International Journal of Analysis and Applications

Volume 18, Number 3 (2020), 396-408

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-18-2020-396

ON THE MIXTURE OF WEIGHTED EXPONENTIAL AND WEIGHTED GAMMA

DISTRIBUTION

TASWAR IQBAL∗, MUHAMMAD ZAFAR IQBAL

Department of Mathematics and Statistics, University of Agriculture Faisalabad, Faisalabad, Pakistan

∗Corresponding author: therisers1@gmail.com

Abstract. In practice, finite mixture models were often used to fit various type of observed phenomena,

specifically those which are random in nature. In this paper, a finite mixture model based on weighted

versions of exponential and gamma distribution is considered and studied. Some mathematical properties

of the resulting model are discussed including moment generating function, skewness, kurtosis, survival

function, hazard rate function, stochastic ordering, order statistics, Bonferroni and Lorenz curves, Renyi

entropy measure and estimation of the model parameters. Two real-life data applications from different

fields exhibit the fact that in certain situations, the proposed mixture model might be a better alternative

than the existing popular models.

1. Introduction

The mixture distributions over time have provided a mathematical based way to model a wide range

of random phenomena. The mixture models are effective and flexible to analyze and interpret random

situations in a possibly heterogeneous populations. In many situations, observed data may be assumed to

have come from a mixture population of two or more distributions. The mixture models are used in medicine,

psychology, finance, engineering, fisheries research, economics, life testing and reliability among others.

In this paper, we perpend a finite mixture of two continuous distributions, a one-parameter exponential

distribution and a two-parameter gamma distribution. In mixture model, the distribution of interest is

Received January 15th, 2020; accepted February 10th, 2020; published May 1st, 2020.

2010 Mathematics Subject Classification. 60E05, 62E15.

Key words and phrases. finite mixture models; weighted exponential distribution; weighted gamma distribution.

©2020 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

396

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-18-2020-396


Int. J. Anal. Appl. 18 (3) (2020) 397

modeled as a mixture of two or more distributions in varying proportions. Thus a mixture model can be

used to model complex situations through proper choice of its mixing components. It can handle situations

where a single parametric family is unable to provide a satisfactory model for local variation in the observed

data. The flexibility and high degree of accuracy of finite mixture models have been the main reason for their

successful applications in a wide range of fields. The concept of finite mixture distribution was pioneered

by Newcomb [1] as a model for outliers. However, the credit for the introduction of statistical modeling

using finite mixture of distributions goes to Pearson [2] while applying the technique in an analysis of crab

morphometry data provided by Weldon [3] and [4] For a comprehensive survey, readers are referred to

Titterington et al. [5] Lindsay [6] Bohning [18] and McLachlan and Peel [19] and the references therein. The

main objective here is to induce greater flexibility in modeling various types of data, specially in situations

where these individual distributions fail to adequately fit the data separately. In the recent years a number

of new life time distributions have been developed and proposed by many researchers which are in general

modification or an extension or generalization of Lindley distribution. The range of such distributions

include Modified One Parameter Lindley distribution [7], Improved Second-Degree Lindley distribution [8],

Amarendra distribution [9], Sujatha distribution [10], Shankar distribution [11] and Akash distribution [12].

Each model has its own limitations and advantages for modeling the lifetime data set. As an extension

of these models, in the context to find a new distribution which is more flexible we have introduced a new

continuous one-parameter distribution. The new proposed model is a two-component mixture of length biased

exponential distribution with parameter θ and length biased gamma distribution with shape parameter 2

and scale parameter θ with their mixing proportions θ
θ+1

and 1
θ+1

respectively.

Suppose we have the following model

f (x) = pf1 (x) + (1 −p) f2 (x) ,

where fj (x) , j = 1, 2 are densities. Clearly, p =
θ
θ+1

and 1−p = 1
θ+1

, are mixture weights (0 < p < 1) , and

f (x) indeed a valid density. Next, we have

f1 (x) = θ
2xe−θx

f2 (x) =
θ3

2
x2e−θx

Our weighted distribution will have density

f(x) =
θ3

θ + 1
[1 +

x

2
]xe−θx x > 0,θ > 0 (1.1)

The density in (1.1) is called the mixture of weighted exponential and weighted gamma (hereafter MWEG

in short) distribution. The cumulative distribution function of MWEG distribution can be obtained as

