A Class of Exponential Rayleigh Distribution and New Modified Weighted Exponential Rayleigh Distribution with Statistical Properties

This paper deals with the mathematical method for extracting the Exponential Rayleighh (ER) distribution based on mixed between the cumulative distribution function of Exponential distribution and the cumulative distribution function of Rayleigh distribution using an application (maximum), as well as derived different statistical properties for 𝐸𝑅 distribution ( Mode, The Median, 𝑟 𝑡ℎ moment, The Variance , Coefficient of Skewness , Coefficient of Kurtosis, Moment Generating Function, Factorial moment generating function, Quantile Function, Characteristic Function ). Then, we present a structure of a new distribution based on a modified weighted version of Azzalini’s named Modified Weighted Exponential Rayleigh (MWER) distribution such that this new distribution is generalization of the ER distribution and provide some special models of the 𝑀𝑊𝐸𝑅 distribution, as well as derived different statistical properties for 𝑀𝑊𝐸𝑅 distribution

produce a more and more realistic and flexible models for data. In 1880, [1] introduced the Rayleigh distribution this distribution with one scale parameter is one of the most widely used distributions. Exponential and Rayleigh were an important distribution in statistics and operation research [2]. [3] mixed Exponential and Rayleigh distributions based on T-X families. [4]. presented the finite mixture of Exponential Rayleigh and Burr Type-XII distribution. [5] mixed multivariate Exponential distribution and the multivariate Rayleigh distribution to obtained Multivariate Rayleigh and Exponential Distributions. [6] introduced a new mixture distribution based on the tail of mixed between Exponential Rayleigh and Exponential Weibull distributions using an application (minimum). There are various methods of inputting the shape parameter of a probability distribution model and they may result in a variety of weighted distributions. The weighted distributions are widely used in reliability, survival, bio-medicine, environment, and many other fields of immense practical interest in mathematics, probability, statistics. These distributions naturally arise as a result of observations created by a random process and recorded with some weight functions [7]. The aim of this paper, two distributions have been introduced Exponential Rayleigh distribution this distribution can be obtained based on mixed between cumulative distribution function of Exponential distribution and the cumulative distribution function of Rayleigh distribution using an application (maximum) and present a new distribution named Modified Weighted Exponential Rayleigh distribution built on a modified weighted version of Azzalini's (1985), this new distribution is a generalization of the Exponential Rayleigh distribution, as well as present the most important statistical properties of these two distributions, finally the conclusion of this paper is determined.

Exponential Rayleigh Distribution
A continuous non-negative random variable is called to have an Exponential distribution with parameter , if its probability density function is given by [8]: Where is a scale parameter. The cumulative distribution function is: ( ; ) = 1 − − ; ≥ 0; > 0 …(2) A continuous non-negative random variable is called to have a Rayleigh distribution with parameter , if its probability density function is given by [9]: Where is a scale parameter. The cumulative distribution function is: ; ≥ 0; > 0 …(4) The Exponential Rayleigh distribution introduced by Mohammed and Hussein in (2019) depends on mixed of the tail (survival) function of Exponential distribution and the tail (survival) function of Rayleigh distribution using an application (minimum) [6]. This distribution can be also generated in another way depending on mixed between the cumulative distribution function of Exponential distribution as in equation (2) and cumulative distribution function of Rayleigh distributions as in equation (4)  ; ≥ 0; , > 0 …(7)  ; ≥ 0; , > 0 …(9) is the hazard rate function given in equation (8), and ( ; , ) is the Survival function was given in equation (7). Since ( ; , ) ≠ 0. Thus dividing the equation (10) by ( ; , ) , we get: 2 2 + 2 + ( 2 − ) = 0 Based on the law of the constitution, we get: Since > 0, the negative value of is ignored. Suppose that = 0 that is a root of equation (11). This root based on the second derivative of the equation:

The Median
The median of distribution is given by: Since > 0, the negative value of will be ignored.

