New Mathematical Properties For Rayleigh distribution

Regression analysis is one of the most commonly statistical techniques used for analyzing data in different fields. And used to fit the relation between the dependent variable and the independent variables require strong assumption to be met in the model. Generalized linear models (GLMs) allow the extension of linear modeling ideas to a wider class of response types, such as count data or binary responses. Many statistical methods exist for such data types, but the advantage of the GLM approach is that it unites a seemingly disparate collection of response types under a common modeling methodology. So, the problem of the current research is to try to provide a new mathematical property for Exponentiated Rayleigh distribution, and it was one of the most important properties that was studied is to determine Harmonic Mean, as well as calculating the Quantile function, Moments of Residual life (MRL), Reversed Residual Life, Mean of Residual life. The study also presented the probability density function (pdf) and cumulative distribution function according to linear representations.


Introduction and Preliminaries
Generalized distributions are 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 Rayleigh distribution are provided. Generalized distributions depend on two things, namely the cumulative distribution function (CDF) and the probability density function

. Rayleigh distribution
The Rayleigh distribution is one of the most used distributions. The Rayleigh distribution was introduced by Rayleigh in 1880 and it has appeared as a special case of the Weibull distribution. It plays a key role in modeling and analyzing life-time data such as project effort loading modeling, survival and reliability analysis, theory of communication physical sciences, technology, diagnostic imaging, applied statistics and clinical research. Let the random variable follows Rayleigh distribution with scale parameter , then its probability density function(pdf) and cumulative distribution function(cdf) take the form

Results and discussion
The main results are studying new subject does not study before and it is very important ,we studied mathematical properties for Exponentiated Rayleigh and we recommend it to researcher to study applied side of this properties.

Harmonic Mean for Exponentiated Rayleigh
The Harmonic mean of Exponentiated Rayleigh distribution is defined as Applying the general definition of the gamma function

Moment Generating Function for Exponentiated Rayleigh
The moment generating function is a function often used to characterize the distribution of a random variable. The moment generating function has great practical relevance because it can be used to easily derive moments; its derivatives at zero are equal to the moments of the random variable

Definition 1.3
Let x be a random variable. If E(x) exists and finites for all real number belongs to a closed interval [-h,h] , with h>0 then we say that X possesses a moment generating function and the function ( ) ∫ ( )

Rehab H. Mahmoud, Salah M. Mohamed
Where a is parameter of GLM ,α is shape parameter of Rayleigh distribution By using expanded binomial assuming that

Characteristic Function: for Exponentiated Rayleigh
Let X is a random variable the characteristic function of X is defined by assuming that

Quantile Function for Exponentiated Rayleigh
Let X denotes the random variable with pdf given in ( ( ) ( ) ∫ ( ) ( ) ). The Similarly, the Moors measure of kurtosis based on octiles is defined by

Raw Moments: For Exponentiated Rayleigh
The r ℎ moment about origin r′ (raw moment) is generally defined as

Moments of Residual Life For Exponentiated Rayleigh (MRL):
The nth moments of residual life is denoted by ,( ) | where assuming that

Mean of Residual life for Exponentiated Rayleigh:
The mean residual of (Rayleigh) distribution is defined by By using expanded binomial assuming that

Reversed Residual Life For Exponentiated Rayleigh
The nth moments of residual life denoted by ,( ) | where Where, a is the parameter of GLM ,α is shape parameter of Rayleigh distribution assuming that

Conclusion
In this research we introduced the Rayleigh distribution and study some new mathematical properties for Exponentiated Rayleigh, including Harmonic Mean, Moment Generating Function, Characteristic Function, Quantile function, Raw Moments, Moments of Residual life (MRL), Reversed Residual Life, Mean of Residual life. Hence, we invite researchers to study more mathematical properties of distributions because of its many applications that can help solve many life problems.