388 

© 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), 

University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons 

Attribution 4.0 International License 

 
 

 

 
               

New lifetime Shanker Weibull distribution: Structure and Properties 

 

Noor Ebadi Ashoor1* , Maysaa Jalil Mohammed2  and Umar Yusuf Madaki3   
1,2Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham), University of   

Baghdad, Baghdad, Iraq. 
3Department of Mathematics and Statistics, Faculty of Science, Yobe State University Damaturu, Nigeria. 

*Corresponding Author. 

 

Received: 20 November 2023             Accepted: 22 January 2024                      Published: 20 April 2025  

 doi.org/10.30526/38.2.3842 

 
 Abstract   

       This paper introduces a new model with two parameters titled “Shanker Weibull distribution". 

This distribution is obtained by merging the Shanker distribution with a scale of one parameter and 

the Weibull distribution with a shape parameter. First, we present the mathematical structure of the 

new distribution, which depends on the survival function for each of the Shanker and Weibull 

distributions. The statistical functions of this new distribution are presented, such as the cumulative, 

probability density, hazard and survival functions. In addition, we discuss the behavior of the 

probability density function and the hazard function by mxamining their shape. We also present the 

statistical properties of this distribution, which include mode, median and moments around the 

origin. As a result of studying the moments around the origin, we obtain the variance and the first 

expected value (mean), the moment generating function, the skewness, the kurtosis, the 

characteristic function, the factorial generating function, the quantile function and the mean time to 

failure. 

Keywords: Shanker distribution, exponential distribution, survival function, moments arond the 

origin, moment generating function. 

 

1. Introduction 

     Although the process of mixing the distributions to create a new distribution is not new, it has 

gone through several stages of development depending on how to choose which distributions to 

mix (1). All of this is done using the data available to the researcher about the properties and nature 

of the work of each distribution (2). Many researchers have combined two distributions. For 

example, in 2015, Rama (3-4) proposed modelling lifetime data with a new one-parameter lifetime 

distribution, the "Shanker distribution", by combining the 'gamma(2,𝛽) distribution'' with the 

''exponential distribution'' where are the (p.d.f.) and the (c.d.f.) of this distribution: 

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
mailto:https://orcid.org/0009-0009-0187-5419
mailto:noor.abadi2103m@ihcoedu.uobaghdad.edu.iq
mailto:https://orcid.org/0000-0002-9155-7159
mailto:maysaa.j.m@ihcoedu.uobaghdad.edu.iq
https://orcid.org/
mailto:uymadaki84@gmail.com


IHJPAS. 2025,38(2) 
 

389 
 

 

𝑓𝑆𝐻(𝑧, 𝛽) =
𝛽2

𝛽2+1
(𝛽 + 𝑧)𝑒−𝛽𝑧;  𝑧 > 0, 𝛽 > 0        (1) 

𝐹𝑆𝐻(𝑧) = 1 −  
(𝛽2+1)+𝛽𝑧

𝛽2+1
𝑒−𝛽𝑧        (2) 

and survival (reliability) function defined as: 

𝑆𝑆𝐻(𝑧) =  
(𝛽2+1)+𝛽𝑧

𝛽2+1
𝑒−𝛽𝑧                                                                            (3)  

Gauss (5), introduced a three-parameter model in 2014, which is a mixture of exponential and 

Weibull distribution. In addition, the three-parameter lifetime model was first introduced in 2015 

by Faton and Ibrahim (6). This is a mixture of Weibull and Rayleigh distribution to obtain a new 

distribution. The structure was based on PDF integration and the bounds of the integration ranged 

from zero to the survival function. Suleman (7) presented a new distribution in 2016, which is a 

mixture of Weibull and Rayleigh distribution. Mohammed, Maysaa and Hussein (8) presented a 

new mixture distribution of Weibull, Rayleigh and exponential distribution in 2018. In addition, 

Mundher et al.  (9) presented an extension of the Burr type X distribution. Also, Maysaa and Iden 

(10) estimated the parameters of the new mixture distribution using classical methods. In addition, 

Ali and Iden (11) estimated the density function using a hybrid Breslow and a semi-symmetric 

wavelet. Also, Hameed et al. (12) presented the estimation of (𝑌<𝑋) in the case of the inverse 

Kumaraswamy distribution. In addition, Maysaa and Ali (13) presented the inverse exponential 

Rayleigh distribution (IERD). There are many applications of continuous distributions, such as 

Wang (14), who presents a new model with a bathtub-shaped failure rate using the Burr XII 

distribution. Moreover, Rama (15) introduced a new distribution that depends on Gama (3,θ). 

Moreover, Kilany (16) introduced a weighted Lomax distribution in 2016. In addition, Oguntunde 

et al.  (17) introduced the inverted weighted exponential distribution. Okagbue, et al.  (18) also 

compared some of the estimation methods. Jebur, et al. 2021 (19) also presented the efficient 

shrinkage estimation method for the generalized inverse Rayleigh distribution. Maysaa and Ali 

(20) estimated the parameters of the new inverse exponential Rayleigh distribution. Also, Jabar et 

al. (21), introduced the truncated inverse generalized Rayleigh distribution. Moreover, Mahmood 

and Iden (22) introduced the comparison between the modified weighted Pareto distribution and 

other distributions. In addition, Pedro (23) presented a new distribution, the Fréchet distribution. 

Finally, Lamyaa and Iden (24) introduced a new distribution in 2023, which is a mixture of 

exponential and Rayleigh distribution. In this paper, we introduce a new mixture of continuous 

distributions, which we call the 'Shanker-Weibull distribution', by mixing both the 'Shanker 

distribution' and the 'Weibull distribution' based on the survival function for both distributions. 

Section 2 describes how we mixed the 'Shanker distribution' with the 'Weibull distribution' to 

obtain the 'Shanker-Weibull distribution' (SHWD) and introduces the basic statistical functions 

such as the probability density function, the commutative density function, the survival function 

and the hazard rate function. Finally, Section 3 presents the statistical and mathematical properties 

of the new distribution, such as mode, median, moments around the origin, mean, variance, 

skewness coefficient, kurtosis coefficient, characteristic function, moment generation function, 

factorial moment generation function and mean time to failure. 

