EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 12, No. 3, 2019, 722-733
ISSN 1307-5543 – www.ejpam.com
Published by New York Business Global

Linear combination and reliability of generalized logistic
random variables

Luan C. de S. M. Ozelim1,∗, Pushpa N. Rathie2

1 Department of Civil and Environmental Engineering, University of Brasilia, Brasilia- DF,
70910-900, Brazil
2 Department of Statistics, University of Brasilia, Brasilia- DF, 70910-900, Brazil

Abstract. Experimental random data, in general, present a skewed behaviour. Thus, asymmet-
rical generalized distributions are of interest. The generalized logistic distributions (GLDs) are
good candidates to model skewed data because their probability density functions (p.d.f.) and
characteristic functions are mathematically simple. In this paper, exact expressions in terms of
the H-function are, for the first time, derived for the p.d.f. and for the cummulative distribution
function of the linear combination of GLDs of type IV with different location, scale and shape
parameters. Also, exact and approximate expressions are derived for R = P (X < Y ). Numerical
examples illustrate the correctness of the expressions derived.

2010 Mathematics Subject Classifications: 33C60

Key Words and Phrases: Generalized Logistic Distribution, Mellin Transform, Fourier Trans-
form, H-function, Reliability

1. Introduction

Modelling nature is by no means a trivial task. At first, scientists have to observe how
a given phenomenon occurs and then, by means of physically justifiable premises, propose
a model which is able to predict the outcomes of such phenomenon when some important
inputs are known. Over the last half century, the scientific community started to model
both the inputs and outcome of such models as random variables. Thus, the algebra of
random variables has become increasingly of interest to not only pure but also applied
scientists.

Instead of modelling failure as stresses overcoming a given strength threshold, a prob-
abilistic approach enables one to calculate the probability of failure. The latter approach
may lead to cost reduction and time consumption in the implementation of a given project.
[13]

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3444

Email addresses: luanoz@gmail.com (L. C. de S. M. Ozelim),
pushpanrathie@yahoo.com (P. N. Rathie)

http://www.ejpam.com 722 c© 2019 EJPAM All rights reserved.



L. C. de S. M. Ozelim, P. N. Rathie / Eur. J. Pure Appl. Math, 12 (3) (2019), 722-733 723

The application of reliability measures of the type R = P (X < Y ), where X is the
stress to which a given structure of strength Y is subjected, have many applications
in various areas, including quality control, engineering statistics, reliability, medicine,
psychology, biostatistics, stochastic precedence, and probabilistic mechanical design [9, 10].
For example, if X represents the maximum pressure caused by flooding and Y represents
the strength of the leg of a bridge on a stream, then R is the probability that the bridge
will resist [7].

On the other hand, for medical applications, let X and Y represent the control and
treatment groups, respectively. Then R measures the treatment effect [7, 18]. Alterna-
tively, in the cases of diagnostic tests used to distinguish between diseased and non-diseased
patients, the area under the receiver operating characteristics (ROC) curve, based on the
sensitivity and the complement to specificity at different cut-off points of the range of
possible test values, is equal to R [16, 18].

Besides, in the ecotoxicological risk assessment literature, R is also employed to quan-
tify risk [8]. Thus, it is clear that studying reliability measures of the type R = P (X < Y )
is fundamental to various areas of science. In this regard, the estimation of R is of interest
to provide information in decision processes. It is common for authors to assume that X
and Y belong to a certain family of probability distributions with unknown parameters
and then to consider the estimation problem of the reliability R [7]. One may check [7] and
the references therein for estimation problems related to exponential, uniform, generalized
exponential, generalized gamma, Burr, gamma and beta distributions.

Being able to build not only estimates but also the exact value of R when the statistical
distribution of the variables involved are known is then important. Besides considering
the random variables (RVs) previously cited, it is important to obtain the exact value of
R for more general RVs. In the present paper, generalized logistic random variables are
studied.

Statistical models of the logistic type have been applied to all sorts of pure and applied
problems in Statistics. This comes from the fact that logistic models are flexible, being
able to mimic normality as well as show skewness in some generalized models. Applications
of this kind of random variable are easily seen in the literature. A complete study has
been recently performed in [12]. For example, regarding applied scientists, [4] employed
generalized logistic models to perform flood analysis in partial duration series.

On the other hand, regarding pure statistics papers, [19] discussed the parameter es-
timation for a certain type of generalized logistic distribution. Also, [3] obtained approxi-
mate maximum likelihood estimation for some generalized logistic distributions. Besides,
[1] performed a comparative study of the techniques available for the estimation of the
generalized logistic distribution parameters.

