EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5981
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A New Method for Generating Continuous
Distributions with Applications

Morad Ahmad1,2, Mohammad A. Amleh2,∗

1 Department of Mathematics, The University of Jordan, Amman, 11942, Jordan
2 Department of Mathematics, Faculty of Science, Zarqa University, Zarqa, Jordan

Abstract. In this paper, a new modifying method has been introduced by adding an extra param-
eter to generate a new family of distributions that has more flexibility and better model fitting. A
special case has been considered; the exponential distribution. All the main properties of the new
modified exponential distribution are derived, including the CDF, PDF, and quantile function. The
maximum likelihood estimation method is used to estimate unknown parameters. The modified
exponential distribution has been applied to two-lifetime data sets to illustrate its efficiency.

2020 Mathematics Subject Classifications: 62E10, 62N02, 62N05

Key Words and Phrases: Reliability function, Maximum likelihood estimation, Exponential
distribution

1. Introduction

The distribution of random variables is one of the key concepts used in statistics to
model and fit data sets. There are several well-known distributions used in the literature.
These distributions show strong effectiveness in many situations but fail in other cases.
Because of this, more general and flexible distributions have appeared recently by many
authors by adding a new parameter to a family of distribution functions to give better
fitting for the empirical data. Recently, Ahmad and Asha[1] introduced the exponential
reliability method which is a new method for generating a new family of distribution
functions. A well-known approach for this purpose is the TX family of distributions,
which was suggested by Aljarrah et al. [2]. This work is similar to the beta-normal
distribution proposed by Eugene et al. [3]. Other researchers made improvements for
classical distributions like the Weibull distribution[4], such as the transmuted modified
Weibull distribution [5] and the transmuted inverse Weibull distribution [6]. More details
on this context can be found in [7] and [8].

This paper aims to insert a new parameter into a family of distribution functions to
produce a new more powerful family that can adjust reliability and has better fit and

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5981

Email addresses: m.ahmad@zu.edu.jo (M. Ahmad), malamleh@zu.edu.jo (M. A. Amleh)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 2 of 16

more flexibility as well as simplicity. This new method is called the alpha-R (α − R)
method. The α − R method can be used easily and effectively for data analysis. First,
we present the modified family and discuss its main properties. Then, the technique
is applied to the exponential distribution. Two data sets are fitted using the classical
distributions and the modified exponential distribution using the alpha-R method, we
compare them using Akaike’s information criterion [9], the Bayesian information criterion
and Kolmogorov-smirnov test. The maximum likelihood estimation method is used to
estimate the parameters. All calculations are obtained using R software.

2. The α−R Family

The cumulative distribution function (CDF) of the α−R family is defined as follows

G(y) = 1−R(y)αF (y), y ∈ R, 0 < α ≤ e (1)

where F(y) and R(y) are the CDF and reliability function of a baseline distribution,
respectively. A new parameter α(0 < α ≤ e) is utilized to extend the use of a random
variable Y . It is clear that if α = 1, then G(y) = F (y). The probability density function
(PDF) corresponding to (1) is given by:

g(y) = f(y)αF (y)[1− (logα)R(y)], y ∈ R, 0 < α ≤ e, (2)

where f(y) represents the PDF of a baseline distribution. Further, it can be shown that
the hazard rate function of the α−R family is given as

z(y) = h(y)[1− (logα)R(y)], y ∈ R, 0 < α ≤ e, (3)

where h(y) is the hazard rate function of the original distribution.

3. α−R Exponential Distribution

In this section, we present a special case of the α−R family. This special distribution
is considered as a generalization of the exponential distribution.

Definition 1. A random variable Y is said to have α−R exponential (α−RE) distribution
if its CDF is given by:

G(y, α, θ) = 1− e−θy × α1−e−θy
, y > 0, θ > 0, 0 < α ≤ e (4)

The Corresponding PDF of the α−RE distribution is defined as:

g(y, α, θ) = θα1−e−θy
(
e−θy − logα× e−2θy

)
, y > 0, θ > 0, 0 < α ≤ e (5)

Plots for the CDF of the α − RE distribution are provided in Figures 1 and 2. In
addition, Figures 3 and 4 display the plots of the corresponding PDFs.



