IHJPAS. 36(2)2023 

390 
This work is licensed under a Creative Commons Attribution 4.0 International License. 

 

   

 

 

 

 

 

 
 

 

 

 

 

 

 
 

 

Abstract 

      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, 𝑟𝑡ℎ 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. 

Keywords: Exponential Rayleigh (ER) distribution, Modified Weighted Exponential Rayleigh 

(MWER) distribution. 

1.Introduction 

Statistical distributions are very important for parametric inferences and are usually applied to 

describe real world phenomena. Although they are useful in several scientific fields, it is observed 

that most common distributions, such as Exponential, Gamma, Weibull, Rayleigh and  Lindley are 

not sufficiently flexible to accommodate various phenomena of nature for example, while 

exponential distribution is frequently defined as flexible, its hazard function is constant. For this 

cause, researchers have focused on the expansion of these common distributions in order to 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: jih.uobaghdad.edu.iq 

 

doi.org/10.30526/36.2.3044 

Article history: Received 18 September 2022, Accepted 12 December 2022, Published in April 2023. 

 

 

 

A Class of Exponential Rayleigh Distribution and New Modified 

Weighted Exponential Rayleigh Distribution with Statistical Properties 
 

Iden Hasan Hussein 
2Department of Mathematic, College of Science for 

Women, University of Baghdad,  Baghdad, Iraq 

Idenalkanani58@gmail.com 

 

Lamyaa Khalid Hussein 

1Department of Mathematics, College of Science, 

Mustansiriyah University, Baghdad, Iraq 

lamyaakhalid8242@gmail.com 

 
Huda Abdullah Rasheed 

3Department of Mathematics, College of Science, 

Mustansiriyah University, Baghdad, Iraq 

iraqalnoor1@gmail.com 

 

https://creativecommons.org/licenses/by/4.0/
mailto:Idenalkanani58@gmail.com
mailto:lamyaakhalid8242@gmail.com
mailto:iraqalnoor1@gmail.com


IHJPAS. 36(2)2023 

391 
 

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. 

2. Exponential Rayleigh Distribution 

A continuous non-negative random variable 𝑍 is called to have an Exponential distribution with 

parameter 𝛼, if its probability density function is given by [8]: 

𝑓(𝑧; 𝛼)𝐸 =     𝛼 𝑒
−𝛼𝑧                               ; 𝑧 ≥ 0;  𝛼 > 0                                                …(1) 

                       zero otherwise. 

 Where 𝛼 is a scale parameter. 

The cumulative distribution function is: 

𝐹(𝑧; 𝛼)𝐸 = 1 −  𝑒
−𝛼𝑧                            ; 𝑧 ≥ 0;  𝛼 > 0                                                   …(2)                                        

A continuous non-negative random variable 𝑌 is called to have a Rayleigh distribution with 

parameter 𝜆, if its probability density function is given by [9]: 

𝑓(𝑦; 𝜆)𝑅 =     𝜆𝑦 𝑒
− 

𝜆

2
 𝑦2

                               ; 𝑦 ≥ 0;  𝜆 > 0                                                   …(3) 

zero otherwise. 

Where 𝜆 is a scale parameter. 

The cumulative distribution function is: 

𝐹(𝑦; 𝜆)𝑅 = 1 −  𝑒
− 

𝜆

2
 𝑦2

                          ; 𝑦 ≥ 0;  𝜆 > 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 



IHJPAS. 36(2)2023 

392 
 

Exponential distribution as in equation (2) and cumulative distribution function of Rayleigh 

distributions as in equation (4) using an application (maximum) as follows:          

 Let 𝑋 = 𝑚𝑎𝑥(𝑍, 𝑌) where 𝑍~𝐸(𝛼), 𝑌~𝑅(𝜆) , 𝑍 and 𝑌 are two independent random variables 

then: 

 

𝐹(𝑥; 𝛼, 𝜆 )𝐸𝑅 = 1 − 𝑝𝑟(𝑚𝑎𝑥(𝑍, 𝑌) > 𝑥)        

𝐹(𝑥; 𝛼, 𝜆 )𝐸𝑅 = 1 − [𝑝𝑟(𝑍 > 𝑥). 𝑝𝑟(𝑌 > 𝑥)]       

𝐹(𝑥; 𝛼, 𝜆 )𝐸𝑅 = 1 − [( ∫ 𝛼 𝑒
−𝛼𝑧 𝑑𝑧

∞

𝑥

). (∫ 𝜆𝑦 𝑒
− 

𝜆
2

 𝑦2
𝑑𝑦

∞

𝑥

)] 

𝐹(𝑥; 𝛼, 𝜆 )𝐸𝑅 = 1 −  𝑒
− (𝛼𝑥 + 

𝜆

2
 𝑥2)

         ; 𝑥 ≥ 0; 𝛼, 𝜆 > 0                                             …(5) 

 

 

Figure 1. plot of the cumulative distribution function of 𝐸𝑅 distribution for 𝜆 = 0.1 and different values of (𝛼 =

0.1,0.2,0.3,0.4,0.5, 0.6, 0.7 )[ MATLAB R2013a ]. 

 

The probability density function of Exponential Rayleigh 𝐸𝑅 distribution is given by: 

𝑓(𝑥; 𝛼, 𝜆 )𝐸𝑅 =     (𝛼 + 𝜆𝑥) 𝑒
− (𝛼𝑥 + 

𝜆

2
 𝑥2)

         ; 𝑥 ≥ 0; 𝛼, 𝜆 > 0                                  …(6) 

                              zero otherwise.  

Where 𝛼 𝑎nd 𝜆 are scale parameters. 

Such that,  

• 𝑓(𝑥; 𝛼, 𝜆 )𝐸𝑅 > 0 

• ∫ 𝑓(𝑥; 𝛼, 𝜆 )𝐸𝑅
∞

0
 𝑑𝑥 =  ∫ (𝛼 + 𝜆𝑥) 𝑒

− (𝛼𝑥 + 
𝜆

2
 𝑥2)

 
∞

0
𝑑𝑥                                                     =

 −  [ 𝑒
− (𝛼𝑥 + 

𝜆

2
 𝑥2)

]
0

∞

                                               = 1 

 
Figure 2. plot of the probability density function of 𝐸𝑅 distribution for 𝜆 = 0.1 and different values of      

( 𝛼 = 0.1, 0.2, 0.3, 0.4,0.5, 0.6, 0.7 )[MATLABR2013a]. 

 

 



IHJPAS. 36(2)2023 

393 
 

 

• The Survival function is given by: 

𝑆(𝑡; 𝛼, 𝜆 )𝐸𝑅 =  𝑒
− (𝛼𝑡 + 

𝜆

2
 𝑡2)

             ; 𝑡 ≥ 0; 𝛼, 𝜆 > 0                           …(7) 

 

 

 
Figure 3. plot of the survival function of 𝐸𝑅 distribution for 𝜆 = 0.1  and different values of                           

( 𝛼 = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 ) [ MATLAB R2013a ]. 

 

The Hazard rate function is obtained by: 

ℎ(𝑡; 𝛼, 𝜆 )𝐸𝑅 = 𝛼 + 𝜆𝑡                   ; 𝑡 ≥ 0; 𝛼, 𝜆 > 0                                      …(8) 

 
Figure 4. plot of hazard rate function of 𝐸𝑅 distribution for 𝜆 = 0.1 and different values of ( 𝛼 =

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 ) [ MATLAB R2013a ]. 