Consider a random variable S. Let S follow a generalized logistic distribution of type
IV, from now on called GLD. Also, let one consider a location parameter µ ∈ R, a scale
parameter σ > 0 and two shape parameters α and β such that α, β > 0. One says
S ∼ GLD(µ, σ, α, β) and that the probability density function (p.d.f) of S is given by:



L. C. de S. M. Ozelim, P. N. Rathie / Eur. J. Pure Appl. Math, 12 (3) (2019), 722-733 724

f(x;σ, µ, α, β) =
1

B(α, β)

e−β(
x−µ
σ

)

σ[1 + e−(
x−µ
σ

)]α+β
, ∀x ∈ R, (1)

where B(α, β) is the Beta function, defined as:

B(α, β) =

∫ 1

0
tα−1(1− t)β−1dt =

Γ(α)Γ(β)

Γ(α+ β)
. (2)

In (2), Γ(x) stands for the Gamma function defined as:

Γ(x) =

∫ ∞
0

tx−1e−tdt. (3)

Despite being constantly considered in pure and applied sciences, no closed-form com-
pact representation for the sum of independent not identically distributed generalized
logistic random variables has been presented yet. In the present paper, the linear com-
bination of generalized logistic random variables with different location, scale and shape
parameters is given, in a compact form, in terms of the H-function. This latter function
is a generalized hypergeometric special function whose importance has been widely recog-
nized. In special, [17] discusses the central role of this function to the study of the algebra
of random variables. Besides the pure statistical applications where the linear combination
itself is sought, reliability models can also be derived based on the latter by noticing that
R = P (X < Y ) = P (X − Y < 0).

As discussed, mathematical procedures are used to estimate R. Regarding logistic
RVs, for example, [14] proposed methods of estimation of the shape parameters of the
generalized logistic distribution and of P (Y < X), when X and Y were independent
random variables from two distributions having the same scale parameters but different
shape parameters.

More recently, [2] derived estimators for R when both the distributions compared are
generalized logistic. The authors considered the estimation of R, when X and Y are both
two-parameter generalized logistic distribution with the same unknown scale but different
shape parameters or with the same unknown shape but different scale parameters. They
also considered the general case when the shape and scale parameters are different. In [2]
the maximum likelihood estimator of R and its asymptotic distribution are obtained and
it is used to construct the asymptotic confidence interval of R. Besides, [5] studied the
estimation of two measures of reliability for generalized half logistic distributions.

One of the purposes of the present paper is to provide exact and approximate formulas

for reliability measures R = P (
N1∑
i=1

Xi <
N2∑
j=1

Yj), when Xi, i = 1, ..., N1 and Yj , j = 1, ..., N2

are GLD random variables with different location, scale and shape parameters. By means
of standard mathematical tools, an exact expression for R is derived in terms of the H-
function. Also, an approximation to R, in terms of elementary functions, is presented.
The latter is based on the series expansion of the H-function.



L. C. de S. M. Ozelim, P. N. Rathie / Eur. J. Pure Appl. Math, 12 (3) (2019), 722-733 725

Considering the linear combination of RVs instead of simply R = P (X < Y ) is of
interest because in many applications, the stress and strength are described in terms of
other random variables. For example, the strength of a given geomaterial may be obtained
by means of the Mohr-Coulomb failure criterion. Mathematically, this criterion may be
represented as [6]:

τ = σ tan(φ) + c (4)

where τ , σ and c are the shear, normal, and cohesion stresses, respectively. When both σ
and c are considered as RVs, it can be seen from (4) that the shear resistance τ is nothing
but the linear combination of the other random variables, as the internal friction angle φ
may be considered constant.

Obtaining the linear combination is, thus, more interesting than simply expressing
P (X < Y ), as the results are more general. For example, in [15], linear combinations,
products and ratios of t Random Variables were studied, therefore all the theoretical
and mathematical basis for studying reliability measures for that specific type of random
variable is presented. To familiarize the reader, the definition of the H-function and its
Mellin Transform are given in the next section.

2. The H-function

The H - function (see [11] ) is defined, as a contour complex integral by

Hm,n
p,q

[
z

∣∣∣∣ (a1, A), . . . , (an, An), (an+1, An+1), . . . , (ap, Ap)
(b1, B1), . . . , (bm, Bm), (bm+1, Bm+1), . . . , (bq, Bq)

]

=
1

2πi

∫
L

m∏
j=1

Γ(bj +Bjs)

n∏
j=1

Γ(1− aj −Ajs)

q∏
j=m+1

Γ(1− bj −Bjs)
p∏

j=n+1

Γ(aj +Ajs)

z−sds, (5)

where Aj and Bj are positive quantities and all the aj and bj may be complex. The
contour L runs from c − i∞ to c + i∞ such that the poles of Γ(bj + Bjs), j = 1, . . . ,m
lie to the left of L and the poles of Γ(1− aj −Ajs), j = 1, . . . , n lie to the right of L.