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 3 of 16

0 2 4 6 8 10
0

0.2

0.4

0.6

0.8

1

y

G
(y
,α

,θ
) α = 0.5, θ = 0.5

α = 0.5, θ = 2

α = 0.75, θ = 0.5

α = 0.75, θ = 2

Figure 1: The CDF curves of the α−R E distribution for the parameter values θ = 0.5 and θ = 2, and α = 0.5
and α = 0.75

0 2 4 6 8 10
0

0.2

0.4

0.6

0.8

1

y

G
(y
,α

,θ
) α = 1.5, θ = 0.5

α = 1.5, θ = 2

α = e, θ = 0.5

α = e, θ = 2

Figure 2: The CDF curves of the α−R E distribution for the parameter values θ = 0.5 and θ = 2, and α = 1.5
and α = e



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 4 of 16

0 2 4 6 8 10
0

1

2

3

y

g
(y
,α

,θ
)

α = 0.5, θ = 0.5
α = 0.5, θ = 2

α = 0.75, θ = 0.5
α = 0.75, θ = 2

Figure 3: The PDF curves of the α−RE distribution for the parameter values θ = 0.5 and θ = 2, and α = 0.5
and α = 0.75

0 2 4 6 8 10
0

0.2

0.4

0.6

0.8

1

y

g
(y
,α

,θ
)

α = 1.5, θ = 0.5
α = 1.5, θ = 2
α = e, θ = 0.5
α = e, θ = 2

Figure 4: The PDF curves of the α−RE distribution for the parameter values θ = 0.5 and θ = 2, and α = 1.5
and α = e

4. Statistical Properties of the α−RE distribution

In this section, some statistical properties of the α − RE distribution are discussed,
including the moments, the quantile function, and the maximum likelihood estimation.



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 5 of 16

4.1. Quantile Function

If the CDF of a distribution Y is H(y), then q = H−1(p), 0 < p < 1 is the corre-
sponding quantile function. Thus, the quantile function of the α − RE distribution can
be obtained by solving the equation:

−θq −
(
1− e−θq

)
logα = log (1− p) , 0 < p < 1, (6)

Or equivalently:
logαe−u + u− log [α (1− p)] = 0, (7)

where u = −θq. Eq.(7) has an analytical solution in terms of the Lambert W function as
follows:

q(p) =
1

θ

[
W

(
−(1− p)

α
logα

)
+ log

[
(1− p)

α

]]
, 0 < p < 1 (8)

where W (.) is the Lambert W function. Here, data from the α−RE distribution can be
obtained by applying its quantile function to a sample from a uniform distribution.

The median of the α−R E distribution is given by:

median =
1

θ

[
W

(
1

2α
log

(
1

α

)
+ log

(
1

4α

))]
.

4.2. Moments

This subsection discusses the computation of the Kth moment of the α − R E distri-
bution. The Kth moment about the origin of the proposed model is derived as follows:

Mk = θ

∫ ∞

0
ykα1−e−θy

(
e−θy − logα× e−2θy

)
dy,

= θ

[∫ ∞

0
α1−e−θy

e−θyyk dy − logα

∫ ∞

0
α1−e−θy

e−2θyyk dy

]
. (9)

It is known that the lower incomplete gamma function is given as:

γ(x, a) =

∫ x

0
ta−1e−tdt,

Consequently, it can be shown that:∫ 1

0
(log t)kta−1e−t dt =

∂k

∂ak
γ(1, a). (10)

Thus, the first integral (I1) in Eq.(9) can be handled by letting t = e−θy, so we get:

I1 =
(−1)kα

(θ)k+1(logα)k+1

∫ 1

0
(log t)k e−t dt =

(−1)kα

(θ logα)k+1

∂k

∂ak
γ(1, a)

∣∣∣
a=1

(11)



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 6 of 16

Similarly, the second integral (I2) in Eq.(9) can be expressed as:

I2 =
(−1)kα

(θ logα)k+1

∫ 1

0
t(log t)ke−t dt

I2 =
(−1)kα

(θ logα)k+1

∂k

∂ak
γ(1, a)

∣∣∣
a=2

(12)

Hence, using Eq.s (11) and (12) the kth moment of the α − RE distribution is given
by:

Mk =
(−1)kα

θk(logα)k+1

[
∂k

∂ak
γ(1, a)

∣∣∣
a=1

− (logα)
∂k

∂ak
γ(1, a)

∣∣∣
a=2

]
(13)

Therefore, the mean of the α−RE distribution is given by:

E(Y ) =
(−1)kα

θ

[
1

(logα)2
∂

∂a
γ(1, a)

∣∣∣
a=1

− 1

logα

∂

∂a
γ(1, a)

∣∣∣
a=2

]
(14)

4.3. Moment Generating Function

The moment generating function (MGF) of Y ∼ α−RE(α, θ) is obtained as:

M(t) = E(etY ) = θ

∫ ∞

0
etyα1−e−θy

(
e−θy − logα× e−2θy

)
dy

By letting u = e−θy, we have:

M(t) = α


∫ 1

0
u−

t
θα−udu︸ ︷︷ ︸
I1

− logα

∫ 1

0
u1−

t
θα−udu︸ ︷︷ ︸
I2


I1 =

1

(logα)1−
t
θ

γ(logα, 1− t

θ
)

I2 =
1

(logα)1−
t
θ

∫ logα

0
u1−

t
θ e−udu

=
1

(logα)1−
t
θ

γ(logα, 2− t

θ
)

Accordingly, the MGF of the α−R E distribution is given by:



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 7 of 16

M(t) =
α

(logα)1−
t
θ

γ(logα, 1− t

θ
)− (logα)

t
θ γ(logα, 2− t

θ
) (15)

4.4. Maximum Likelihood Estimation

The foremost method and extensively utilized technique for the purpose of estimating
parameters is the maximum likelihood method. In this subsection, we utilize the em-
ployment of the maximum likelihood method to estimate the parameters of the α − RE
distribution.

Assume that Y1, Y2, ...., Yn is a random sample from α − RE distribution, the corre-
sponding likelihood function is:

L(α, θ, yi) = θn αn−
∑n

i=1 e
−θyi ·

n∏
i=1

(
e−θyi − logα× e−2θyi

)
(16)

The log-likelihood function is given by:

l(α, θ, yi) = n log θ +

(
n−

n∑
i=1

e−θyi

)
logα+

n∑
i=1

log
(
e−θyi − logα× e−2θyi

)
(17)

In order to obtain the maximum likelihood estimators (MLEs) of α and θ, the partial
derivatives of l in Eq.(17) are provided:

∂l

∂α
=

(
n−

∑n
i=1 e

−θyi
)

α
+

n∑
i=1

e−2θyi

α (e−θyi − logα× e−2θyi)
(18)

∂l

∂θ
=

n

θ
+ logα

n∑
i=1

yie
−θyi +

n∑
i=1

−yie
−θyi + 2 logα yie

−2θyi

e−θyi − logα× e−2θyi
(19)

By equating the non-linear Eq.s (18) and (19) to zero and solving subsequently, we
obtain the MLEs of α and θ. In fact, numerical solutions are required to find out such
MLEs. Newton-Raphson method via R software can be applied in this context.

5. Simulation Experiment

In this section, a simulation experiment is performed to assess the efficiency and preci-
sion of the MLEs of the parameters θ and α of the α−RE distribution. First, we generate
a random sample Y1, Y2, . . . , Yn from α−RE distribution. Using Eq.(18) and Eq.(19) we
compute the estimates. The average bias (AB), and the mean squared error (MSE), of
the suggested estimators are used to compare the performance of such estimators. The
AB(β̂) of an estimator β̂ of any parameter β indicates the typical disparity between an
outcome and its observed counterpart. It quantifies the precision of a model’s predic-



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 8 of 16

tions, with diminished mean bias implying greater proximity between obtained and actual
values.

AB(β̂) =
1

N

N∑
i=1

(
β̂i − β

)
,

where β̂i is the estimate achieved in iteration i, i = 1, . . . , N . The MSE(β̂) reflects
the average squared difference between an estimated value and the actual value of β.
Precisely, it measures how accurate a model’s efficiency is. The lower the MSE, the closer
the estimated values are to the actual values, and the better the model’s performance,
formally

MSE(β̂) =
1

N

N∑
i=1

(
β̂i − β

)2
.

A simulation experiment is conducted based on different sample sizes and parameter
values. For this, we generate random samples of α − RE distribution by considering the
true values of θ and α to be 1.5 and 2, respectively. Samples from α − RE distribution
are randomly generated under this setup with 1000 replications of the simulation process.
Using these random samples, ABs andMSEs of the estimators are calculated. We propose
several sample sizes n = 50, 100, 200, 500, 1000. We display the results in Table 1.