 

The Reverse hazard rate function is given by: 

 ∅(𝑡; 𝛼, 𝜆)𝐸𝑅 =
(𝛼+𝜆𝑡) 𝑒

− (𝛼𝑡 + 
𝜆
2

 𝑡2)
 

1− 𝑒
− (𝛼𝑡 + 

𝜆
2

 𝑡2)
          ; 𝑡 ≥ 0; 𝛼, 𝜆 > 0                                     …(9) 

 

Figure 5. plot of the reverse hazard rate function of 𝐸𝑅 distribution for 𝜆 = 0.1and different values of               

( 𝛼 = 0.1, 0.2, 0.3, 0.4,0.5, 0.6, 0.7 ) [ MATLAB R2013a ]. 

 

 

2.1 Some Statistical Properties of 𝑬𝑹 Distribution 

2.1.1 The Mode 

 We can give the mode of 𝐸𝑅 distribution as follows: 

t

t 

T
h

e
 S

u
rv

iv
a
l 

F
u

n
c
ti

o
n



IHJPAS. 36(2)2023 

394 
 

𝜕𝑓(𝑥; 𝛼, 𝜆)𝐸𝑅
𝜕𝑥

= − (𝛼 + 𝜆𝑥)2 𝑒
− (𝛼𝑥 + 

𝜆
2

 𝑥2)
+  𝜆  𝑒

− (𝛼𝑥 + 
𝜆
2

 𝑥2)
 

 [− (𝛼 + 𝜆𝑥)2 +  𝜆]  𝑒
− (𝛼𝑥 + 

𝜆

2
 𝑥2)

= 0                                                                                …(10) 

It is clear that 
𝜕𝑓(𝑥;𝛼,𝜆)𝐸𝑅

𝜕𝑥
= [ − {ℎ(𝑥; 𝛼, 𝜆)𝐸𝑅 }

2 + ℎ′(𝑥; 𝛼, 𝜆)𝐸𝑅 ] 𝑆(𝑥; 𝛼, 𝜆)𝐸𝑅 , Where ℎ(𝑥; 𝛼, 𝜆)𝐸𝑅 

is the hazard rate function given in equation (8), and 𝑆(𝑥; 𝛼, 𝜆)𝐸𝑅 is the Survival function was 

given in equation (7). Since 𝑆(𝑥; 𝛼, 𝜆)𝐸𝑅 ≠ 0. Thus dividing the equation (10)  by 𝑆(𝑥; 𝛼, 𝜆)𝐸𝑅 , we 

get: 

𝜆2𝑥2 + 2 𝜆𝛼𝑥 + (𝛼2 − 𝜆) = 0 

Based on the law of the constitution, we get: 

𝑥 =
−(2 𝜆𝛼) ∓ √(2 𝜆𝛼)2− 4 𝜆2(𝛼2−𝜆)

2𝜆2 
                                                                                          …(11)  

Since 𝑥 > 0, the negative value of 𝑥 is ignored. Suppose that 𝑥 = 𝑥0 that is a root of equation 

(11). This root based on the second derivative of the equation:  

If  
𝜕2𝑓(𝑥;𝛼,𝜆)𝐸𝑅

𝜕𝑥2
⃒𝑥=𝑥0 < 0  the root is the local maximum. If  

𝜕2𝑓(𝑥;𝛼,𝜆)𝐸𝑅

𝜕𝑥2
⃒𝑥=𝑥0 > 0 the root is the 

local minimum. If  
𝜕2𝑓(𝑥;𝛼,𝜆)𝐸𝑅

𝜕𝑥2
⃒𝑥=𝑥0 = 0 the point is inflection. 

 

2.1.2 The Median 

The median of 𝐸𝑅 distribution is given by: 

1 −  𝑒
− (𝛼𝑥 + 

𝜆
2

 𝑥2)
=

1

2
 

𝜆 𝑥2 + 2 𝛼𝑥 − 2 ln 2 = 0 

Based on the law of the constitution, we get: 

𝑥 =
−(2𝛼) ∓ √4𝛼2+ 8 𝜆 𝑙𝑛 2

2 𝜆 
                                                                                                      …(12) 

Since 𝑥 > 0, the negative value of 𝑥 will be ignored.                                                       

 

2.1.3 The Moment about the Origin  

 The 𝑟𝑡ℎ moment about the origin can be obtained by: 

𝐸(𝑋𝑟 )𝐸𝑅 = ∫ 𝑥
𝑟 (𝛼 + 𝜆𝑥) 𝑒

− (𝛼𝑥 + 
𝜆

2
 𝑥2)

 
∞

0
𝑑𝑥                                                                      …(13) 

Let 

𝐾(𝑟, 𝛼, 𝜆) = 𝑥𝑟  𝑒
− (𝛼𝑥 + 

𝜆

2
 𝑥2)

                                                                                                …(14) 

By Maclaurin series: 

 𝑒−𝛼𝑥 = ∑
(−𝛼)𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛                                                                                                           …(15) 

Substituting equation (15) in equation (14), we get: 

 𝐾(𝑟, 𝛼, 𝜆) = ∑
(−𝛼)𝑛

𝑛!
∞
𝑛=0  𝑥

𝑟+𝑛 𝑒
− 

𝜆

2
 𝑥2

                                                                                  …(16) 

Substituting equation (16) in equation (13) we get: 

𝐸(𝑋𝑟 )𝐸𝑅 = ∑
(−𝛼)𝑛

𝑛!
∞
𝑛=0 [∫ 𝛼 𝑥

𝑟+𝑛  𝑒
− 

𝜆

2
 𝑥2

 
∞

0
𝑑𝑥 +  ∫ 𝜆 𝑥𝑟+𝑛+1  𝑒

− 
𝜆

2
 𝑥2

 
∞

0
𝑑𝑥]                           …(17) 

Now, solve the first integral as follows:  

𝐿1 = ∫ 𝛼 𝑥
𝑟+𝑛  𝑒

− 
𝜆

2
 𝑥2

 
∞

0
𝑑𝑥 =  

𝛼 2
 
𝑟+𝑛−1

2

𝜆 
𝑟+𝑛+1

2

 𝛤( 
𝑟+𝑛+1

2
 )                                                               …(18)                    

Now, solve the second integral as follows:  



IHJPAS. 36(2)2023 

395 
 

𝐿2 = ∫ 𝜆 𝑥
𝑟+𝑛+1  𝑒

− 
𝜆

2
 𝑥2

 
∞

0
𝑑𝑥 =  

 2
 
𝑟+𝑛

2

𝜆 
𝑟+𝑛

2

 𝛤( 
𝑟+𝑛+2

2
 )                                                                 …(19) 

Substituting equations (18) and (19) in equation (17) we get: 

𝐸(𝑋𝑟 )𝐸𝑅 = ∑
(−𝛼)𝑛

𝑛!
∞
𝑛=0

 2 
𝑟+𝑛

2

 𝜆 
𝑟+𝑛

2

[ 
𝛼 

√2𝜆
 𝛤 ( 

𝑟+𝑛+1

2
 ) +  𝛤 ( 

𝑟+𝑛+2

2
 )]                                              …(20) 

The Mean: Let 𝑟 = 1 in equation (20) we get the first moment which is called the mean, thus: 

𝐸(𝑋)𝐸𝑅 = ∑
(−𝛼)𝑛

𝑛!
∞
𝑛=0

 2 
1+𝑛

2

 𝜆 
1+𝑛

2

[ 
𝛼 

√2𝜆
 𝛤 ( 

𝑛+2

2
 ) +  𝛤 (

𝑛+3

2
 )]                                                        …(21) 

The Variance: The general form of 𝑣(𝑋) of 𝐸𝑅 distribution is given by: 