 

 

 



IHJPAS. 2025,38(2) 
 

390 
 

2. Materials and Methods 

2.1. Structure of New Shanker Weibull distribution 

Equations (1and 2) represent the pdf and cdf of the Shanker distribution. In addition, the pdf, cdf, 

and the survival function of standard Weibull distribution are(25) 

fW(v) = λvλ−1e−vλ
, λ > 0, v > 0                                                                                       (4) 

FW(v) = 1 − e−vλ
   

Sw(v) = e−vλ
    (5) 

Let  Z, and V are independent random variables and 𝑋 =  𝑚𝑖𝑛 ( 𝑍, 𝑉 ) 

𝑆(𝑥) = 𝑝𝑟(𝑋 > 𝑥) = 𝑝𝑟(𝑚𝑖𝑛(𝑍 , 𝑉) > 𝑥),  

because Z and V are independent random variables then : 

𝑆(𝑆𝐻𝑊𝐷) (𝑥) =  𝑝𝑟(𝑍 > 𝑥). 𝑝𝑟(𝑉 > 𝑥) = (1 − 𝑝𝑟(𝑍 ≤ 𝑥)) ∙ (1 − 𝑝𝑟(𝑉 ≤ 𝑥))

= (1 − 𝐹𝑠ℎ (𝑧)) ∙ (1 − 𝐹𝑤 (𝑣)) = 𝑆𝑠ℎ (𝑥)  ∙  𝑆𝑤 (𝑥) 

From equations (3) and (5) we get:   

  𝑆(𝑆𝐻𝑊𝐷) (𝑥) =
(𝛽2+1)+𝛽𝑥

𝛽2+1
𝑒−(𝛽𝑥+𝑥𝜆)                                                                                              (6) 

Where x > 0, and 𝛽 > 0 is the scale parameter, 𝜆 > 0 is the shape parameter. 

The (cdf) of the new (𝑆𝐻𝑊𝐷) distribution is: 

𝐹(𝑥) = 1 − ( 
(𝛽2+1)+𝛽𝑥

𝛽2+1
𝑒−(𝛽𝑥+𝑥𝜆))   (7) 

Therefore, the pdf of the new (𝑆𝐻𝑊𝐷) is: 

𝑓(𝑥) = (
(𝛽2+1)+𝛽𝑥

𝛽2+1
(𝛽 + 𝜆𝑥𝜆−1) −

𝛽

(𝛽2+1)
) 𝑒−(𝛽𝑥+𝑥𝜆)   (8) 

f(x) is a probability density function, since,𝛽 > 0, 𝜆 > 0, and 𝑥 > 0, which implies 

that f(x) > 0. 

Now, to prove that∫ 𝑓(𝑥) 𝑑𝑥 = 1
∞

0
,  

∫ f(x)dx = ∫ (
𝛽x + 𝛽2 + 1

𝛽2 + 1
(𝛽 + λxλ−1) −

𝛽

𝛽2 + 1
) e−(𝛽x+xλ) dx

∞

0

∞

0

 

∫ f(x)dx = Lim
y→∞

(∫ (
𝛽x + 𝛽2 + 1

𝛽2 + 1
(𝛽 + λxλ−1) −

𝛽

𝛽2 + 1
) e−(𝛽x+xλ) dx

y

0

)

∞

0

= Lim
y→∞

1

𝛽2 + 1
(∫(𝛽x + 𝛽2 + 1)(𝛽 + λxλ−1)e−(𝛽x+xλ) dx

y

0

− ∫ 𝛽 𝑒−(𝛽𝑥+𝑥𝜆) dx

y

0

) 

𝑢 = (𝛽𝑥 + 𝛽2 + 1)  ⇨ 𝑑𝑢 = 𝛽𝑑𝑥 

𝑑𝑣 = (𝛽 + 𝜆𝑥𝜆−1)𝑒−(𝛽𝑥+𝑥𝜆)𝑑𝑥 ⇨ 𝑣 = − 𝑒−(𝛽𝑥+𝑥𝜆) 

∫ 𝑓(𝑥) 𝑑𝑥 = 𝐿𝑖𝑚
𝑦→∞

1

𝛽2 + 1
((−(𝛽𝑥 + 𝛽2 + 1)𝑒−(𝛽𝑥+𝑥𝜆))

𝑦
|
0

+ ∫  𝛽 𝑒−(𝛽𝑥+𝑥𝜆) 𝑑𝑥

𝑦

0

∞

0

− ∫ 𝛽 𝑒−(𝛽𝑥+𝑥𝜆)𝑑𝑥

𝑦

0

) =  − 𝐿𝑖𝑚
𝑦→∞

(
(𝛽𝑦 + 𝛽2 + 1)

𝛽2 + 1)𝑒(𝛽𝑦+𝑦𝜆)
−

𝛽2 + 1

𝛽2 + 1
) 



IHJPAS. 2025,38(2) 
 

391 
 

By using L’Hopital’s - Role for the above limit, 

∫ 𝑓(𝑥) 𝑑𝑥

∞

0

= − 𝐿𝑖𝑚
𝑦→∞

(
𝛽 

(𝛽2 + 1)(𝛽 + 𝜆𝑦𝜆−1)𝑒(𝛽𝑦+𝑦𝜆)
− 1) = −

𝛽

∞
+ 1 = 1 

Based on what was stated above, the hazard function can be defined as follows: 

ℎ(𝑥) =
𝑓(𝑥)

𝑆(𝑥)
=

((𝛽𝑥 + 𝛽2 + 1)(𝛽 + 𝜆𝑥𝜆−1) − 𝛽)
1

(𝛽2 + 1)
𝑒−(𝛽𝑥+𝑥𝜆)

(𝛽𝑥 + 𝛽2 + 1)
1

(𝛽2 + 1)
𝑒−(𝛽𝑥+𝑥𝜆)

 

ℎ(𝑥) = (𝛽 + 𝜆𝑥𝜆−1) −
𝛽

(𝛽𝑥+𝛽2+1)
  (9) 

2.1.1. The shapes of the new Shanker Weibull distribution 

Knowing the shape of the (SHWD) helps us understand the behavior and approach of distribution 

functions in dealing with data, to understand this mathematically, especially through the limit 

values of the probability density and hazard functions when(𝑥 → 0 &𝑥 → ∞). 

