Some Mathematical Properties for Kumaraswamy Fréchet distribution

In this research, some mathematical properties for Kumaraswamy Fréchet distribution was presented, include entropy, the Shannon entropy, probability weighted moments, moments of residual life and mean of residual life. the properties were concluded for the Kumaraswamy Fréchet distribution using the probability density function (pdf) and cumulative distribution function according to linear representations.


INTRODUCTION AND PRLIMINARIES
Generalized distributions are very important in the scope of probability distribution, and it contains many mathematical properties that make the distribution more elastic. In this chapter the definitions and new properties of Kumaraswamy Fréchet distribution are provided [11] in 1972 . The baseline of generalized distributions of Kumaraswamy distribution depends on the probability density function (pdf) and cumulative distribution function as the equation (1.1) and (1.2) respectively are given by: 1 (1.1) where g(x) is density function for distribution and G(x) corresponding cumulative function for the base line.

. Fréchet Distribution
The Fréchet distribution was introduced in 1927, Fréchet distribution known as inverse Weibull distribution is a special case of the generalizes extreme value distribution. Where the Fréchet distribution was used model maximum values in a data set. It is one of four extreme value Bassant W. Farouk, Salah M. Mohamed distribution "EVDs" in common use. The other three are the Gumbel distribution, the Weibull Distribution and the Generalized Extreme Value Distribution [17] in 2018. This distribution is used to model a wide range of phenomena like flood analysis, horse racing, human lifespans, maximum rainfalls, and river discharges in hydrology. Equations (1.3) and (1.4) respectively are represent the probability density function (pdf) and cumulative distribution function (cdf) of the Fréchet distribution with parameter α, β for x ≥ 0. f the following graph represents pdf of Kumaraswamy Fréchet (Kw-Fr) distribution by using R program .

2.1linear representation
In this subsection the researcher used the following power series expansion that given by the following formula to find the linear representation of Kumaraswamy Fréchet distribution [5] in 2009.
To make pdf of Kumaraswamy Fréchet "kw-fr" in a linear representation form to be able to use in several properties, so we have.
then transfer the last term of the equation to linear representation as the following.
Therefore, we can obtain the pdf of KW-fr as the following therefore, the pdf of kw-fr by using linear representation is

Probability weighted moments
Probability weighted moments (PWMs) are introduced and shown to be potentially useful in expressing the parameters of these distributions [12] in 2019 . This method is considered as a generalization of the traditional moments of a probability distribution for a random variable X. The probability weighted moments of a random variable expressible in an inverse form is defined by: =∫ Where s,r are real numbers and x(F) is the inverse cumulative distribution function. Moreover, the PWMs of a Kumaraswamy Fréchet take the following form: By substitution with pdf of kw-fr distribution at (1.6) get By using factorial rule get [1-exp[-) β ] b as the following can represented by

Bassant W. Farouk, Salah M. Mohamed
Entropy is one of the most popular measures of uncertainty. It is used to measure the randomness of system; also, it is a measure of information [17] in 2018As former mathematical work, statistical entropy was introduced by Shannon in 1948 as a basic concept in information theory measuring the amount of information in a random variable X. Entropy function represented by the following:

Shannon entropy
∫ By using equation (2.1) and substitute in Shannon entropy's function get the following.
To simplify the previous function, suppose that separated into four parts as the following First part represented by I [1+β] ⸫II= Add (1) and subtract (1) from x as and transfer the log get the following By using the integration get the following II= ∑ ∑ ( ) ) Using integration rule to find ) that given by )= ∫ By using pdf of kw-fr distribution in equation (1.5) to get ) as the following )= ∑ ∫ By simplify Put ∫ dv is a gamma formula

Moment of residual life
In life testing situations, the additional lifetime given that component has survived until time t is called the residual life function (RLF) of the component [2] in 2017. Moment of residual represented as the following:

Mean of residual life
The mean of residual life is an important application of the moments of residual lifetime function which has been employed in life lengths studies by various authors. It has also been shown that the exponential function determines the distribution function uniquely [16] in 2016. It is also well known that a constant MRL function implies that the distribution is exponential and represented as the following:

CONCLUSION
In this research paper, we introduce some mathematical properties of the Kumaraswamy Fréchet distribution based on linear representation. These properties are probability weighted moments, entropy, Shannon entropy, moment of residual life and Mean of Residual. The introduced mathematical properties are new for Kumaraswamy Fréchet distribution. Therefore, we invite researchers to study more mathematical properties of the distributions because of its many applications which can contribute to solving many life problems.