 𝑣(𝑋)𝐸𝑅 = 𝐸(𝑋
2)𝐸𝑅 − [𝐸(𝑋)𝐸𝑅 ]

2  

𝑣(𝑋)𝐸𝑅 =  [∑
(−𝛼)𝑛

𝑛!
∞
𝑛=0

 2 
2+𝑛

2

 𝜆 
2+𝑛

2

[ 
𝛼 

√2𝜆
 𝛤 ( 

𝑛+3

2
 ) +  𝛤 ( 

𝑛+4

2
 )]]  –  [∑

(−𝛼)𝑛

𝑛!
∞
𝑛=0

 2 
1+𝑛

2

 𝜆 
1+𝑛

2

[ 
𝛼 

√2𝜆
 𝛤 ( 

𝑛+2

2
 ) +

𝛤 ( 
𝑛+3

2
 )]]

2

                                                                                                                      …(22)          

2.1.4 Coefficient of Skewness  

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

𝐶. 𝑆𝐸𝑅 =
𝐸(𝑋3)

𝐸𝑅

[𝐸(𝑋2)𝐸𝑅]
3
2

=

∑
(−𝛼)𝑛 

𝑛!
 ∞𝑛=0
 2 

3+𝑛
2

 𝜆 
3+𝑛

2

 [ 
𝛼 

√2𝜆
 𝛤( 

𝑛+4

2
 )+ 𝛤( 

𝑛+5

2
 )]

[∑
(−𝛼)𝑛

𝑛!
 ∞𝑛=0
 2 

2+𝑛
2

 𝜆 
2+𝑛

2

 [ 
𝛼 

√2𝜆
 𝛤( 

𝑛+3

2
 )+ 𝛤( 

𝑛+4

2
 )]]

3
2

                                           …(23) 

2.1.5 Coefficient of Kurtosis   

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

𝐶. 𝐾𝐸𝑅 =
𝐸(𝑋4)

𝐸𝑅

[𝐸(𝑋2)𝐸𝑅]
2

− 3 =

∑
(−𝛼)𝑛

𝑛!
 ∞𝑛=0
 2 

4+𝑛
2

 𝜆 
4+𝑛 

2

 [ 
𝛼 

√2𝜆
 𝛤( 

𝑛+5

2
 )+ 𝛤( 

𝑛+6

2
 )]

[∑
(−𝛼)𝑛

𝑛!
 ∞𝑛=0
 2 

2+𝑛
2

 𝜆 
2+𝑛 

2

 [ 
𝛼 

√2𝜆
 𝛤( 

𝑛+3

2
 )+ 𝛤( 

𝑛+4

2
 )]]

2 − 3                         …(24)                

 

2.1.6  Moment Generating Function  

The moment generating function of 𝐸𝑅 distribution can be derived as follows: 

𝑀𝑋 (𝑡)𝐸𝑅 = 𝐸(𝑒
𝑥𝑡 ) = ∫ 𝑒 𝑥𝑡  (𝛼 + 𝜆𝑥) 𝑒

− (𝛼𝑥 + 
𝜆
2

 𝑥2)
 

∞

0

𝑑𝑥 

𝑀𝑋 (𝑡)𝐸𝑅 = ∫  (𝛼 + 𝜆𝑥) 𝑒
− ((𝛼−𝑡) 𝑥 + 

𝜆

2
 𝑥2)

 
∞

0
𝑑𝑥                                                                …(25) 

 

Let 

𝑊((𝛼 − 𝑡), 𝜆) =  𝑒
− ((𝛼−𝑡) 𝑥 + 

𝜆

2
 𝑥2)

                                                                                   …(26) 

By Maclaurin series: 

 𝑒−(𝛼−𝑡)𝑥 = ∑
(−(𝛼−𝑡))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛                                                                                          …(27) 

Substituting equation (27) in equation (26) we get: 



IHJPAS. 36(2)2023 

396 
 

 𝑊((𝛼 − 𝑡), 𝜆) = ∑
(−(𝛼−𝑡))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛  𝑒
− 

𝜆

2
 𝑥2

                                                                      …(28) 

Substituting equation (28) in equation (25) we get: 

𝑀𝑋 (𝑡)𝐸𝑅 = ∑
(−(𝛼−𝑡))𝑛

𝑛!
∞
𝑛=0 [∫ 𝛼 𝑥

𝑛   𝑒
− 

𝜆

2
 𝑥2

 
∞

0
𝑑𝑥 + ∫ 𝜆 𝑥𝑛+1  𝑒

− 
𝜆

2
 𝑥2

 
∞

0
𝑑𝑥]                     …(29) 

Now, solve the first integral as follows:  

𝐿1 = ∫ 𝛼 𝑥
𝑛   𝑒

− 
𝜆

2
 𝑥2

 
∞

0
𝑑𝑥 =  

𝛼 2
 
𝑛−1

2

𝜆 
𝑛+1

2

 𝛤( 
𝑛+1

2
 )                                                                   …(30)                    

Now, solve the second integral as follows:  

𝐿2 = ∫ 𝜆 𝑥
𝑛+1  𝑒

− 
𝜆

2
 𝑥2

 
∞

0
𝑑𝑥  =  

 2
 
𝑛
2

𝜆 
𝑛
2

 𝛤( 
𝑛+2

2
 )                                                                   …(31) 

Substituting equations (30) and (31) in equation (29) yields: 

𝑀𝑋 (𝑡)𝐸𝑅 = ∑
(−(𝛼−𝑡))𝑛

𝑛!
∞
𝑛=0

 2 
𝑛
2

 𝜆 
𝑛
2

[ 
𝛼 

√2𝜆
 𝛤 ( 

𝑛+1

2
 ) +  𝛤 ( 

𝑛+2

2
 )]                                            …(32) 

 

2.1.7 Factorial Moment Generating Function  

The factorial moment generating function of 𝐸𝑅 distribution can be obtained as follows: 

𝑀(𝑡)𝐸𝑅 = 𝐸(𝑡
𝑥 ) = ∫ 𝑡𝑥  (𝛼 + 𝜆𝑥) 𝑒

− (𝛼𝑥 + 
𝜆
2

 𝑥2)
 

∞

0

𝑑𝑥 

𝑀(𝑡)𝐸𝑅 = ∫  (𝛼 + 𝜆𝑥) 𝑒
− ((𝛼−ln (𝑡)) 𝑥 + 

𝜆

2
 𝑥2)

 
∞

0
𝑑𝑥                                                           …(33) 

Let 

𝐴((𝛼 − ln(𝑡)) , 𝜆) =  𝑒
− ((𝛼−𝑙𝑛(𝑡)) 𝑥 + 

𝜆

2
 𝑥2)

                                                                      …(34) 

By Maclaurin series:  

 𝑒−(𝛼−ln (𝑡))𝑥 = ∑
(−(𝛼−ln (𝑡)))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛                                                                               …(35) 

Substituting equation (35) in equation (34) we get: 

𝐴((𝛼 − ln(𝑡)), 𝜆) = ∑
(−(𝛼−ln (𝑡)))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛  𝑒
− 

𝜆

2
 𝑥2

                                                           …(36) 

Substituting equation (36) in equation (33) we get: 

𝑀(𝑡)𝐸𝑅 = ∑
(−(𝛼−ln (𝑡)))𝑛

𝑛!
∞
𝑛=0 [∫ 𝛼 𝑥

𝑛   𝑒
− 

𝜆

2
 𝑥2

 
∞

0
𝑑𝑥 + ∫ 𝜆 𝑥𝑛+1  𝑒

− 
𝜆

2
 𝑥2

 
∞

0
𝑑𝑥]                  …(37)      