𝑙𝑖𝑚
𝑥→0

𝑓(𝑥) = 𝑙𝑖𝑚
𝑥→0

((𝛽𝑥 + 𝛽2 + 1)(𝛽 + 𝜆𝑥𝜆−1) − 𝛽)
1

(𝛽2 + 1)
𝑒−(𝛽𝑥+𝑥𝜆) =  

𝛽3

(𝛽2 + 1)
 

𝑙𝑖𝑚
𝑥→∞

𝑓(𝑥) = 𝑙𝑖𝑚
𝑥→∞

((𝛽𝑥 + 𝛽2 + 1)(𝛽 + 𝜆𝑥𝜆−1) − 𝛽)
1

(𝛽2 + 1)
𝑒−(𝛽𝑥+𝑥𝜆)

=  𝑙𝑖𝑚
𝑥→∞

(𝛽𝑥 + 𝛽2 + 1)(𝛽 + 𝜆𝑥𝜆−1)

(𝛽2 + 1)𝑒(𝛽𝑥+𝑥𝜆)
− 𝑙𝑖𝑚

𝑥→∞

𝛽

(𝛽2 + 1)𝑒(𝛽𝑥+𝑥𝜆)

= 𝑙𝑖𝑚
𝑥→∞

(𝛽𝑥 + 𝛽2 + 1)(𝛽 + 𝜆𝑥𝜆−1)

(𝛽2 + 1)𝑒(𝛽𝑥+𝑥𝜆)
−

𝛽

∞
= 𝑙𝑖𝑚

𝑥→∞

(𝛽𝑥 + 𝛽2 + 1)(𝛽 + 𝜆𝑥𝜆−1)

(𝛽2 + 1)𝑒(𝛽𝑥+𝑥𝜆)
 

Applied L'Hospital's Rule: 

𝑙𝑖𝑚
𝑥→∞

𝑓(𝑥) = 𝑙𝑖𝑚
𝑥→∞

(𝛽𝑥 + 𝛽2 + 1)(𝛽 + 𝜆𝑥𝜆−1)(𝜆(𝜆 − 1)𝑥𝜆−2) − 𝛽(𝛽 + 𝜆𝑥𝜆−1)

(𝛽2 + 1)(𝛽 + 𝜆𝑥𝜆−1)𝑒(𝛽𝑥+𝑥𝜆)
 

As we continue to derive for 𝑥 the result of the numerator will be equal constant as considered an 

integer value for 𝑥 and as contained to derive the denominator it always contains the exponential 

part which is equal to ∞ as(𝑥 →∞). The final result of the limit is a constant divide by ∞ which 

equal to zero 

Lim
x→∞

f(x) = 0. 

Lim
𝑥→0

ℎ(𝑥) = Lim
𝑥→0

((𝛽 + 𝜆𝑥𝜆−1) −
𝛽

𝛽𝑥 + 𝛽2 + 1
) =

𝛽3

𝛽2 + 1
 

Lim
𝑥→∞

ℎ(𝑥) = Lim
𝑥→∞

((𝛽 + 𝜆𝑥𝜆−1) −
𝛽

𝛽𝑥 + 𝛽2 + 1
) = ∞ 

 

 



IHJPAS. 2025,38(2) 
 

392 
 

  
Figure 1. The shape of pdf for new Shanker Weibull distribution with different values of β and 𝜆. 

 

  
Figure 2. The shape of cdf for new Shanker Weibull distribution with different values of β and𝜆. 

  
Figure 3. The shape of S(x) for new Shanker Weibull distribution with different values of λ and𝛽. 

 



IHJPAS. 2025,38(2) 
 

393 
 

  
Figure 4. The shape of h(x) for new Shanker Weibull distribution with different values of λ and𝛽. 

 

In the special case, if 𝜆 = 1, then the distribution becomes the Shanker exponential distribution, 

also if 𝛽=0, then the distribution becomes Weibull distribution.  

2.2. Mathematical and statistical properties of SHWD  

2.2.1. The Mode 

The mod is defined as the point at which the probability density function achieves its maximum 

value, the mode of (SHWD) is obtained as follows: 

𝑓(𝑥) = (𝛽2𝑥 + 𝛽𝜆𝑥𝜆 + 𝛽3 + 𝛽2𝜆𝑥𝜆−1 + 𝜆𝑥𝜆−1)
1

(𝛽2 + 1)
𝑒−(𝛽𝑥+𝑥𝜆) 

𝜕𝑓(𝑥)

𝜕𝑥
= ((𝛽2 + 𝛽𝜆2𝑥𝜆−1 + 𝛽2𝜆(𝜆 − 1)𝑥𝜆−2 + 𝜆(𝜆 − 1)𝑥𝜆−2)

− (𝛽2𝑥 + 𝛽𝜆𝑥𝜆 + 𝛽3 + 𝛽2𝜆𝑥𝜆−1 + 𝜆𝑥𝜆−1)(𝛽 + 𝜆𝑥𝜆−1))
1

(𝛽2 + 1)
𝑒−(𝛽𝑥+𝑥𝜆)

= 0 

Since that 
𝑒

−(𝛽𝑥+𝑥𝜆)

(𝛽2+1)
≠ 0 for each selected value of β, λ, and x, This leads to the following: 

(𝛽2 + 𝛽𝜆2𝑥𝜆−1 + 𝛽2𝜆(𝜆 − 1)𝑥𝜆−2 + 𝜆(𝜆 − 1)𝑥𝜆−2)

− (𝛽2𝑥 + 𝛽𝜆𝑥𝜆 + 𝛽3 + 𝛽2𝜆𝑥𝜆−1 + 𝜆𝑥𝜆−1)(𝛽 + 𝜆𝑥𝜆−1) = 0 

This is a polynomial of degree (𝜆 − 2) and if 𝜆 = 1that will imply the following: 

((𝛽2 + 𝛽) − (𝛽2𝑥 + 𝛽𝑥 + 𝛽3 + 𝛽2 + 1)(𝛽 + 1)) = 0 

(𝛽2 + 𝛽)−. (𝛽3𝑥 + 𝛽2𝑥 + 𝛽4 + 𝛽3 + 𝛽 + 𝛽2𝑥 + 𝛽𝑥 + 𝛽3 + 𝛽2 + 1) = 0 

−(𝛽3 + 2𝛽2 + 𝛽)𝑥 = 𝛽4 + 2𝛽3 + 1 

𝑥 =
−(𝛽4 + 2𝛽3 + 1) 