F(x) = 1 −
[θ2x2 + (θ + 1)2θx + 2(θ + 1)]e−θx

2(θ + 1)
x > 0,θ > 0 (1.2)



Int. J. Anal. Appl. 18 (3) (2020) 398

(a) (b)

Figure 1. Probability density function of MWEG distribution at different values of θ

(a) (b)

Figure 2. Cumulative distribution function of MWEG distribution at different values of θ

2. Properties of the MWEG distribution

The mathematical properties of the proposed MWEG distribution are discussed as follows

2.1. Hazard Function. The hazard function is defined as the ratio of probability density function to the

survival function, it is also known as the hazard rate or force of mortality. The hazard function associated

with the MWEG distribution is

hf (x) =
f(x)

1 −F(x)

hf (x) =
2θ3x

(
1 + x

2

)
(θ2x2 + (2θ2 + 2θ) x + 2θ + 2)

hf (x) =
θ3x(2 + x)

θ2x2 + (θ + 1)2θx + 2(θ + 1)
(2.1)



Int. J. Anal. Appl. 18 (3) (2020) 399

(a) (b)

Figure 3. Hazard function of MWEG distribution at different values of θ

2.2. Survival Function. Let X be the non-negative random variable with pdf, f(x) given by (1.1). The

reliability function S(x) corresponding to the finite mixture of 2 components of (1.1) is given by

S(x) = pS1(x) + (1 −p)S2(x)

S(x) =
[θ2x2 + (θ + 1)2θx + 2(θ + 1)]e−θx

2(θ + 1)

Where Sj(x) , is the reliability function corresponding to the j-th component in the mixture, j = 1, 2. One

can write the hazard rate function of a mixture in terms of the hazard rate functions of the two components

as follows.

hf (x) = B(x)h1(x) + [1 −B(x)]h2(x), Where B(x) = pS1(x)/ [pS1(x) + (1 −p)S2(x)]

and hj(x) are hazard rate function for the j-th component, j=1,2. On differentiating the hazard function, we

get

h′f (x) = B(x)h
′
1(x) + (1 −B(x))h

′
2(x) −B(x)(1 −B(x)) (h1(x) −h2(x))

2

Where prime denote the derivatives with respect to x. Now, from the above, it follows that if h′j(x) < 0, for

all x, (j = 1, 2) then h′(x) < 0, for all x. Therefore, a mixture with decreasing hazard rate components has

decreasing hazard rate. However, if the components have increasing hazard rates, their mixture need not

have increasing hazard rate.

2.3. Moments and Related Measures. Moments plays a very important role to understand some

important features of a distribution. In statistics, moments can be used to study central tendency,

dispersion, skewness and kurtosis of any distribution.

The moment generating function of MWEG distribution is

Mx(t) =
θ3

θ + 1

∫ ∞
0

[
1 +

x

2

]
xe−(θ−t)xdx

Mx(t) =
θ3(θ + t + 1)

(θ + 1) (θ3 + 3tθ2 + 3t2θ + t3)



Int. J. Anal. Appl. 18 (3) (2020) 400

Mx(t) =
θ3(θ + t + 1)

(θ + 1)(θ + t)3
(2.2)

By using the equation (2.2) the first four moments about the origin of MWEG distribution are given by

µ′1 =
2θ + 3

θ(θ + 1)

µ′2 =
6(θ + 2)

θ2(θ + 1)

µ′3 =
12(2θ + 5)

θ3(θ + 1)

µ′4 =
120(θ + 3)

θ4(θ + 1)

By using the relationship between moments about origin and moments about mean, the moments about

mean of MWEG distribution are given by

µ2 =
2θ2 + 6θ + 3

θ2(θ + 1)2

µ3 =
2
(
2θ3 + 9θ2 + 9θ + 3

)
θ3 (θ3 + 3θ2 + 3θ + 1)

µ4 =
24θ4 + 144θ3 + 252θ2 + 180θ + 45

)
θ4 (θ4 + 4θ3 + 6θ2 + 4θ + 1)

Skewness and Kurtosis plays a vital role in explaining the shape and tail property of a distribution. The

expressions of Skewness and Kurtosis are given by

√
β1 =

µ3

µ
3/2
2

=
2
(
2θ3 + 9θ2 + 9θ + 3

)
(2θ2 + 6θ + 3)