Now, based on equations (30) and (31) we get: 

𝑀(𝑡)𝐸𝑅 = ∑
(−(𝛼−ln(𝑡)))𝑛

𝑛!
∞
𝑛=0

 2 
𝑛
2

 𝜆 
𝑛
2

[ 
𝛼 

√2𝜆
 𝛤 ( 

𝑛+1

2
 ) +  𝛤 ( 

𝑛+2

2
 )]                                        …(38) 

 

2.1.8 Characteristic Function   

The Characteristic function of 𝐸𝑅 distribution can be derived as follows: 

∅𝑋 (𝑖𝑡)𝐸𝑅 = 𝐸(𝑒
𝑖𝑡𝑥 ) = ∫ 𝑒𝑖𝑡𝑥  (𝛼 + 𝜆𝑥) 𝑒

− (𝛼𝑥 + 
𝜆
2

 𝑥2)
 

∞

0

𝑑𝑥 

∅𝑋 (𝑖𝑡)𝐸𝑅 = ∫  (𝛼 + 𝜆𝑥) 𝑒
− ((𝛼−it) 𝑥 + 

𝜆

2
 𝑥2)

 
∞

0
𝑑𝑥                                                               …(39) 

Let 

𝑃((𝛼 − it), 𝜆) =  𝑒
− ((𝛼−𝑖𝑡) 𝑥 + 

𝜆

2
 𝑥2)

                                                                                  …(40) 

By Maclaurin series: 



IHJPAS. 36(2)2023 

397 
 

 𝑒−(𝛼−it)𝑥 = ∑
(−(𝛼−it))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛                                                                                         …(41) 

Substituting equation (41) in equation (40) gives: 

𝑃((𝛼 − it), 𝜆) = ∑
(−(𝛼−it))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛  𝑒
− 

𝜆

2
 𝑥2

                                                                      …(42) 

Substituting equation (42) in equation (39) gives: 

∅𝑋 (𝑖𝑡)𝐸𝑅 = ∑
(−(𝛼−it))𝑛

𝑛!
∞
𝑛=0 [∫ 𝛼 𝑥

𝑛   𝑒
− 

𝜆

2
 𝑥2

 
∞

0
𝑑𝑥 + ∫ 𝜆 𝑥𝑛+1  𝑒

− 
𝜆

2
 𝑥2

 
∞

0
𝑑𝑥]                    …(43) 

Now, based on equations (30) and (31) we get: 

∅𝑋 (𝑖𝑡)𝐸𝑅 = ∑
(−(𝛼−it))𝑛

𝑛!
∞
𝑛=0

 2 
𝑛
2

 𝜆 
𝑛
2

[ 
𝛼 

√2𝜆
 𝛤 ( 

𝑛+1

2
 ) +  𝛤 ( 

𝑛+2

2
 )]                                           …(44) 

2.1.9 Quantile Function  

The quantile function of  𝐸𝑅 random variable is defined as a solution of 𝑝(𝑥 ≤ 𝑥(𝑞)) =

𝐹(𝑥(𝑞))𝐸𝑅  w.r.t. 𝑥(𝑞), Therefore, via using the inverse transformation to equation (5), it can be 

found as: 

𝑥(𝑞) = 𝐹
−1(𝑞)                ; 𝑥(𝑞) > 0;  0 < 𝑞 < 1 

𝑞 = 1 −  𝑒
− (𝛼𝑥(𝑞) + 

𝜆
2

 𝑥(𝑞)
2 )

 

𝑙𝑛 ( 1 − 𝑞) = −(𝛼𝑥(𝑞)  +  
𝜆

2
 𝑥(𝑞)

2 ) 

 𝜆 𝑥(𝑞)
2 + 2𝛼𝑥(𝑞)  + 2 𝑙𝑛 ( 1 − 𝑞) = 0 

Based on law of the constitution, we get: 

𝑥(𝑞) =
−(2𝛼)∓√4𝛼2−8 𝜆 𝑙𝑛 (1−𝑞)

2 𝜆 
                                                                                        …(45) 

Since 𝑥(𝑞) > 0, the negative values of 𝑥(𝑞) will be ignored. 

 

3. Modified Weighted Exponential Rayleigh Distribution  

This section discusses adding a shape parameter to Exponential Rayleigh 𝐸𝑅 distribution and 

generating a Modified Weighted Exponential Rayleigh 𝑀𝑊𝐸𝑅 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 𝑔(𝑥) be a probability density function and �̅�(𝑥) be corresponding reliability (survival) 

function such that the cumulative distribution function 𝐺(𝑥) exist. Then the modified weighted 

model of distribution is given by: 

𝑓(𝑥)𝑀𝑊 = 𝑀 𝑔(𝑥) �̅�(𝜃𝑥)                                                              

Where, 𝑀 is the normalizing constant and 𝜃 > 0 is the shape parameter.  

In our work this parameter 𝜃 does not depend on the degree of the random variable X. Now, 

consider a probability density function of 𝐸𝑅 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 𝑀 = 1 + 𝜃 extract the probability density function of Modified 

Weighted Exponential Rayleigh 𝑀𝑊𝐸𝑅 distribution as follows: 

𝑓(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅 = 𝑀(𝛼 + 𝜆𝑥) 𝑒
− (𝛼𝑥 + 

𝜆
2

 𝑥2)
 𝑒

− (𝛼𝜃𝑥 + 
𝜆𝜃
2

 𝑥2)
 

𝑓(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅 = (1 + 𝜃)(𝛼 + 𝜆𝑥) 𝑒
− (𝛼(1+𝜃)𝑥 + 

𝜆(1+𝜃)
2

 𝑥2)
 

𝑓(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅 =      (𝛼(1 + 𝜃) + 𝜆(1 + 𝜃)𝑥) 𝑒
− (𝛼(1+𝜃)𝑥 + 

𝜆(1+𝜃)

2
 𝑥2)

     ; 𝑥 ≥ 0        …(46) 

                                       zero otherwise . 



IHJPAS. 36(2)2023 

398 
 

 

 

 

 

 

Where 𝛼, 𝜆 > 0 are scale parameters and 𝜃 > 0 is the shape parameter. 

Such that, 

• 𝑓(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅 > 0  

• ∫ 𝑓(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅  𝑑𝑥
∞

0
 

               = ∫ (𝛼(1 + 𝜃) + 𝜆(1 + 𝜃)𝑥) 𝑒
− (𝛼(1+𝜃)𝑥 + 

𝜆(1+𝜃)
2

 𝑥2)
 𝑑𝑥

∞

0

 

              = − [ 𝑒
− (𝛼(1+𝜃)𝑥 + 

𝜆(1+𝜃)
2

 𝑥2)
]

0

∞

 

              = 1 

 

Figure 6. plot of the probability density function of 𝑀𝑊𝐸𝑅 distribution for 𝜆 = 𝜃 = 0.1 and different values of 

( 𝛼 = 0.1, 0.2, 0.3, 0.4,0.5, 0.6, 0.7 ) [ MATLAB R2013a ]. 

 

The cumulative distribution function of 𝑀𝑊𝐸𝑅 can be obtained by: 

𝐹(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅 = 1 −  𝑒
− (𝛼(1+𝜃)𝑥 + 

𝜆(1+𝜃)

2
 𝑥2)

        ; 𝑥 ≥ 0;  𝛼, 𝜆, 𝜃 > 0                         …(47) 

 

Figure 7. plot of the cumulative distribution function of 𝑀𝑊𝐸𝑅 distribution for 𝜆 = 𝜃 = 0.1 and different values of 

(𝛼 = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 ) [ MATLAB R2013a ]. 