(𝛽3 + 2𝛽2 + 𝛽)
 

2.2.2. Quantile Function  

Let 𝑋 be a continuous random variable distributed with "SHWD" then the Quantile function is 

defined as given: 

Let 𝐹(𝑥) = 𝑢, 𝑢~𝑈(0,1), then     𝐹(𝑥) = 𝑈 , 𝑥 = 𝐹−1(𝑢) 

(1 −
𝛽𝑥+𝛽2+1

𝛽2+1
∙ 𝑒−(𝛽𝑥+𝑥𝜆)) = 𝑢  

− (1 −
𝛽𝑥+𝛽2+1

𝛽2+1
∙ 𝑒−(𝛽𝑥+𝑥𝜆)) = 𝑢 − 1  

1 − 𝑢 =
𝛽𝑥+𝛽2+1

𝛽2+1
∙ 𝑒−(𝛽𝑥+𝑥𝜆) , 𝜆=1 



IHJPAS. 2025,38(2) 
 

394 
 

For Lambert form (26-27) 

(1 − 𝑢)(𝛽2 + 1) = (𝛽𝑥 + 𝛽2 + 1) ∙ 𝑒−(𝛽𝑥+𝑥)  

Where, 𝑥 =
𝛽𝑏−(−𝛽−1)(𝛽2+1)

𝛽(−𝛽−1)
                                

Hence the value of  𝑏 = (−𝛽 − 1)𝑥 +
(𝛽2+1)(−𝛽−1)

𝛽
                    

(1 − 𝑢)(𝛽2 + 1) = (𝛽(
(𝛽𝑏−(−𝛽−1)(𝛽2+1)

𝛽(−𝛽−1)
) + 𝛽2 + 1) 𝑒

(−𝛽−1)((𝛽𝑏−(−𝛽−1)(𝛽2+1))

𝛽(−𝛽−1)   

(1 − 𝑢)(𝛽2 + 1) = (
𝛽𝑏

−𝛽−1
−

(−𝛽−1)(𝛽2+1)

(−𝛽−1)
+ 𝛽2 + 1)𝑒

𝛽𝑏−(−𝛽−1)(𝛽2+1)

𝛽   

(1 − 𝑢)(𝛽2 + 1)(−𝛽 − 1) =  𝛽𝑏 ∙ 𝑒
𝛽𝑏
𝛽 ∙ 𝑒

−(−𝛽−1)(𝛽2+1)
𝛽  

(1 − 𝑢)(𝛽2 + 1)(−𝛽 − 1) =  𝛽𝑏 ∙ 𝑒𝑏 ∙ 𝑒
−(−𝛽−1)(𝛽2+1)

𝛽  

(1 − 𝑢)(𝛽2 + 1)(−𝛽 − 1)𝑒
(−𝛽−1)(𝛽2+1)

𝛽 =  𝛽𝑏 ∙ 𝑒𝑏 

(1 − 𝑢)(𝛽2 + 1)(−𝛽 − 1)

𝛽
𝑒

(−𝛽−1)(𝛽2+1)

𝛽 =  𝑏 ∙ 𝑒𝑏 

𝑊 ( 𝑏𝑒𝑏 =
(1 − 𝑢)(𝛽2 + 1)(−𝛽 − 1)

𝛽
. 𝑒

(−𝛽−1)(𝛽2+1)

𝛽 ) 

𝑏 = 𝑊 (
(1 − 𝑢)(𝛽2 + 1)(−𝛽 − 1)

𝛽
. 𝑒

(−𝛽−1)(𝛽2+1)

𝛽 )  

Since  

𝑏 = (−𝛽 − 1)𝑥 +
(𝛽2 + 1)(−𝛽 − 1)

𝛽
 

(−𝛽 − 1)𝑥 +
(𝛽2+1)(−𝛽−1)

𝛽
= 𝑊 (

(1−𝑢)(𝛽2+1)(−𝛽−1)

𝛽
. 𝑒

(−𝛽−1)(𝛽2+1)

𝛽 )  

(−𝛽 − 1)𝑥 = 𝑊 (
(1−𝑢)(𝛽2+1)(−𝛽−1)

𝛽
. 𝑒

(−𝛽−1)(𝛽2+1)

𝛽 ) −
(𝛽2+1)(−𝛽−1)

𝛽
  

𝑥 =

𝑊(
(1−𝑢)(𝛽2+1)(−𝛽−1)

𝛽
∙𝑒

(−𝛽−1)(𝛽2+1)

𝛽 )−
(−𝛽−1)(𝛽2+1)

𝛽

(−𝛽−1)
.                                                                        (10) 

A special case from quantity function (10) is the median whene 𝑢 =
1

2
. 

𝑥 =

𝑊(
(1−

1
2

)(𝛽2+1)(−𝛽−1)

𝛽
∙𝑒

(−𝛽−1)(𝛽2+1)
𝛽 )−

(−𝛽−1)(𝛽2+1)

𝛽

(−𝛽−1)
  

𝑥 =

𝑊(
(

1
2

)(𝛽2+1)(−𝛽−1)

𝛽
∙𝑒

(−𝛽−1)(𝛽2+1)
𝛽 )−

(−𝛽−1)(𝛽2+1)

𝛽

(−𝛽−1)
          

2.2.3. Moments about the origin  

One of the most important distribution properties is moments, as they play a role in the 

determination of many other distribution properties, such as (mean, variance, skewness, and 

kurtosis). 

The moments of (SHWD) can be determined as follows: 



IHJPAS. 2025,38(2) 
 

395 
 

𝑀𝑟
′ (𝑥) = ∫ 𝑥𝑟

∞

0

𝑓(𝑥)𝑑𝑥

= ∫ 𝑥𝑟
∞

0

(𝛽𝑒−(𝛽𝑥+𝑥𝜆) + 𝜆𝑥𝜆−1𝑒−(𝛽𝑥+𝑥𝜆) +
𝛽2𝑥 

𝛽2 + 1
𝑒−(𝛽𝑥+𝑥𝜆)

+
𝛽𝜆 𝑥𝜆

𝛽2 + 1
𝑒−(𝛽𝑥+𝑥𝜆) −

𝛽 

𝛽2 + 1
𝑒−(𝛽𝑥+𝑥𝜆)) 𝑑𝑥 

Let  𝑒−𝑥𝜆
= ∑

(−1)𝑗

𝑗!