3/2

β2 =
µ4
µ22

=
3
(
8θ4 + 48θ3 + 84θ2 + 60θ + 15

)
θ2(θ + 1)2 (2θ2 + 6θ + 3)

The index of dispersion and coefficient of variation are thus obtained as

C.V =
σ

µ′1
=

√
2θ2 + 6θ + 3

(2θ + 3)

γ =
σ2

µ′1
=

2θ2 + 6θ + 3

θ(θ + 1)(2θ + 3)

The condition under which MWEG distribution is over-dispersed, under-dispersed, and equi-dispersed

has been given along Amarendra, Ishita, Akash, Lindley and exponential distributions in table 1



Int. J. Anal. Appl. 18 (3) (2020) 401

Table 1. Over-dispersion, under-dispersion and equi-dispersion MWEG, Ishita, Amaren-

dra, Akash, Lindley and exponential distributions for parameterθ.

Distribution Over-dispersion Under-dispersion Equi-dispersion

MWEG θ < 1.10073 θ > 1.10073 θ = 1.10073

Ishita θ < 1.535653152 θ > 1.535653152 θ = 1.535653152

Amarendra θ < 1.525763580 θ > 1.525763580 θ = 1.525763580

Akash θ < 1.515400063 θ > 1.515400063 θ = 1.515400063

Lindley θ < 1.170086487 θ > 1.170086487 θ = 1.170086487

Exponential θ < 1 θ > 1 θ = 1

2.4. Mean Residual Life Function. The mean residual function of the MWEG distribution is given as

m(x) =
1

1 −F(x)

∫ ∞
x

[1 −F(t)]dt

m(x) =
2(θ + 1)

[θ2x2 + (θ + 1)2θx + 2(θ + 1)] e−θx

∫ ∞
x

[
θ2t2 + (θ + 1)2θt + 2(θ + 1)

2(θ + 1)

]
e−θtdt

m(x) =
θ2x2 + (θ + 2)2θx + 2(2θ + 3)

θ [θ2x2 + (θ + 1)2θx + 2(θ + 1)]
(2.3)

(a) (b)

Figure 4. Mean residual life function of MWEG distribution at different values of θ

2.5. Stochastic Orderings. In probability theory, stochastic ordering is considered an important tool for

assessing the comparative behavior of a positive continuous random variable. A random variable X is said

to be smaller than a random variable Z, in the

i. Stochastic order (X ≤st Z) , if FX(x) ≥ FZ(x) for all x.

ii. Hazard rate order (X ≤hr Z) , if hX(x) ≥ hZ(x) for all x.

iii. Mean residual life order (X ≤mrl Z) , if mX(x) ≤ mz(x) for all x.



Int. J. Anal. Appl. 18 (3) (2020) 402

iv. Likelihood ratio order (X ≤lr Z) , if
fX(x)
fZ(x)

decreases in x.

Shaked and Shanthikumar [13] proposed the following well-known results for demonstrating the stochastic

ordering of distributions

X ≤lr Z ⇒ X ≤hr Z ⇒ X ≤mrl Z ⇒ X ≤st Z (2.4)

In the following theorem, it has shown that MWEG distribution is being ordered with respect to the

strongest “Likelihood ratio” ordering

Theorem

Let X ∼ MWEG distribution (θ1) and Z ∼ MWEG distribution (θ2) , if θ1 ≥ θ2, then X ≤lr Z and hence

X ≤hr Z ⇒ X ≤mrl Z and X ≤st Z

Proof

We have

fX(x) =
θ31

θ1 + 1

[
x +

x2

2

]
e−θ1x

fz(x) =
θ32

θ2 + 1

[
x +

x2

2

]
e−θ2x

fX(x)

fz(x)
=

θ31
θ32

[
x + x

2

2

]
e−θ1x

θ32
θ2+1

[
x + x

2

2

]
e−θ2x

fX(x)

fz(x)
=
θ31 (θ2 + 1)

θ32 (θ1 + 1)
e−(θ1−θ2)x

log

[
fx(x)

fz(x)

]
= log

[
θ31 (θ2 + 1)

θ32 (θ1 + 1)

]
− (θ1 −θ2) x

This implies that

d

dx
log

fX(x)

fz(x)
= −(θ1 −θ2)

Hence for θ1 ≥ θ2, ddx log
fx(x)
fz(x)