The Survival function of 𝑀𝑊𝐸𝑅 is given by: 

𝑆(𝑡; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅 =  𝑒
− (𝛼(1+𝜃)𝑡 + 

𝜆(1+𝜃)

2
 𝑡2)

         ; 𝑡 ≥ 0;  𝛼, 𝜆, 𝜃 > 0                                 …(48) 



IHJPAS. 36(2)2023 

399 
 

 

Figure 8. plot of the survival function of 𝑀𝑊𝐸𝑅 distribution for 𝜆 = 𝜃 = 0.1  and different values of         ( 𝛼 =

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 ) [ MATLAB R2013a ]. 

The Hazard rate function of 𝑀𝑊𝐸𝑅 is given by: 

ℎ(𝑡; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅 = 𝛼(1 + 𝜃) + 𝜆(1 + 𝜃)𝑡         ; 𝑡 > 0;  𝛼, 𝜆, 𝜃 > 0                                …(49) 

 

Figure 9. plot of hazard rate function of 𝑀𝑊𝐸𝑅 distribution for 𝜆 = 𝜃 = 0.1 and different values of              (𝛼 =

0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7) [ MATLAB R2013a ]. 

 

 

The Reverse hazard rate function of 𝑀𝑊𝐸𝑅 is given by: 

∅(𝑡; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅 =
(𝛼(1+𝜃)+𝜆(1+𝜃)𝑡) 𝑒

− (𝛼(1+𝜃)𝑡 + 
𝜆(1+𝜃)

2
 𝑡2)

 

1− 𝑒
− (𝛼(1+𝜃)𝑡 + 

𝜆(1+𝜃)
2

 𝑡2)
        ; 𝑡 ≥ 0;  𝛼, 𝜆, 𝜃 > 0            …(50) 

 

Figure 10. plot of the reverse hazard rate function of 𝑀𝑊𝐸𝑅 distribution for 𝜆 = 𝜃 = 0.1  and different values of 

( 𝛼 = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 ) [ MATLAB R2013a ]. 

 

 

 



IHJPAS. 36(2)2023 

400 
 

3.1 Special Models 

 In this section, we provide special models of the 𝑀𝑊𝐸𝑅 distribution: 

1. When 𝛼 = 𝜃 = 0 the probability density function of Modified Weighted Exponential 

Rayleigh 𝑀𝑊𝐸𝑅 distribution reduces to give the probability density function of Rayleigh 

distribution [11]: 

      𝑓(𝑥; 𝜆)𝑅 =   𝜆𝑥 𝑒
− 

𝜆

2
 𝑥2

                               ; 𝑥 ≥ 0;  𝜆 > 0                    

                  zero otherwise. 

 Where 𝜆 is scale parameter. 

2. When 𝜆 = 𝜃 = 0 the probability density function of Modified Weighted Exponential 

Rayleigh 𝑀𝑊𝐸𝑅 distribution reduces to give the probability density function of 

Exponential distribution [7]: 

     𝑓(𝑥; 𝛼)𝐸 =    𝛼 𝑒
−𝛼𝑥                                ; 𝑥 ≥ 0;  𝛼 > 0    

                                  zero otherwise . 

 Where 𝛼 is scale parameter.                    

3. When 𝜃 = 0 the probability density function of Modified Weighted Exponential Rayleigh 

𝑀𝑊𝐸𝑅 distribution reduces to give the probability density function of Exponential 

Rayleigh 𝐸𝑅 distribution: 

 

             𝑓(𝑥; 𝛼, 𝜆 )𝐸𝑅 =    (𝛼 + 𝜆𝑥) 𝑒
− (𝛼𝑥 + 

𝜆

2
 𝑥2)

         ; 𝑥 ≥ 0; 𝛼, 𝜆 > 0           

                                        zero otherwise . 

 Where 𝛼 𝑎nd 𝜆 are scale parameters. 

4. When 𝜆 = 0 the probability density function of Modified Weighted Exponential Rayleigh 

𝑀𝑊𝐸𝑅 distribution reduces to give the probability density function of New Weighted 

Exponential distribution [12]: 

              𝑓(𝑥; 𝛼, 𝜃)𝑁𝑊𝐸 =      𝛼(1 + 𝜃) 𝑒
− 𝛼(1+𝜃)𝑥      ; 𝑥 ≥ 0;  𝛼, 𝜃 > 0   

                                            zero otherwise . 

Where 𝛼 is scale parameter and 𝜃 is shape parameter. 

5. When 𝛼 = 0 the probability density function of Modified Weighted Exponential Rayleigh 

𝑀𝑊𝐸𝑅 distribution reduces to give the probability density function of new distribution 

named Modified Weighted Rayleigh distribution, this distribution is obtain depending on 

definition of modified weighted version of Azzalini’s (1985) as follows: 

Consider a probability density function of  Rayleigh distribution  as in equation (3) and the survival 

function:  

𝑆(𝑥; 𝜆)𝑅 =  𝑒
− 

𝜆
2

 𝑥2
 

depending on definition of modified weighted version of Azzalini’s (1985) and put 𝑀 = 1 + 𝜃 , 

define the probability density function of Modified Weighted Rayleigh 𝑀𝑊𝑅 distribution as 

follows: 

 

 



IHJPAS. 36(2)2023 

401 
 

𝑓(𝑥; 𝜆, 𝜃)𝑀𝑊𝑅 = 𝑀𝜆𝑥 𝑒
− 

𝜆
2

 𝑥2
 𝑒

− 
𝜆𝜃
2

 𝑥2
 

𝑓(𝑥; 𝜆, 𝜃)𝑀𝑊𝑅 =    𝜆(1 + 𝜃)𝑥 𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

     ; 𝑥 ≥ 0;  𝜆, 𝜃 > 0      

                               zero otherwise . 

Where 𝜆 is a scale parameter and 𝜃 is the shape parameter. 

Such that; 

• 𝑓(𝑥; 𝜆, 𝜃)𝑀𝑊𝑅 > 0  

•  ∫ 𝑓(𝑥; 𝜆, 𝜃)𝑀𝑊𝑅  𝑑𝑥 = ∫ 𝜆(1 + 𝜃)𝑥 𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

 𝑑𝑥
∞

0

∞

0
 

                                                          = − [ 𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

]
0

∞

 

                                                          = 1 

Table 1. Special models of the Modified Weighted Exponential Rayleigh MWER distribution 

Distribution 𝑓(𝑥) 𝐹(𝑥) 𝑆(𝑡) ℎ(𝑡) 

𝑅 𝜆𝑥𝑒
− 

𝜆
2

 𝑥2
 1 −  𝑒

− 
𝜆
2

 𝑥2
  𝑒

− 
𝜆
2

 𝑡2
 𝜆𝑡 

𝐸 𝛼 𝑒−𝛼𝑥 1 −  𝑒−𝛼𝑥  𝑒−𝛼𝑡 𝛼 

𝐸𝑅 (𝛼 + 𝜆𝑥) 𝑒
− (𝛼𝑥 + 

𝜆
2

 𝑥2)
  1 −  𝑒

− (𝛼𝑥 + 
𝜆
2

 𝑥2)
  𝑒

− (𝛼𝑡 + 
𝜆
2

 𝑡2)
 𝛼 + 𝜆𝑡 

𝑁𝑊𝐸 𝛼(1 + 𝜃) 𝑒− 𝛼(1+𝜃)𝑥 1 −  𝑒− 𝛼(1+𝜃)𝑥  𝑒− 𝛼(1+𝜃)𝑡 𝛼(1 + 𝜃) 