∞
𝑗=0 𝑥𝜆𝑗  (28) 

 

∫ 𝑥𝑟

∞

0

 𝑓(𝑥) 𝑑𝑥 = ∑
(−1)𝑗

𝑗!

∞

𝑗=0

(∫ 𝛽𝑥𝑟+𝜆𝑗

∞

0

 𝑒−𝛽𝑥𝑑𝑥 + ∫ 𝜆𝑥𝑟+𝜆(𝑗+1)−1 𝑒−𝛽𝑥𝑑𝑥

∞

0

+ ∫
𝛽2

𝛽2 + 1
𝑥𝑟+𝜆𝑗+1

∞

0

 𝑒−𝛽𝑥𝑑𝑥 + ∫
𝛽𝜆 

𝛽2 + 1
𝑥𝑟+𝜆(𝑗+1)

∞

0

 𝑒−𝛽𝑥𝑑𝑥

− ∫
𝛽 

𝛽2 + 1
𝑥𝑟+𝜆𝑗  𝑒−𝛽𝑥𝑑𝑥

∞

0

) 

Let 𝛽𝑥 = 𝑦, then 𝑥 =
𝑦

𝛽
 and 𝑑𝑥 =

𝑑𝑦

𝛽
, and since Γ(𝛽) = ∫ 𝑥𝛽−1∞

0
𝑒−𝑥𝑑𝑥. 

𝑀𝑟
′ (𝑥) = ∑

(−1)𝑗

𝑗!

∞

𝑗=0

(
1

𝛽𝑟+𝜆𝑗
∫ 𝑦𝑟+𝜆𝑗

∞

0

 𝑒−𝑦𝑑𝑦 +
𝜆

𝛽𝑟+𝜆(𝑗+1)
∫ 𝑦𝑟+𝜆(𝑗+1)−1 𝑒−𝑦𝑑𝑦

∞

0

+
𝛽

(𝛽2 + 1)𝛽𝑟+𝜆𝑗+1
∫ 𝑦𝑟+𝜆𝑗+1

∞

0

 𝑒−𝑦𝑑𝑦 +
𝜆

(𝛽2 + 1)𝛽𝑟+𝜆(𝑗+1)
∫ 𝑦𝑟+𝜆(𝑗+1)

∞

0

 𝑒−𝑦𝑑𝑦

−
1 

(𝛽2 + 1)𝛽𝑟+𝜆𝑗
∫ 𝑦𝑟+𝜆𝑗 𝑒−𝑦𝑑𝑦

∞

0

) 

𝑀𝑟
′ (𝑥) = ∑

(−1)𝑗

𝑗!

∞
𝑗=0 (

𝛤(𝑟+𝜆𝑗+1)

𝛽𝑟+𝜆𝑗 +
𝜆∙Γ(𝑟+𝜆(𝑗+1))

𝛽𝑟+𝜆(𝑗+1) +
Γ(𝑟+𝜆𝑗+1)−Γ(𝑟+𝜆𝑗+2)

(𝛽2+1)𝛽𝑟+𝜆𝑗 +
𝜆 ∙Γ(𝑟+𝜆(𝑗+1)+1)

(𝛽2+1)𝛽𝑟+𝜆(𝑗+1))                   (11) 

As a direct result, it can be found𝐸(𝑥) = 𝑀1
′ (𝑥),  𝐸(𝑥2) = 𝑀2

′ (𝑥), and 𝑣𝑎𝑟(𝑥) as follows: 

𝐸(𝑥) = ∑
(−1)𝑗

𝑗!

∞

𝑗=0

(
Γ(𝜆𝑗+2)

𝛽𝜆𝑗+1
+

𝜆 ∙ Γ(𝜆(𝑗+1)+1)

𝛽𝜆(𝑗+1)+1
+

Γ(𝜆𝑗+2) − Γ(𝜆𝑗+3)

(𝛽2 + 1)𝛽𝜆𝑗+1
+

𝜆 ∙ Γ(𝜆(𝑗+1)+2)

(𝛽2 + 1)𝛽𝜆(𝑗+1)+1
) 

𝐸(𝑥2) = ∑
(−1)𝑗

𝑗!

∞

𝑗=0

(
𝛤(𝜆𝑗+3)

𝛽𝜆𝑗+2
+

𝜆 ∙ 𝛤(𝜆(𝑗+1)+2)

𝛽𝜆(𝑗+1)+2
+

𝛤(𝜆𝑗+3) − 𝛤(𝜆𝑗+4)

(𝛽2 + 1)𝛽𝜆𝑗+2
+

𝜆 ∙ 𝛤(𝜆(𝑗+1)+3)

(𝛽2 + 1)𝛽𝑟+𝜆(𝑗+1)+2
) 

𝑣𝑎𝑟(𝑥) = 𝑀2
′ (𝑥) − (𝑀1

′ (𝑥))
2
 

 

 

 

 

 

 



IHJPAS. 2025,38(2) 
 

396 
 

2.2.4. Coefficients of Skewness and Kurtosis 

Depending on the moment, the coefficient skewness (𝐶. 𝑆) and kurtosis(𝐶. 𝐾)  can be found 

through the following formulas(29): 

𝑀3
′ (𝑥) = ∑

(−1)𝑗

𝑗!

∞

𝑗=0

(
𝛤(𝜆𝑗+4)

𝛽𝜆𝑗+3
+

𝜆 ∙ 𝛤(𝜆(𝑗+1)+3)

𝛽𝜆(𝑗+1)+3
+

𝛤(𝜆𝑗+4) − 𝛤(𝜆𝑗+5)

(𝛽2 + 1)𝛽𝜆𝑗+3
+

𝜆 ∙ 𝛤(𝜆(𝑗+1)+4)

(𝛽2 + 1)𝛽𝑟+𝜆(𝑗+1)+3
) 

𝐶. 𝑆 =
𝑀3

′

(𝑀2
′ )

3
2

 

𝑀4
′ (𝑥) = ∑

(−1)𝑖

𝑖!

