specifically, the Kullback–Leibler divergence of from , denoted is a measure of the information gained when one revises ones beliefs from the prior probability distribution to the posterior probability distribution . More exactly, it is the amount of information that is lost when is used to approximate , defined operationally as the expected extra number of bits required to code samples from using a code optimized for rather than the code optimized for .

The relative entropy for a random variable can be found as follows,

Let,

And,

And,

And,

And,

By using equation (10) we get,

then,

By using equation (17) we get,

By applying of equation in section 0.314 of Gradshteyn and Ryzhik (2000) [3] for power series raised to power, we obtain for any m positive integer

Where the coefficients satisfy the recurrence relation

let

Since then

And,

By using equation (10) we get,

let , then,

By using (11) we get,

Since,

By using equation (10) we get,

By using expansion series of incomplete Gamma function

Since, then,

let

let

And,

let

Since,

Since,

By using equation (10) we get,

By using eq. expansion of incomplete gamma (17) we get,

let

Since,

then,

So

(19)

2.2. Stress-Strength Reliability

Let y and x be the stress and strength random variable, independent of each other, follow respectively [0,1] and [0,1] then,

Since, and

then,

By using equation (10) we get,

then,

By using expansion of incomplete gamma function (17) we get,

let then,

(20)

3. Summary and Conclusions

In statistical analysis a lot of distributions are used to represent set(s) of data. Recently, new distributions are derived to extend some of well-known families of distributions, such that the new distributions are more flexible than the others to model real data. The composing of some distributions with each other's in some way has been in the foreword of data modeling.

In this paper, we presented a new family of continuous distributions based on [0,1] truncated Fréchet distribution. [0,1] truncated Fréchet Generalized Gamma ([0,1]TFGG) distribution is discussed as special case. Properties of [0,1] TFGG is derived. We provide form for characteristic function, rth raw moment, mean, variance, skewness, kurtosis, mode, median, reliability function, hazard rate function, Shannon entropy function and Relative entropy function. This paper deals also with the determination of stress-strength R=p[y<x] when x (strength) and y (stress) are two independent [0,1] TFGG distribution with different parameters.