𝑀𝑊𝑅 𝜆(1 + 𝜃)𝑥 𝑒
− 

𝜆(1+𝜃)
2

 𝑥2
  1 − 𝑒

− 
𝜆(1+𝜃)

2
 𝑥2

  𝑒
− 

𝜆(1+𝜃)
2

 𝑡2
 𝜆(1 + 𝜃)𝑡 

 

 

3.2 Some Statistical Properties of 𝑴𝑾𝑬𝑹 Distribution 

3.2.1 The Mode  

 

 The mode of  𝑀𝑊𝐸𝑅 distribution can be derived as follows: 

𝜕𝑓(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅
𝜕𝑥

= − (𝛼(1 + 𝜃) + 𝜆(1 + 𝜃)𝑥)2 𝑒
− (𝛼(1+𝜃)𝑥 + 

𝜆(1+𝜃)
2

 𝑥2)

+  𝜆 (1 + 𝜃) 𝑒
− (𝛼(1+𝜃)𝑥 + 

𝜆(1+𝜃)
2

 𝑥2)
 

 [− (𝛼(1 + 𝜃) + 𝜆(1 + 𝜃)𝑥)2 +  𝜆(1 + 𝜃)]  𝑒
− (𝛼(1+𝜃)𝑥 + 

𝜆(1+𝜃)

2
 𝑥2)

= 0                                 

…(51) 

It is clear that 
𝜕𝑓(𝑥;𝛼,𝜆,𝜃)𝑀𝑊𝐸𝑅

𝜕𝑥
= [ − {ℎ(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅 }

2 +

ℎ′(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅 ] 𝑆(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅  

Where ℎ(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅  is the hazard rate function was given in equation (49), and 

𝑆(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅  is the survival function was given in equation (48). Since 𝑆(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅 ≠

0 Thus dividing the equation (51)  by 𝑆(𝑥; 𝛼, 𝜆, 𝜃)𝑀𝑊𝐸𝑅  yeilds: 

𝜆2(1 + 𝜃)2𝑥2 + 2 𝜆 𝛼(1 + 𝜃)2𝑥 + 𝛼2(1 + 𝜃)2 − 𝜆(1 + 𝜃) = 0 



IHJPAS. 36(2)2023 

402 
 

Based on the law of the Constitution, we get: 

𝑥 =
−2 𝜆 𝛼(1+𝜃)2 ∓ √4𝛼2𝜆2(1+𝜃)4− 4 𝜆2(1+𝜃)2(𝛼2(1+𝜃)2−𝜆(1+𝜃))

2𝜆2 (1+𝜃)2
                                            …(52)  

The value of 𝑥 is ignor when 𝑥 < 0, suppose that 𝑥 = 𝑥0 that is a root of equation (52),  

If  
𝜕2𝑓(𝑥;𝛼,𝜆,𝜃)𝑀𝑊𝐸𝑅

𝜕𝑥2
⃒𝑥=𝑥0 < 0  the root is the local maximum. If  

𝜕2𝑓(𝑥;𝛼,𝜆,𝜃)𝑀𝑊𝐸𝑅

𝜕𝑥2
⃒𝑥=𝑥0 > 0 the 

root is the local minimum. If  
𝜕2𝑓(𝑥;𝛼,𝜆,𝜃)𝑀𝑊𝐸𝑅

𝜕𝑥2
⃒𝑥=𝑥0 = 0 the point is inflection. 

 

3.2.2 The Median 

The median of 𝑀𝑊𝐸𝑅 distribution can be obtained as follows: 

1 −  𝑒
− (𝛼(1+𝜃)𝑥 + 

𝜆(1+𝜃)
2

 𝑥2)
=

1

2
 

𝜆(1 + 𝜃) 𝑥2 + 2 𝛼(1 + 𝜃)𝑥 − 2 ln 2 = 0 

 

Based on law of the Constitution, we get: 

𝑥 =
−2𝛼(1+𝜃) ∓ √4𝛼2(1+𝜃)2+ 8 𝜆(1+𝜃) 𝑙𝑛 2

2 𝜆(1+𝜃) 
                                                                               …(53) 

The value of 𝑥 is ignor when 𝑥 < 0. 

 

 

3.2.3 The Moment about the Origin  

The 𝑟𝑡ℎ moment about the origin can be defined as: 

𝐸(𝑋𝑟 )𝑀𝑊𝐸𝑅  = ∫ 𝑥
𝑟 (𝛼(1 + 𝜃) + 𝜆(1 + 𝜃)𝑥) 𝑒

− (𝛼(1+𝜃)𝑥 + 
𝜆(1+𝜃)

2
 𝑥2)

 
∞

0
𝑑𝑥                          …(54) 

Let 

𝐷(𝑟, 𝛼, 𝜆, 𝜃) = 𝑥𝑟  𝑒
− (𝛼(1+𝜃)𝑥 + 

𝜆(1+𝜃)

2
 𝑥2)

                                                                                …(55) 

 

By Maclaurin series: 

 𝑒−𝛼(1+𝜃)𝑥 = ∑
(−𝛼(1+𝜃))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛                                                                                            …(56) 

Substituting equation (56) in equation (55) we get: 

𝐷(𝑟, 𝛼, 𝜆, 𝜃) = ∑
(−𝛼(1+𝜃))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑟+𝑛 𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

                                                                   …(57) 

Substituting equation (57) in equation (54) we get: 

𝐸(𝑋𝑟 )𝑀𝑊𝐸𝑅 = ∑
(−𝛼(1+𝜃))

𝑛

𝑛!
∞
𝑛=0 [ ∫ 𝛼(1 + 𝜃)𝑥

𝑟+𝑛  𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

 𝑑𝑥
∞

0
  + ∫ 𝜆(1 +

∞

0

𝜃)𝑥𝑟+𝑛+1  𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

 𝑑𝑥]               …(58) 

Now, solve the first integral as follows:  

𝐿1 = ∫ 𝛼(1 + 𝜃)𝑥
𝑟+𝑛  𝑒

− 
𝜆(1+𝜃)

2
 𝑥2

 𝑑𝑥 
∞

0
 =  

𝛼 2
 
𝑟+𝑛−1

2

𝜆 
𝑟+𝑛+1

2   (1+𝜃)
𝑟+𝑛−1

2

 𝛤( 
𝑟+𝑛+1

2
 )                             …(59)                    

Now, solve the second integral as follows:  

𝐿2 = ∫ 𝜆(1 +  𝜃) 𝑥
𝑟+𝑛+1  𝑒

− 
𝜆(1+𝜃)

2
 𝑥2

𝑑𝑥 
∞

0
 =  

 2
 
𝑟+𝑛

2

𝜆 
𝑟+𝑛

2   (1+ 𝜃)
𝑟+𝑛

2

 𝛤( 
𝑟+𝑛+2

2
 )                              …(60) 

 

 

 



IHJPAS. 36(2)2023 

403 
 

Substituting equations (59) and (60) in equation (58) we get: 

𝐸(𝑋𝑟 )𝑀𝑊𝐸𝑅 = ∑
(−𝛼(1+ 𝜃))𝑛

𝑛!
∞
𝑛=0

 2 
𝑟+𝑛

2

 𝜆 
𝑟+𝑛

2   (1+ 𝜃)
𝑟+𝑛

2

[ 
𝛼 √𝜃 

√2𝜆
 𝛤 ( 

𝑟+𝑛+1

2
 ) +

𝛤 ( 
𝑟+𝑛+2

2
 )]                                                                                                                                         …(61) 