∞

𝑖=0

(
𝛤(𝜆𝑗+5)

𝛽𝜆𝑗+4
+

𝜆 ∙ 𝛤(𝜆(𝑗+1)+4)

𝛽𝜆(𝑗+1)+4
+

𝛤(𝜆𝑗+5) − 𝛤(𝜆𝑗+6)

(𝛽2 + 1)𝛽𝜆𝑗+4
+

𝜆 ∙ 𝛤(𝜆(𝑗+1)+5)

(𝛽2 + 1)𝛽𝜆(𝑗+1)+4
) 

𝐶. 𝐾 =
𝑀4

′

(𝑀2
′ )2

− 3 

 

Table 1. The first - fourth moments, variance, skewness, and kurtosis for the distribution 

𝛃 𝛌 𝛍𝟏
′  𝛍𝟐

′  𝛍𝟑
′  𝛍𝟒

′  𝐊 𝐒 𝐯𝐚𝐫 

2 
0.5 0.3222 0.2719 0.3665 0.6762 6.1462 2.5849 0.1681 

0.3 0.2879 0.2668 0.3952 0.7962 8.1838 2.8673 0.1839 

0.5 
0.9 0.8598 1.5289 4.0855 14.5190 3.2110 2.1610 0.7897 

1.2 0.8255 1.1479 2.1874 5.2296 0.9687 1.7785 0.4664 

2.5 
2 0.3579 0.2177 0.1765 0.1739 0.6687 1.7376 0.0896 

3 0.3754 0.2303 0.1792 0.1620 0.0543 1.6218 0.0893 

 

Since the values of skewness are positive, the distribution is skewed to the right according to the 

given data values. The flatness depends on the values of the given characteristics. Sometimes it is 

flattened for values above 3, and sometimes it is peaked for values below 3. 

2.2.5. Characteristic Function 

𝜙𝑥(𝑖𝑡) = 𝐸(𝑒𝑖𝑡𝑥) = ∫ 𝑒𝑖𝑡𝑥

∞

0

 𝑓(𝑥) 𝑑𝑥

= ∫ 𝑒𝑖𝑡𝑥

∞

0

(1 +
𝛽𝑥 

𝛽2 + 1
) (𝛽 + 𝜆𝑥𝜆−1) −

𝛽 

𝛽2 + 1
)  𝑒−(𝛽𝑥+𝑥𝜆) 𝑑𝑥 

𝐿𝑒𝑡 𝑒−𝑥𝛽
= ∑

(−1)𝑛

𝑛!

∞

𝑛=0

𝑥𝜆𝑛 

𝜙𝑥(𝑖𝑡) = ∑
(−1)𝑛

𝑛!

∞

𝑛=0

(∫ 𝛽𝑥𝛽𝑛

∞

0

 𝑒−(𝛽−𝑖𝑡)𝑥𝑑𝑥 + ∫ 𝜆𝑥𝜆𝑛+𝜆−1

∞

0

 𝑒−(𝛽−𝑖𝑡)𝑥𝑑𝑥

+ ∫ (
𝛽2

𝛽2 + 1
) 𝑥𝜆𝑛+1 𝑒−(𝛽−𝑖𝑡)𝑥𝑑𝑥

∞

0

+ ∫ (
𝛽𝜆 

𝛽2 + 1
) 𝑥𝜆𝑛+𝜆

∞

0

 𝑒−(𝛽−𝑖𝑡)𝑥𝑑𝑥

− ∫ (
𝛽 

𝛽2 + 1
) 𝑥𝜆𝑛

∞

0

 𝑒−(𝛽−𝑖𝑡)𝑥𝑑𝑥) 



IHJPAS. 2025,38(2) 
 

397 
 

𝐿𝑒𝑡 (𝛽 − 𝑖𝑡)𝑥 = 𝑦, 𝑎𝑛𝑑 𝑥 =
𝑦

𝛽 − 𝑖𝑡
𝑡ℎ𝑒𝑛 𝑑𝑥 =

𝑑𝑦

𝛽 − 𝑖𝑡
 

𝜙𝑥(𝑖𝑡) = ∑
(−1)𝑛

𝑛!

∞

𝑛=0

(
𝛽

(𝛽 − 𝑖𝑡)𝜆𝑛+1
∫ 𝑦𝜆𝑛

∞

0

 𝑒−𝑦𝑑𝑦 +
𝜆

(𝛽 − 𝑖𝑡)𝜆𝑛+𝜆
∫ 𝑦𝜆𝑛+𝜆−1

∞

0

 𝑒−𝑦𝑑𝑦

+
𝛽2

(𝛽2 + 1)(𝛽 − 𝑖𝑡)𝜆𝑛+2
∫ 𝑦𝜆𝑛+1

∞

0

 𝑒−𝑦𝑑𝑦

+
𝛽𝜆 

(𝛽2 + 1)(𝛽 − 𝑖𝑡)𝜆𝑛+𝜆+1
∫ 𝑦𝜆𝑛+𝜆

∞

0

 𝑒−𝑦𝑑𝑦

−
𝛽2

(𝛽2 + 1)(𝛽 − 𝑖𝑡)𝜆𝑛+1
∫ 𝑦𝜆𝑛

∞

0

 𝑒−𝑦𝑑𝑦) 

𝜙𝑥(𝑖𝑡) = ∑
(−1)𝑛

𝑛!

∞

𝑛=0

(
𝛽 ∙ 𝛤(𝜆𝑛+1)

(𝛽 − 𝑖𝑡)𝜆𝑛+1
+

𝜆 ∙ 𝛤(𝜆𝑛+𝜆)

(𝛽 − 𝑖𝑡)𝜆𝑛+𝜆
+

𝛽2 ∙ 𝛤(𝜆𝑛+2)

(𝛽2 + 1)(𝛽 − 𝑖𝑡)𝜆𝑛+2

+
𝛽𝜆 ∙ 𝛤(𝜆𝑛+𝜆+1)

(𝛽2 + 1)(𝛽 − 𝑖𝑡)𝜆𝑛+𝜆+1
−

𝛽2 ∙ 𝛤(𝜆𝑛+1)

(𝛽2 + 1)(𝛽 − 𝑖𝑡)𝜆𝑛+1
) 

2.2.6. Moment Generating Function 

𝑀𝑥(𝑡) = 𝐸(𝑒𝑡𝑥) = ∫ 𝑒𝑡𝑥

∞

0

 𝑓(𝑥) 𝑑𝑥

= ∫ 𝑒𝑡𝑥

∞

0

(1 +
𝛽𝑥 

𝛽2 + 1
) (𝛽 + 𝜆𝑥𝜆−1) −

𝜆 

𝜆2 + 1
)  𝑒−(𝛽𝑥+𝑥𝜆) 𝑑𝑥 

𝐿𝑒𝑡  𝑒−𝑥𝛽
= ∑

(−1)𝑘

𝑘!