< 0, it means that X ≤lr Z and hence X ≤hr Z ⇒ X ≤mrl Z and X ≤st Z

2.6. Order Statistics. Let X1,X2,X3 . . .Xn be a random sample of size n from MWEG distribution. Let

X(1),X(2),X(3) . . .X(n) denotes the corresponding order statistics. The probability density function and the



Int. J. Anal. Appl. 18 (3) (2020) 403

cumulative distribution function of the Kth order statistic, say Z = X(k) are given below

fz(x) =n


 n− 1

k − 1


F(x)k−1(1 −F(x))n−kf(x)

and

Fz(x) =

n∑
j=k

n−j∑
l=0


 n

j




 n− j

l


 (−1)lF(x)j+l

Respectively for different values of k = 1, ...,n Hence the probability density function and cumulative

distribution function of Kth order statistics are given as

fz(x) =
θ3

θ+1

[
1 + x

2

]
xe−θxn


 n− 1

k − 1




(
1 −

(
θ2x2 +

(
2θ2 + 2θ

)
x + 2θ + 2

)
e−θx

2(θ + 1)

)k−1 ((
θ2x2 +

(
2θ2 + 2θ

)
x + 2θ + 2

)
e−θx

2(θ + 1)

)n−k
It can be written as

fz(x) =
n!θ3(2+θ)xe−θx

2(θ+1)(n−k)!(k−1)!
∑n−k
l=0


 n−k

l


 (−1)l (1 − (θ2x2+(2θ2+2θ)x+2θ+2)e−θx)e−θx

2(θ+1)

)k+l−1
Similarly

Fz(x) =
∑n
j=k

∑n−j
l=0


 n− j

j




 n− j

l


 (−1)l (1 − (θ2x2+(2θ2+2θ)x+2θ+2)e−θx

2(θ+1)

)j+l

2.7. Bonferroni and Lorenz Curves. The Bonferroni and Lorenz curves [14] and Bonferroni and Gini

indices have applications in many fields like insurance, medicines, demography and also in economics for

studying the patterns of income and poverty. The Bonferroni and Lorenz curves may be defined as

Bp =
1

Pµ

∫ q
0

xf(x)dx

Bp =
1

Pµ

∫ ∞
0

xf(x)dx−
∫ ∞
q

xf(x)dx

B(P) =
1

Pµ

[
µ−

∫ ∞
q

xf(x)dx

]

B(P) =
1

P

[
1 −

1

µ

∫ ∞
q

xf(x)dx

]
(2.5)

and

LP =
1

µ

∫ q
0

xf(x)dx =
1

µ

[
xf(x)dx−

∫ ∞
q

xf(x)dx

]



Int. J. Anal. Appl. 18 (3) (2020) 404

L(P) =
1

µ

[
µ−

∫ ∞
q

xf(x)dx

]

L(P) = 1 −
1

µ

∫ ∞
q

xf(x)dx (2.6)

Where E(X) = µ and q = F−1(P) By using (1.1) and (1.2) , we can define the Bonfernoni and Gini indices

as

B = 1 −
∫ 1
0

B(P)dP (2.7)

B = 1 −
∫ 1
0

B(P)dP

and

G = 1 − 2
∫ 1
0

L(P)dP (2.8)

Now by using pdf of MWEG distribution, we get

∫ ∞
q

xf(x)dx =

∫ ∞
q

θ3

θ + 1

[
1 +

x

2

]
xe−θxdx

∫ ∞
q

xf(x)dx =
θ3
(
q3 + 2q2

)
+ θ2

(
3q2 + 4q

)
+ θ(6q + 4) + 6

2θ(θ + 1)
e−θq (2.9)

Using above (2.9) in (2.5) and (2.6), we get

1

µ

∫ ∞
q

xf(x)dx =
θ3
(
q3 + 2q2

)
+ θ2

(
3q2 + 4q

)
+ θ(6q + 4) + 6

2(2θ + 3)
e−θq

B(P) =
1

P

[
1 −

θ3
(
q3 + 2q2

)
+ θ2

(
3q2 + 4q

)
+ θ(6q + 4) + 6

2(2θ + 3)
e−θq

]
(2.10)

and

L(P) = 1 −
θ3
(
q3 + 2q2

)
+ θ2

(
3q2 + 4q

)
+ θ(6q + 4) + 6

2(2θ + 3)
e−θq (2.11)