The Mean: Let 𝑟 = 1 in equation (61) we get the first moment which is called the mean, thus: 

𝐸(𝑋)𝑀𝑊𝐸𝑅 = ∑
(−𝛼(1+ 𝜃))𝑛

𝑛!
∞
𝑛=0

 2 
1+𝑛

2

 𝜆 
1+𝑛

2   (1+ 𝜃)
1+𝑛

2

[ 
𝛼 √𝜃  

√2𝜆
 𝛤 ( 

𝑛+2

2
 ) +  𝛤 (

𝑛+3

2
 )]                        …(62) 

The Variance: The general form of 𝑣(𝑋) of 𝑀𝑊𝐸𝑅 distribution is defined as: 

  𝑣(𝑋)𝑀𝑊𝐸𝑅 = [∑
(−𝛼(1+ 𝜃))𝑛

𝑛!
∞
𝑛=0

 2 
2+𝑛

2

 𝜆 
2+𝑛

2   (1+ 𝜃)
2+𝑛

2

[ 
𝛼 √𝜃 

√2𝜆
 𝛤 ( 

𝑛+3

2
 ) +  𝛤 ( 

𝑛+4

2
 )]] − 

                              [∑
(−𝛼(1+ 𝜃))𝑛

𝑛!
∞
𝑛=0

 2 
1+𝑛

2

 𝜆 
1+𝑛

2   (1+ 𝜃)
1+𝑛

2

[ 
𝛼 √𝜃  

√2𝜆
 𝛤 ( 

𝑛+2

2
 ) +  𝛤 (

𝑛+3

2
 )] ]

2

                   …(63) 

3.2.4 Coefficient of Skewness  

The general form of the Coefficient of Skewness (𝐶. 𝑆) of 𝑀𝑊𝐸𝑅 distribution can be obtained 

by:  

𝐶. 𝑆𝑀𝑊𝐸𝑅 =

∑
(−𝛼(1+ 𝜃))𝑛 

𝑛!
 ∞𝑛=0

 2 
3+𝑛

2

 𝜆 
3+𝑛

2
  (1+ 𝜃)

3+𝑛
2

 [ 
𝛼 √𝜃 

√2𝜆
 𝛤( 

𝑛+4

2
 )+ 𝛤( 

𝑛+5

2
 )]

[∑
(−𝛼(1+ 𝜃))𝑛

𝑛!
∞
𝑛=0

 2 
2+𝑛

2

 𝜆 
2+𝑛

2  (1+ 𝜃)
2+𝑛

2

[ 
𝛼 √𝜃 

√2𝜆
 𝛤( 

𝑛+3

2
 )+ 𝛤( 

𝑛+4

2
 )]]

3
2

                                            …(64) 

 

 

3.2.5 Coefficient of Kurtosis   

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

𝐶. 𝐾𝑀𝑊𝐸𝑅 =

∑
(−𝛼(1+ 𝜃))𝑛

𝑛!
∞
𝑛=0

 2 
4+𝑛

2

 𝜆 
4+𝑛 

2
  (1+ 𝜃)

4+𝑛
2

 [ 
𝛼 √𝜃 

√2𝜆
 𝛤( 

𝑛+5

2
 )+ 𝛤( 

𝑛+6

2
 )]

[∑
(−𝛼(1+ 𝜃))𝑛

𝑛!
∞
𝑛=0

 2 
2+𝑛

2

 𝜆 
2+𝑛

2   (1+ 𝜃)
2+𝑛

2

[ 
𝛼 √𝜃 

√2𝜆
 𝛤( 

𝑛+3

2
 )+ 𝛤( 

𝑛+4

2
 )]]

2 − 3                                    …(65)                                                                     

3.2.6 Moment Generating Function  

 The moment generating function of 𝑀𝑊𝐸𝑅 distribution can be found as follows:  

 𝑀𝑋 (𝑡)𝑀𝑊𝐸𝑅 = ∫ 𝑒
𝑥𝑡  (𝛼(1 + 𝜃) + 𝜆(1 + 𝜃)𝑥) 𝑒

− (𝛼(1+𝜃)𝑥 + 
𝜆(1+𝜃)

2
 𝑥2)∞

0
𝑑𝑥  

𝑀𝑋 (𝑡)𝑀𝑊𝐸𝑅 = ∫  (𝛼(1 + 𝜃) + 𝜆(1 + 𝜃)𝑥) 𝑒
− ((𝛼(1+𝜃)−𝑡) 𝑥 + 

𝜆(1+𝜃)

2
 𝑥2)

 
∞

0
𝑑𝑥                        …(66) 

Let 

𝑇((𝛼(1 + 𝜃) − 𝑡), 𝜆) =  𝑒
− ((𝛼(1+𝜃)−𝑡) 𝑥 + 

𝜆(1+𝜃)

2
 𝑥2)

                                                              …(67) 

By Maclaurin series: 

 𝑒−(𝛼(1+𝜃)−𝑡)𝑥 = ∑
(−(𝛼(1+𝜃)−𝑡))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛                                                                                …(68) 

Substituting equation (68) in equation (67) we get: 

𝑇((𝛼(1 + 𝜃) − 𝑡), 𝜆) = ∑
(−(𝛼(1+𝜃)−𝑡))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛  𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

                                                   …(69) 

Substituting equation (69) in equation (66) we get: 



IHJPAS. 36(2)2023 

404 
 

𝑀𝑋 (𝑡)𝑀𝑊𝐸𝑅 = ∑
(−(𝛼(1+𝜃)−𝑡))𝑛

𝑛!
∞
𝑛=0 [∫ 𝛼(1 + 𝜃) 𝑥

𝑛   𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

 
∞

0
𝑑𝑥 + ∫ 𝜆(1 +

∞

0

𝜃) 𝑥𝑛+1  𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

 𝑑𝑥]                   …(70) 

Now, solve the first integral as follows:  

𝐿1 = ∫ 𝛼(1 + 𝜃) 𝑥
𝑛   𝑒

− 
𝜆(1+𝜃)

2
 𝑥2

 
∞

0
𝑑𝑥 =  

𝛼 2
 
𝑛−1

2

𝜆 
𝑛+1

2  (1+𝜃)
𝑛−1

2

 𝛤( 
𝑛+1

2
 )                                          …(71)                    

Now, solve the second integral as follows:  

𝐿2 = ∫ 𝜆(1 + 𝜃) 𝑥
𝑛+1  𝑒

− 
𝜆(1+𝜃)

2
 𝑥2

 
∞

0
𝑑𝑥 =  

 2
 
𝑛
2

𝜆 
𝑛
2  (1+𝜃)

𝑛
2

 𝛤( 
𝑛+2

2
 )                                              …(72) 

Substituting equations (71) and (72) in equation (70) yields: 

𝑀𝑋 (𝑡)𝑀𝑊𝐸𝑅 = ∑
(−(𝛼(1+𝜃)−𝑡))𝑛

𝑛!
∞
𝑛=0

 2 
𝑛
2

 𝜆 
𝑛
2  (1+𝜃)

𝑛
2

[ 
𝛼 √𝜃 

√2𝜆
 𝛤 ( 

𝑛+1

2
 ) +  𝛤 ( 

𝑛+2

2
 )]                         …(73) 

 

3.2.7 Factorial Moment Generating Function 

The factorial moment generating function of 𝑀𝑊𝐸𝑅 distribution can be obtained as follows: 

𝑀(𝑡)𝑀𝑊𝐸𝑅 = 𝐸(𝑡
𝑥 ) = ∫ 𝑡𝑥  (𝛼(1 + 𝜃) + 𝜆(1 + 𝜃)𝑥) 𝑒

− (𝛼(1+𝜃)𝑥 + 
𝜆(1+𝜃)