∞

𝑘=0

𝑥𝜆𝑘  

𝑀𝑥(𝑡) = ∑
(−1)𝑘

𝑘!

∞

𝑘=0

(∫ 𝛽𝑥𝜆𝑘

∞

0

 𝑒−(𝛽−𝑡)𝑥𝑑𝑥 + ∫ 𝜆𝑥𝜆𝑘+𝜆−1

∞

0

 𝑒−(𝛽−𝑡)𝑥𝑑𝑥

+ ∫ (
𝛽2

𝛽2 + 1
) 𝑥𝜆𝑘+1 𝑒−(𝛽−𝑡)𝑥𝑑𝑥

∞

0

+ ∫ (
𝛽𝜆 

𝛽2 + 1
) 𝑥𝜆𝑘+𝜆

∞

0

 𝑒−(𝛽−𝑡)𝑥𝑑𝑥

− ∫ (
𝛽 

𝛽2 + 1
) 𝑥𝜆𝑘

∞

0

 𝑒−(𝛽−𝑡)𝑥𝑑𝑥) 

𝐿𝑒𝑡 (𝛽 − 𝑡)𝑥 = 𝑦, 𝑎𝑛𝑑 𝑥 =
𝑦

𝛽 − 𝑡
𝑡ℎ𝑒𝑛 𝑑𝑥 =

𝑑𝑦

𝛽 − 𝑡
 



IHJPAS. 2025,38(2) 
 

398 
 

𝑀𝑥(𝑡) = ∑
(−1)𝑘

𝑘!

∞

𝑘=0

(
𝛽

(𝛽 − 𝑡)𝜆𝑘+1
∫ 𝑦𝜆𝑘

∞

0

 𝑒−𝑦𝑑𝑦 +
𝛽

(𝛽 − 𝑡)𝜆𝑘+𝜆
∫ 𝑦𝜆𝑘+𝜆−1

∞

0

 𝑒−𝑦𝑑𝑦

+
𝛽2

(𝛽2 + 1)(𝛽 − 𝑡)𝜆𝑘+2
∫ 𝑦𝜆𝑘+1

∞

0

 𝑒−𝑦𝑑𝑦

+
𝛽𝜆 

(𝛽2 + 1)(𝛽 − 𝑡)𝜆𝑘+𝜆+1
∫ 𝑦𝜆𝑘+𝜆

∞

0

 𝑒−𝑦𝑑𝑦

−
𝛽2

(𝛽2 + 1)(𝛽 − 𝑡)𝜆𝑘+1
∫ 𝑦𝜆𝑘

∞

0

 𝑒−𝑦𝑑𝑦) 

𝑀𝑥(𝑡) = ∑
(−1)𝑘

𝑘!

∞

𝑘=0

(
𝛽 ∙ 𝛤(𝜆𝑘+1)

(𝛽 − 𝑡)𝜆𝑘+1
+

𝜆 ∙ 𝛤(𝜆𝑘+𝛽)

(𝛽 − 𝑡)𝜆𝑘+𝛽
+

𝛽2 ∙ 𝛤(𝜆𝑘+2)

(𝛽2 + 1)(𝛽 − 𝑡)𝜆𝑘+2

+
𝛽𝜆 ∙ 𝛤(𝜆𝑘+𝜆+1)

(𝛽2 + 1)(𝛽 − 𝑡)𝜆𝑘+𝜆+1
−

𝛽2 ∙ 𝛤(𝜆𝑘+1)

(𝛽2 + 1)(𝛽 − 𝑡)𝜆𝑘+1
) 

2.2.7. Factorial Moments Generating Function 

ℳ𝑥(𝑡) = 𝐸(𝑡𝑥) = ∫ 𝑡𝑥

∞

0

 𝑓(𝑥) 𝑑𝑥

= ∫ 𝑒𝑥𝑙𝑛𝑡

∞

0

(1 +
𝛽𝑥 

𝛽2 + 1
) (𝛽 + 𝜆𝑥𝜆−1) −

𝛽 

𝛽2 + 1
)  𝑒−(𝛽𝑥+𝑥𝜆) 𝑑𝑥

= ∫ (1 +
𝛽𝑥 

𝛽2 + 1
) (𝛽 + 𝜆𝑥𝜆−1) −

𝛽 

𝛽2 + 1
)  𝑒−((𝛽−𝑙𝑛 𝑡)𝑥+𝑥𝜆) 𝑑𝑥

∞

0

 

𝐿𝑒𝑡  𝑒−𝑥𝛽
= ∑

(−1)𝑠

𝑠!

∞

𝑠=0

𝑥𝜆𝑠 

ℳ𝑥(𝑡) = ∑
(−1)𝑠

𝑠!

∞

𝑠=0

(∫ 𝛽𝑥𝜆𝑠

∞

0

 𝑒−(𝛽−𝑙𝑛𝑡)𝑥𝑑𝑥 + ∫ 𝜆𝑥𝜆𝑠+𝜆−1

∞

0

 𝑒−(𝛽−𝑙𝑛𝑡)𝑥𝑑𝑥

+ ∫ (
𝛽2

𝛽2 + 1
) 𝑥𝜆𝑠+1 𝑒−(𝛽−𝑙𝑛𝑡)𝑥𝑑𝑥

∞

0

+ ∫ (
𝛽𝜆

𝛽2 + 1
) 𝑥𝜆𝑠+𝜆

∞

0

 𝑒−(𝛽−𝑙𝑛𝑡)𝑥𝑑𝑥

− ∫ (
𝛽 

𝛽2 + 1
) 𝑥𝜆𝑠

∞

0

 𝑒−(𝛽−𝑙𝑛𝑡)𝑥𝑑𝑥) 

𝐿𝑒𝑡 (𝛽 − 𝑙𝑛𝑡)𝑥 = 𝑦, 𝑎𝑛𝑑 𝑥 =
𝑦

𝛽 − 𝑙𝑛𝑡
𝑡ℎ𝑒𝑛 𝑑𝑥 =

𝑑𝑦

𝛽 − 𝑙𝑛𝑡
 



IHJPAS. 2025,38(2) 
 

399 
 

ℳ𝑥(𝑡) = ∑
(−1)𝑠

𝑠!