By using (2.10) and (2.11) in (2.7) and (2.8), we get

B = 1 −
θ3
(
q3 + 2q2

)
+ θ2

(
3q2 + 4q

)
+ θ(6q + 4) + 6

2(2θ + 3)
e−θq (2.12)

and

G = −1 +
θ3
(
q3 + 2q2

)
+ θ2

(
3q2 + 4q

)
+ θ(6q + 4) + 6

2θ + 3
e−θq (2.13)



Int. J. Anal. Appl. 18 (3) (2020) 405

2.8. Renyi Entropy. Entropy of a random variable X can be defined as a measure of variation of uncer-

tainty. Renyi entropy [15] is considered as a very popular entropy measure. If X is a continuous random

variable having probability density function f(x), then Renyi entropy is defined as

TR(y) =
1

1 −γ
log

{∫
fγ(x)dx

}
where γ > 0 also γ 6= 1

Thus, the Renyi entropy for the MWEG distribution ( 1) can be obtained as

TR(γ) =
1

1 −γ
log

∫ ∞
0

θ3γ

(θ + 1)γ

[
1 +

x

2

]γ
xγe−sγxdx

=
1

1 −γ
log


 θ3γ

(θ + 1)γ

∫ ∞
0

∞∑
j=0


 γ

j


(x

2

)j
xγe−θγxdx




=
1

1 −γ
log


 θ3γ

(θ + 1)γ

∞∑
j=0


 γ

j


(x

2

)j ∫ ∞
0

xγe−θγxdx




=
1

1 −γ
log


 θ3γ

(θ + 1)γ

∞∑
j=0


 γ

j


(1

2

)j
Γ(γ + j + 1)

(θγ)γ+j+1




=
1

1 −γ
log


θ2γ−j−1

(θ + 1)γ

∞∑
j=0

Γ(γ + j + 1)

2jγγ+j+1


 γ

j






3. Estimation of Parameters

This section consists of estimation of the unknown parameters of proposed model by using the methods

of moments and maximum likelihood.

3.1. Maximum Likelihood Estimation. Let X1,X2,X3 . . .Xn be an iid random sample from MWEG,

then the likelihood function of the MWEG distribution is given by,

L =

(
θ3

θ + 1

)n n∏
i=1

[
xi +

x2i

2

]
e−θ

∑n
i=1

xi

and the log-likelihood function can be written as

ln L = n ln

(
θ3

θ + 1

)
+

n∑
i=1

[
xi +

x2i

2

]
−nθx̄

d

dθ
(ln L) =

3n

θ
−

n

θ + 1
−nx̄

Where x̄ is the sample mean. We can find the maximum likelihood estimate of θ by simply equating

d
dθ

(ln L) = 0, and it can be find by solving the following nonlinear equation

nx̄θ(θ + 1) − 2nθ − 3n = 0 (3.1)



Int. J. Anal. Appl. 18 (3) (2020) 406

3.2. Method of Moment. Let X1,X2,X3 . . .Xn be a random sample of size n from MWEG distribution.

By equating the first population moment about origin to the sample mean, the method of moment estimate

θ̂ of θ is the same as the ML estimate as given in (3.1)

4. Real Data Application

The goodness of fit of MWEG distribution has been checked by using several lifetime data sets from

engineering and medical science. In this section, we have used two real-life data sets to compare the goodness

of fit by using ML estimate of the parameter of MWEG distribution with the exponential, Akash, Lindley,

Ishita and Modified One Parameter Lindley distributions and have proved that MWEG distribution provides

better estimate for modeling lifetime data sets as compared to its competing models.

Data Set 1: The following dataset acts the breaking stress of carbon fibers of 50 mm in length Nichols and

Padgett [16]

3.70, 2.12, 2.95, 4.70, 1.25, 3.22, 1.69, 3.27, 2.87, 1.47, 3.11, 3.65, 2.74, 3.15, 2.97, 2.03, 4.38, 3.39, 3.28, 3.09,

1.87, 3.15, 4.90, 4.42, 2.73, 1.08, 3.39, 1.89, 1.84, 2.81, 4.20, 3.33, 2.55, 3.31, 1.57, 2.41, 2.50, 2.56, 2.96, 2.88,

0.39, 3.68, 2.48, 0.85, 1.61, 3.31, 2.67, 3.19, 3.60, 1.80, 2.35, 2.82, 2.05, 3.65, 3.75, 2.43, 2.79, 2.85, 2.93, 3.22,