Table 1: ABs and MSEs of the parameter estimates when θ = 1.5, α = 2

n AB(θ̂) MSE(θ̂) AB(α̂) MSE(α̂)

50 0.0242 0.0630 0.0417 0.2188

100 0.0191 0.02823 0.0374 0.0904

200 0.0144 0.0133 0.0129 0.0427

500 0.0027 0.0060 0.0053 0.01750

1000 0.0019 0.00297 0.0017 0.0093

It can be observed that the AB(β̂) and MSE(β̂), of the estimators decrease as the
sample size increases, which means all estimators tend to the true value of the parameter.

6. Real Data Applications

In this section, we examine the adaptability of the new distribution using real-life
applications and compare the goodness of fit of the α−RE distribution to other existing
distributions.

The goodness of fit of the suggested distribution is compared with the following dis-



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 9 of 16

tributions: Two-parameter exponential distribution (Johnson et al.)[10], with PDF:

fExp(x) = θe−θ(x−b), x > b, θ > 0.

Gamma distribution (Johnson et al.)[10], with PDF:

fGam(x) =
βαxα−1e−βx

Γ(α)
, x > 0, α > 0, β > 0.

Weibull distribution (Johnson et al.)[10], with PDF:

fWei(x) =
α

β

(
x

β

)α−1

e
−
(

x
β

)α

, x > 0, α > 0, β > 0.

The performance of each model is obtained according to many criteria, including
−2 lnL, Kolmogorov-Smirnov statistic (KSS), and its corresponding p-value, (Chakravarti
et al.)[11], Akaike information criterion (AIC), see (Akaike)[12], and Bayesian information
criterion (BIC), (Konishi et al.)[13]. We choose the best model so that it has lower values
of −2 lnL, KSS, AIC, and BIC and higher p-value. Here, L represents the likelihood
function. The values of KSS,AIC,BIC are given respectively:

KSS = sup x|Fn(x)− F0(x)|

AIC = −2 lnL+ 2m

BIC = −2 lnL+m ln(n)

where n is the sample size, m represents the number of parameters, and Fn(x) stands for
the empirical distribution function. We consider two data sets that are widely used in the
literature for the purpose of fitting new distributions.

6.1. Example 1

We have a dataset that contains information about the lifespan of Kevlar 373/epoxy
samples exposed to constant pressure at 90% of their stress capacity until each sample
reaches the point of failure. This dataset has been utilized by Abdul-Moniem and Seham
[14]. The data is recorded as follows:



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 10 of 16

Example 1

0.0251 0.0886 0.0891 0.2501 0.3113 0.3451 0.4763 0.5650 0.5671

0.6566 0.6748 0.6751 0.6753 0.7696 0.8375 0.8391 0.8425 0.8645

0.8851 0.9113 0.9120 0.9836 1.0483 1.0596 1.0773 1.1733 1.2570

1.2766 1.2985 1.3211 1.3503 1.3551 1.4595 1.4880 1.5728 1.5733

1.7083 1.7263 1.7460 1.7630 1.7746 1.8275 1.8375 1.8503 1.8808

1.8878 1.8881 1.9316 1.9558 2.0048 2.0408 2.0903 2.1093 2.1330

2.2100 2.2460 2.2878 2.3203 2.3470 2.3513 2.4951 2.5260 2.9911

3.0256 3.2678 3.4045 3.4846 3.7433 3.7455 3.9143 4.8073 5.4005

5.4435 5.5295 6.5541 9.0960

The results of −2 lnL, KSS, p− value, AIC, and BIC for α−RE, gamma, weibull,
and exponential distributions are given in Table 2.

Table 2: −2 lnL, KSS, p− value, AIC, and BIC statistic and the of the fitted distributions

Distribution −2 lnL KSS p-value AIC BIC

α−RE 242.5566 0.08946 0.5472 246.5567 251.2182

Gamma 244.499 0.09614 0.4554 248.4987 253.1602

Weibull 245.049 0.1101 0.2937 249.0494 253.7109

Exponential 185.3168 0.1663 0.0299 256.2289 258.5594

It can be observed that the α−RE distribution has minimum values of: −2 lnL, KSS, p−
value, AIC, and BIC and the largest p − value among all other distribution. Conse-
quently, the α−RE distribution is more adequate to fit this real data.

