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. [2] will be our guide. Eugene et al. (2002) 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) 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 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 two sections, we will assume that G are Uniform and Exponential distributions respectively.

2. [0,1] Truncated Fréchet Uniform Distribution

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

(8)

(9)

So, the reliability and hazard rate functions are respectively

The rth raw moment can be derived as follows,

let then,

(10)

And then, the characteristic function is

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

(11)

(12)

The mode and the median can be derived as,

(13)

(14)

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

(15)

Also, the kurtosis is,

(16)

The quantile function of [0,1] TFU random variable can be derived as,

(17)

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

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

2.1. Shannon and Relative Entropies

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

Let

let then,

Since where

and is an Euler constant, then

For, let then,

since then,

let then,

then the Shannon entropy is

(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

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

since, then

Since,

And,

And,

let then,

so the relative entropy is,

(19)

2.2. Stress-Strength Reliability

Inferences about R = P[Y < X], where X and Y are two independent random variables, is very common in the reliability literature. For example, if X is the strength of a component which is subject to a stress Y, then R is a measure of system performance and arises in the context of mechanical reliability of a system. The system fails if and only if at any time the applied stress is greater than its strength. Let Y and X be the stress and the strength random variables, independent of each other, follow respectively and then,

since then,

let so,

(20)

3. [0,1] Truncated Fréchet Exponential Distribution

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

(21)

(22)

So, the reliability and hazard rate functions are respectively

The rth raw moment can be derived as follows,

by using poisson series, we get,

By using the series expansion we get,

and then,

(23)

And then, the characteristic function is

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

(24)

(25)

The mode has not closed form, whilst the median can be derived as,

Since, then,

(26)

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

(27)

Also, the kurtosis is,

(28)

The quantile function of [0,1] TFE random variable can be derived as,

(29)

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

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

3.1. Shannon and Relative Entropies

The Shannon entropy of random variable can be found as follows,

Let,

And,

since where

is an Eualer constant.

since And

And,

since then,

By using the series expansion we get,

then,

(30)

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

Let,

And,

And,

And,

Since, then,

by using expansion series we get,

and then,

by using the for formala we get,

and then,

since then,

And,

By using, we get,

Then, the relative entropy is,

(31)

3.2. Stress-Strength Reliability

Let Y and X be the stress and the strength random variables, independent of each other, follow respectively and then,

since then,

So, by using, we get,

And using, then,

(32)

Where,

4. Summary and Conclusions

In statistical analysis, a lot of distributions are used to represent set(s) 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 Uniform ([0,1] TFU ) and [0,1] truncated Fréchet Exponential ([0,1] TFE ) distributions are discussed as special cases. Properties of [0,1] TFU and [0,1] TFE is derived. We provide forms 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 reliability R = P[Y < X] when X (strength) and Y (stress) are two independent [0,1] TFU ([0,1] TFE) distributions with different parameters.