3.11, 2.53, 2.55, 2.59, 2.03, 1.61

Data set 2: The following data represent the survival times (in days) of 72 guinea pigs infected with virulent

tubercle bacilli, observed and reported by Bjerkedal [17]. The data are as follows:

0.1, 0.33, 0.44, 0.56, 0.59, 0.72, 0.74, 0.77, 0.92, 0.93, 0.96, 1, 1, 1.02, 1.05, 1.07, 1.07, 1.08, 1.08, 1.08,1.09,

1.12, 1.13, 1.15, 1.16, 1.2, 1.21, 1.22, 1.22, 1.24, 1.3, 1.34, 1.36, 1.39, 1.44, 1.46, 1.53, 1.59, 1.6, 1.63, 1.63,1.68,

1.71, 1.72, 1.76, 1.83, 1.95, 1.96, 1.97, 2.02, 2.13, 2.15, 2.16, 2.22, 2.3, 2.31, 2.4, 2.45, 2.51, 2.53, 2.54,2.54,

2.78, 2.93, 3.27, 3.42, 3.47, 3.61, 4.02, 4.32, 4.58, 5.55

In order to compare MWEG, exponential, Lindley, Akash, Ishita and modified one parameter Lindley dis-

tributions, values of −2ln L , AIC (Akaike Information Criterion), AICC (Akaike Information Criterion

Corrected), BIC (Bayesian Information Criterion) and K-S Statistic ( Kolmogorov-Smirnov Statistic) for

two real data sets have been computed and presented in the following table.



Int. J. Anal. Appl. 18 (3) (2020) 407

Table 2. MLE’s,−2lnL, AIC, AICC, BIC, and K-S Statistics of the fitted distributions of

data sets 1 and 2

Model MLE of θ̂ SE(θ̂) -2lnL AIC CAIC BIC K-S
D

A
T

A
S

E
T

1

MWEG 1.2832 0.1751 50.8754 52.8754 53.0976 53.8711 0.3058

MOPLD 1.5213 0.1523 59.7066 61.7066 61.9288 62.7023 0.3569

Ishita 1.0948 0.1217 60.1647 62.1647 62.3869 63.1604 0.3507

Exponential 0.5263 0.1177 65.4742 67.6742 67.8964 68.6699 0.4395

Lindely 0.8162 0.1361 60.4991 62.4991 62.7213 63.4948 0.3911

Akash 1.1569 0.1456 59.5226 61.5226 61.7448 62.5183 0.3705

D
A

T
A

S
E

T
2

MWEG 1.3697 0.0989 193.103 195.102 195.16 197.379 0.153

MOPLD 1.5852 0.0839 220.004 222.004 222.061 224.281 0.231

Ishita 1.1598 0.0677 216.632 218.632 218.689 220.909 0.229

Exponential 0.5655 0.0666 226.074 228.074 228.131 230.351 0.295

Lindely 0.8683 0.0766 213.857 215.857 215.914 218.134 0.247

Akash 1.2159 0.081 214.678 216.678 216.735 218.954 0.234

The best distribution corresponds to lower values of −2lnL,AIC,AICC,BIC and K-S statistic. It can be

easily seen from above table that MWEG distribution provide better fit as compare to exponential, Lindley,

Akash, Ishita and modified one parameter Lindley distributions.

5. Conclusion

In this paper, we consider a simple mixture of two absolutely continuous distributions weighted exponential

and weighted gamma distribution. Some structural properties of the resulting distribution are discussed. The

resulting model appears to be a reasonable choice in the sense of modeling lifetime data sets, in particular,

where the popular choices (e.g., Exponential, gamma, Lindley, Ishita and/or Akash distribution) fail to

adequately model the observed phenomena. We sincerely hope that this mixture model will find many more

applications in different fields affecting human life.

Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication

of this paper.

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	1. Introduction
	2. Properties of the MWEG distribution
	2.1. Hazard Function
	2.2. Survival Function
	2.3. Moments and Related Measures
	2.4. Mean Residual Life Function
	2.5. Stochastic Orderings
	2.6. Order Statistics
	2.7. Bonferroni and Lorenz Curves
	2.8. Renyi Entropy

	3. Estimation of Parameters
	3.1. Maximum Likelihood Estimation
	3.2. Method of Moment 

	4. Real Data Application 
	5. Conclusion
	References