Table 3: The Parameter Estimates and Standard Errors of the distributions considered.

Distribution Estimates of the parameters Std Error

α−RE θ = 0.805 0.0846
α = 2.250 0.2797

Gamma α = 1.6406 0.2439
β = 1.1941 0.2072

Weibull α = 1.3258 0.1138
β = 2.1336 0.1946

Exponential θ = 0.5104 0.0586

In Table 3, we present the MLEs of the parameters of the α − RE and the other
existing distributions with their corresponding standard error.



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 11 of 16

Figure 5: presents plots of the estimated PDFs with the data set.



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 12 of 16

Figure 6: shows the CDFs for the four distribution functions.

Figures 5 and 6 display the PDF and the CDF of the α − RE distribution alongside
alternative distributions in the first data set, respectively. Based on the plotted data, it
is evident that the α − RE distribution provides a superior match to the data compared
to other rival distributions.

6.2. Example 2

The second data set represents the survival times of 121 patients suffering from breast
cancer collected from a large hospital in a period from 1929-1938 [15]. The times are
reported as follows:



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 13 of 16

Example 2

0.3 0.3 4.0 5.0 5.6 6.2 6.3 6.6 6.8

7.4 7.5 8.4 8.4 10.3 11.0 11.8 12.2 12.3

13.5 14.4 14.4 14.8 15.5 15.7 16.2 16.3 16.5

16.8 17.2 17.3 17.5 17.9 19.8 20.4 20.9 21.0

21.0 21.1 23.0 23.4 23.6 24.0 24.0 27.9 28.2

29.1 30.0 31.0 31.0 32.0 35.0 35.0 37.0 37.0

37.0 38.0 38.0 38.0 39.0 39.0 40.0 40.0 40.0

41.0 41.0 41.0 42.0 43.0 43.0 43.0 44.0 45.0

45.0 46.0 46.0 47.0 48.0 49.0 51.0 51.0 51.0

52.0 54.0 55.0 56.0 57.0 58.0 59.0 60.0 60.0

60.0 61.0 62.0 65.0 65.0 67.0 67.0 68.0 69.0

78.0 80.0 83.0 88.0 89.0 90.0 93.0 96.0 103.0

105.0 109.0 109.0 111.0 115.0 117.0 125.0 126.0 127.0

129.0 129.0 139.0

The results of −2 lnL, KSS, p− value, AIC, and BIC for α−RE, gamma, weibull,
and exponential distributions are given in Table 4.

Table 4: −2lnL,KSS, p− values,AIC, and BIC statistic and the of the fitted distributions

Distribution −2 lnL KSS p− value AIC BIC

α−RE 1142.6 0.0577 0.8188 1146.6 1152.2

Gamma 1144.5 0.0805 0.4174 1148.5 1154.1

Weibull 1142.7 0.06762 0.6427 1146.7 1152.3

Exponential 987.6 0.3781 0 982.56 988.15

It can be observed that, the α−RE distribution has minimum values of:−2lnL,KSS,AIC,
and BIC and the largest p−value among all other distribution. Consequently, the α−RE
distribution is more adequate to fit this real data.



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 14 of 16

Table 5: The Parameter Estimates and Standard Errors of the distributions considered.

Distribution Estimates of the parameters Std Error

α−RE θ = 0.0329 0.0031
α = 2.134 0.265

Gamma α = 1.515 0.177
β = 29.95 4.15

Weibull α = 1.317 0.0951
β = 48.77 3.537

Exponential b = 0.3 0.0586
θ = 22.1

In Table 5, we present the MLEs of the parameters of the α−RE and the other existing
distributions with their corresponding standard error.

Figure 7: presents plots of the estimated PDFs with the data set.



M. Ahmad, M. A. Amleh / Eur. J. Pure Appl. Math, 18 (2) (2025), 5981 15 of 16

Figure 8: shows the CDFs for the four distribution functions.

Figures 7 and 8 display the PDF and the CDF of the α − RE distribution alongside
alternative distributions in the first data set, respectively. Based on the plotted data, it
is evident that the α − RE distribution provides a superior match to the data compared
to other rival distributions.

Acknowledgements

This research was carried out during the first author’s sabbatical leave from The Uni-
versity of Jordan to Zarqa University. The authors thank the editor and anonymous
reviewers for their comments that helped improve the quality of this work.

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