∞

𝑠=0

(
𝛽

(𝛽 − 𝑙𝑛𝑡)𝜆𝑠+1
∫ 𝑦𝜆𝑠

∞

0

 𝑒−𝑦𝑑𝑦 +
𝜆

(𝛽 − 𝑙𝑛𝑡)𝜆𝑠+𝜆
∫ 𝑦𝜆𝑠+𝜆−1

∞

0

 𝑒−𝑦𝑑𝑦

+
𝛽2

(𝛽2 + 1)(𝛽 − 𝑙𝑛𝑡)𝜆𝑠+2
∫ 𝑦𝜆𝑠+1

∞

0

 𝑒−𝑦𝑑𝑦

+
𝛽𝜆 

(𝛽2 + 1)(𝛽 − 𝑙𝑛𝑡)𝜆𝑠+𝜆+1
∫ 𝑦𝜆𝑠+𝜆

∞

0

 𝑒−𝑦𝑑𝑦

−
𝛽

(𝛽2 + 1)(𝛽 − 𝑙𝑛𝑡)𝜆𝑠+1
∫ 𝑦𝜆𝑠

∞

0

 𝑒−𝑦𝑑𝑦) 

ℳ𝑥(𝑡) = ∑
(−1)𝑠

𝑠!

∞

𝑠=0

(
𝛽 ∙ 𝛤(𝜆𝑠+1)

(𝛽 − 𝑙𝑛𝑡)𝜆𝑠+1
+

𝜆 ∙ 𝛤(𝜆𝑠+𝜆)

(𝛽 − 𝑙𝑛𝑡)𝜆𝑠+𝜆
+

𝛽2 ∙ 𝛤(𝜆𝑠+2)

(𝛽2 + 1)(𝛽 − 𝑙𝑛𝑡)𝜆𝑠+2

+
𝛽𝜆 ∙ 𝛤(𝜆𝑠+𝜆+1)

(𝛽2 + 1)(𝛽 − 𝑙𝑛𝑡)𝜆𝑠+𝜆+1
−

𝛽 ∙ 𝛤(𝜆𝑠+1)

(𝛽2 + 1)(𝛽 − 𝑙𝑛𝑡)𝜆𝑠+1
) 

2.2.8. Mean time to failure 

Let 𝑇 denote the lifetime of a component, the𝑆(𝑆𝐻𝑊𝐷)(𝑥) = Pr (𝑇 > 𝑥) [30] 

𝑆(𝑆𝐻𝑊𝐷)(𝑥) = 𝑃𝑟 (𝑇 > 𝑥) 

Since 𝑆(𝑆𝐻𝑊𝐷)(𝑥) → 0 at 𝑡 → ∞, then is given by: 

𝑀𝑇𝑇𝐸 = ∫ 𝑆(𝑆𝐻𝑊𝐷)(𝑥)
∞

0

𝑑𝑥 = ∫ (1 +
𝛽𝑥 

𝛽2 + 1
)𝑒−(𝛽𝑥+𝑥𝜆)

∞

0

𝑑𝑥 

Recall that, 𝑒− 𝑥𝛽
= ∑

(−1)𝑗

𝑗!

∞
𝑗=0  𝑥𝜆𝑗 

𝑀𝑇𝑇𝐸 = ∑
(−1)𝑗

𝑗!

∞

𝑗=0

(∫  𝑥𝜆𝑗

∞

0

∙  𝑒−𝛽𝑥𝑑𝑥 +
𝛽

𝛽2 + 1
∫  𝑥𝜆𝑗+1

∞

0

∙  𝑒−𝛽𝑥𝑑𝑥) 

Let  𝛽𝑥 = 𝑦 then 𝑥 =
𝑦

𝛽
 and 𝑑𝑥 =

𝑑𝑦

𝛽
 

𝑀𝑇𝑇𝐸 = ∑
(−1)𝑗

𝑗!

∞

𝑗=0

(∫ (
𝑦

𝛽
)

𝜆𝑗
∞

0

∙  𝑒−𝑦
𝑑𝑦

𝛽
+

𝛽

𝛽2 + 1
∫ (

𝑦

𝛽
)

𝜆𝑗+1
∞

0

∙  𝑒−𝑦
𝑑𝑦

𝛽
)

= ∑
(−1)𝑗

𝑗!

∞

𝑗=0

(
1

𝛽𝜆𝑗+1
∫  𝑦𝜆𝑗

∞

0

∙  𝑒−𝑦𝑑𝑦 +
1

(𝛽2 + 1)𝛽𝜆𝑗+1
∫  𝑦𝜆𝑗+1

∞

0

∙  𝑒−𝑦𝑑𝑦) 

𝑀𝑇𝑇𝐸 = ∑
(−1)𝑗

𝑗!

∞

𝑗=0

(
1

𝛽𝜆𝑗+1
∙ 𝛤(𝜆𝑗+1) +

1

(𝛽2 + 1)𝛽𝜆𝑗+1
∙ 𝛤(𝜆𝑗+2)) 

 

3. Conclusion 

       A new statistical lifetime distribution was introduced, which is a mixture of Shanker and 

Weibull distribution depending on their survival functions and is called "Shanker-Weibull 

distribution". In addition, the shape of pdf, cdf and the hazard function were discussed. In addition, 

the basic statistical and mathematical properties of the new distribution such as mode, median, the 

rth moments around the origin and the moment generating function were described.   



IHJPAS. 2025,38(2) 
 

400 
 

 

Acknowledgments 

     Our researcher extends his Sincere thanks to the editor and members of the preparatory 

committee of the Ibn AL-Haitham Journal of Pure and Applied Sciences. 

 

Conflict of Interest  

     There are no conflicts of interest. 

 

Funding  

     There is no funding for the article. 

 

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