The Moment about the Origin
The ℎ moment about the origin can be obtained by: …(14) By Maclaurin series: Substituting equation (15) in equation (14), we get: Substituting equation (16) in equation (13) we get: Now, solve the first integral as follows: Now, solve the second integral as follows: Substituting equations (18) and (19) in equation (17) we get: The Mean: Let = 1 in equation (20) we get the first moment which is called the mean, thus: The Variance: The general form of ( ) of distribution is given by:

Coefficient of Skewness
The general form of the Coefficient of Skewness ( . ) of distribution is given by:

Coefficient of Kurtosis
The general form of the Coefficient of Kurtosis ( . ) of distribution is given by:

Factorial Moment Generating Function
The factorial moment generating function of distribution can be obtained as follows: Now, based on equations (30) and (31) we get: )] …(38)

Quantile Function
The quantile function of random variable is defined as a solution of ( ≤ ( ) ) = ( ( ) ) w.r.t. ( ) , Therefore, via using the inverse transformation to equation (5), it can be found as: Based on law of the constitution, we get: Since ( ) > 0, the negative values of ( ) will be ignored.

Modified Weighted Exponential Rayleigh Distribution
This section discusses adding a shape parameter to Exponential Rayleigh distribution and generating a Modified Weighted Exponential Rayleigh distribution as follows: The general definition for extracting modified weighted non-negative models depending on a modified weighted version of Azzalini's (1985) can be summarized by [10]: Let ( ) be a probability density function and ̅ ( ) be corresponding reliability (survival) function such that the cumulative distribution function ( ) exist. Then the modified weighted model of distribution is given by: ( ) = ( ) ̅ ( ) Where, is the normalizing constant and > 0 is the shape parameter. In our work this parameter does not depend on the degree of the random variable X. Now, consider a probability density function of distribution as in equation (6) and the survival function as in equation (7), according to the previous definition for extracting modified weighted non-negative models, put = 1 + extract the probability density function of Modified Weighted Exponential Rayleigh distribution as follows:

Special Models
In this section, we provide special models of the distribution: 1. When = = 0 the probability density function of Modified Weighted Exponential Rayleigh distribution reduces to give the probability density function of Rayleigh distribution Where is a scale parameter and is the shape parameter. Such that;  Based on the law of the Constitution, we get: The value of is ignor when < 0, suppose that = 0 that is a root of equation (52)

The Median
The median of distribution can be obtained as follows: Based on law of the Constitution, we get: The value of is ignor when < 0.

The Moment about the Origin
The ℎ moment about the origin can be defined as: The Mean: Let = 1 in equation (61) we get the first moment which is called the mean, thus: The Variance: The general form of ( ) of distribution is defined as: )] ] 2 …(63)

Coefficient of Skewness
The general form of the Coefficient of Skewness ( . ) of distribution can be obtained by: )]] 3 2 …(64)

Coefficient of Kurtosis
The general form of the Coefficient of Kurtosis ( . ) of distribution is given by: )]] 2 − 3 …(65)

Moment Generating Function
The moment generating function of distribution can be found as follows:

Factorial Moment Generating Function
The factorial moment generating function of distribution can be obtained as follows:

Characteristic Function
The characteristic function of distribution can be found as follows:

Quantile Function
The quantile function of random variable is defined as a solution of ( ≤ ( ) ) = ( ( ) ) w.r.t. ( ) , Therefore, via using the inverse transformation to equation (47), it can be found as: The values of ( ) will be ignored when ( ) < 0.

4.Conclusions
In this paper, introduce Exponential Rayleigh distribution depending on mixed between cumulative distribution function of Exponential and Rayleigh distribution, as well as introduce a new class depending on a modified weighted version of Azzalini's (1985) named Modified Weighted Exponential Rayleigh distribution, such that the Exponential Rayleigh distribution is special case of Modified Weighted Exponential Rayleigh distribution and provide some special models of the distribution. Different statistical properties such as the mode, the median, the ℎ moment about the origin, the moment generating function, factorial moment generating function, the characteristic function and quantile function are discuses and study for these two distributions.