2
 𝑥2)

 
∞

0
𝑑𝑥                 

𝑀(𝑡)𝑀𝑊𝐸𝑅 = ∫  (𝛼(1 + 𝜃) + 𝜆(1 + 𝜃)𝑥) 𝑒
− ((𝛼(1+𝜃)−ln (𝑡)) 𝑥 + 

𝜆(1+𝜃)

2
 𝑥2)

 
∞

0
𝑑𝑥                     …(74) 

Let 

𝑈((𝛼(1 + 𝜃) − ln(𝑡)) , 𝜆) =  𝑒
− ((𝛼(1+𝜃)−𝑙𝑛(𝑡)) 𝑥 + 

𝜆(1+𝜃)

2
 𝑥2)

                                                  …(75) 

 

By Maclaurin series: 

 𝑒−(𝛼(1+𝜃)−ln (𝑡))𝑥 = ∑
(−(𝛼(1+𝜃)−ln (𝑡)))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛                                                                      …(76) 

Substituting equation (76) in equation (75) we get: 

𝑈((𝛼(1 + 𝜃) − ln(𝑡)), 𝜆) = ∑
(−(𝛼(1+𝜃)−ln (𝑡)))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛  𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

                                       …(77) 

Substituting equation (77) in equation (74) we get: 

𝑀(𝑡)𝑀𝑊𝐸𝑅 = ∑
(−(𝛼(1+𝜃)−ln (𝑡)))𝑛

𝑛!
∞
𝑛=0 [∫ 𝛼(1 + 𝜃) 𝑥

𝑛   𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

 
∞

0
𝑑𝑥 + ∫ 𝜆(1 +

∞

0

𝜃) 𝑥𝑛+1  𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

 𝑑𝑥]                                                                                                                        … (78)     

Now, based on equations (71) and (72) we get: 

𝑀(𝑡)𝑀𝑊𝐸𝑅 = ∑
(−(𝛼(1+𝜃)−ln(𝑡)))𝑛

𝑛!
∞
𝑛=0

 2 
𝑛
2

 𝜆 
𝑛
2  (1+𝜃)

𝑛
2

[ 
𝛼 √𝜃 

√2𝜆
 𝛤 ( 

𝑛+1

2
 ) +  𝛤 ( 

𝑛+2

2
 )]                     …(79) 

3.2.8 Characteristic Function  

The characteristic function of 𝑀𝑊𝐸𝑅 distribution can be found as follows: 

∅𝑋 (𝑖𝑡)𝑀𝑊𝐸𝑅 = 𝐸(𝑒
𝑖𝑡𝑥 ) = ∫ 𝑒𝑖𝑡𝑥  (𝛼(1 + 𝜃) + 𝜆(1 + 𝜃)𝑥) 𝑒

− (𝛼(1+𝜃)𝑥 + 
𝜆(1+𝜃)

2
 𝑥2)

 

∞

0

𝑑𝑥 

∅𝑋 (𝑖𝑡)𝑀𝑊𝐸𝑅 = ∫  (𝛼(1 + 𝜃) + 𝜆(1 + 𝜃)𝑥) 𝑒
− ((𝛼(1+𝜃)−it) 𝑥 + 

𝜆(1+𝜃)

2
 𝑥2)

 
∞

0
𝑑𝑥                     …(80) 

Let 

𝐶((𝛼(1 + 𝜃) − it), 𝜆) =  𝑒
− ((𝛼(1+𝜃)−𝑖𝑡) 𝑥 + 

𝜆(1+𝜃)

2
 𝑥2)

                                                            …(81) 

 



IHJPAS. 36(2)2023 

405 
 

By Maclaurin series: 

 𝑒−(𝛼(1+𝜃)−it)𝑥 = ∑
(−(𝛼(1+𝜃)−it))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛                                                                             …(82) 

Substituting equation (82) in equation (8 1) gives: 

𝐶((𝛼(1 + 𝜃) − it), 𝜆) = ∑
(−(𝛼(1+𝜃)−it))𝑛

𝑛!
∞
𝑛=0 𝑥

𝑛  𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

                                               …(83) 

Substituting equation (83) in equation (80) gives: 

∅𝑋 (𝑖𝑡)𝑀𝑊𝐸𝑅 = ∑
(−(𝛼(1+𝜃)−it))𝑛

𝑛!
∞
𝑛=0 [∫ 𝛼(1 + 𝜃) 𝑥

𝑛  𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

 
∞

0
𝑑𝑥 +   ∫ 𝜆(1 +

∞

0

𝜃) 𝑥𝑛+1  𝑒
− 

𝜆(1+𝜃)

2
 𝑥2

 𝑑𝑥]            …(84) 

Now, based on equations (71) and (72) we get: 

∅𝑋 (𝑖𝑡)𝑀𝑊𝐸𝑅 = ∑
(−(𝛼(1+𝜃)−it))𝑛

𝑛!
∞
𝑛=0

 2 
𝑛
2

 𝜆 
𝑛
2  (1+𝜃)

𝑛
2

[ 
𝛼 √𝜃

√2𝜆
 𝛤 ( 

𝑛+1

2
 ) +  𝛤 ( 

𝑛+2

2
 )]                        …(85) 

3.2.9 Quantile Function  

The quantile function of  𝑀𝑊𝐸𝑅 random variable is defined as a solution of 𝑝(𝑥 ≤ 𝑥(𝑞)) =

𝐹(𝑥(𝑞))𝑀𝑊𝐸𝑅  w.r.t. 𝑥(𝑞), Therefore, via using the inverse transformation to equation (47), it can 

be found as: 

𝑥(𝑞) = 𝐹
−1(𝑞)                ; 𝑥(𝑞) > 0;  0 < 𝑞 < 1                                   

𝑞 = 1 −  𝑒
− (𝛼(1+𝜃)𝑥(𝑞) + 

𝜆(1+𝜃)
2

 𝑥(𝑞)
2 )

 

1 − 𝑞 =  𝑒
− (𝛼(1+𝜃)𝑥(𝑞) + 

𝜆(1+𝜃)
2

 𝑥(𝑞)
2 )

 

𝑙𝑛 ( 1 − 𝑞) = −(𝛼(1 + 𝜃)𝑥(𝑞)  +  
𝜆(1 + 𝜃)

2
 𝑥(𝑞)

2 ) 

𝜆 (1 + 𝜃)𝑥(𝑞)
2 + 2𝛼(1 + 𝜃)𝑥(𝑞)  + 2 𝑙𝑛 ( 1 − 𝑞) = 0 

 

Based on law of the constitution, we get: 

𝑥(𝑞) =
−2𝛼(1+𝜃)∓√4𝛼2(1+𝜃)2− 8 𝜆 (1+𝜃) 𝑙𝑛 (1−𝑞)

2 𝜆 (1+𝜃)
                                                                     …(86)  

The values of 𝑥(𝑞) will be ignored when 𝑥(𝑞) < 0.  

 

4.Conclusions 

In this paper, introduce Exponential Rayleigh 𝐸𝑅 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 𝑀𝑊𝐸𝑅 distribution, such that the Exponential Rayleigh 𝐸𝑅 

distribution is special case of Modified Weighted Exponential Rayleigh 𝑀𝑊𝐸𝑅 distribution and 

provide some special models of the 𝑀𝑊𝐸𝑅 distribution. Different statistical properties such as the 

mode, the median, the 𝑟𝑡ℎ 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. 

 

 

 

 

 



IHJPAS. 36(2)2023 

406 
 

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