The Mellin transform of the H -function is given by

∫ ∞
0

xs−1Hm,n
p,q

[
cx

∣∣∣∣ (ap, Ap)
(bq, Bq)

]
dx =

c−s
m∏
j=1

Γ(bj +Bjs)

n∏
j=1

Γ(1− aj −Ajs)

q∏
j=m+1

Γ(1− bj −Bjs)
p∏

j=n+1

Γ(aj +Ajs)

z−s. (6)

In the next section, one shall proceed to obtain the p.d.f and c.d.f of the linear com-
bination of independent GLD random variables by means of the H-function.



L. C. de S. M. Ozelim, P. N. Rathie / Eur. J. Pure Appl. Math, 12 (3) (2019), 722-733 726

3. The Linear Combination of N GLDs

A brief description of the problem is presented and its solution is derived in this section.

3.1. Problem Statement

Let Xi ∼ GLD(µi, σi, αi, βi). Then, one seeks the probability density function of the
random variable:

Z =

N∑
i=1

biXi (7)

where bi, i = 1, ..., N are real numbers.
At first, one may obtain the p.d.f. of the linearly scaled distributions biXi. This can

be easily done by considering the Jacobian rule as presented in [17]. Thus, when Xi ∼
GLD(µi, σi, αi, βi), the transformed random variable biXi has a p.d.f. mathematically
represented as:

fbiXi(x;σi, µi, αi, βi, bi) =
1

|bi|σiB(αi, βi)

e
−βi(

x−biµi
biσi

)

[1 + e
−(x−biµi

biσi
)
]αi+βi

, ∀x ∈ R, (8)

This way, by using (7) and (8), the variable Z can be rewritten as the sum of the
random variables Yi, with Yi = biXi. To get a closed form exact probability density
function of the random variable Z, one shall first proceed to obtain of the characteristic
functions of the random variables Yi, i = 1, ..., N .

3.2. The Characteristic Function of Yi, i = 1, ..., N

The characteristic function of a given random variable can be obtained by calculating
the Fourier transform of its probability density function. The characteristic function of a
GLD random variable is widely known in the literature [12]. On the other hand, in order
to better understand the scaling process performed to generate the variable Yi = biXi, the
characteristic function of Yi shall be obtained explicitly. This way, by means of (8), the
characteristic function φi(t) (CF) for the random variables Yi, i = 1, ..., N is given by:

φi(t) =
1

σi|bi|B(αi, βi)

∞∫
−∞

ejtx
e
− (x−µibi)

σibi(
1 + e

− (x−µibi)
σibi

)αi+βi dx, (9)

where j =
√
−1.

On substituting r = e
− (x−µibi)

σibi /

(
1 + e

− (x−µibi)
σibi

)
, the integral in (9) becomes:



L. C. de S. M. Ozelim, P. N. Rathie / Eur. J. Pure Appl. Math, 12 (3) (2019), 722-733 727

φi(t) =
ejtbiµi

B(αi, βi)

1∫
0

rβi−jtσibi−1(1− r)αi+jtσibi−1dr. (10)

By considering the Beta function definition in (2), (10) becomes:

φi(t) =
ejtbiµi

B(αi, βi)
B(βi − jtσibi, αi + jtσibi). (11)

Since the characteristic function of the sum of independent random variables is the
product of the individual characteristic functions [17], the CF of the random variable Z,
φZ(t) , is given by:

φZ(t) =
N∏
i=1

[
ejtbiµi

B(αi, βi)
B(βi − jtσibi, αi + jtσibi)

]
. (12)

The probability density function of the random variable Z is obtained by the Fourier
transform as described in the next subsection.

3.3. The Probability Density Function of the Linear Combination of GLD
Variables

Being the CF of the random variable Z known, by means of the inversion formula
for Fourier transform, one shall get that the probability density function of Z, fZ(x).
By considering the alternative representation of the Beta function in terms of Gamma
functions presented in (2), fZ(x) can be expressed as:

fZ(x;σ, µ, α, β, b) =
1

2π

∞∫
−∞

e
−jt

(
x−

N∑
i=1

biµi

)
N∏
i=1

[
Γ(βi − jtσibi)Γ(αi + jtσibi)

Γ(αi)Γ(βi)

]
dt, (13)

where σ, µ, α,β and b represent the vectors of scale, mean and shape parameters and
coefficients, respectively.

It is possible to transform the real integral in (13) into a contour integral by the
variable change jt = s. In order to represent the equivalent contour integral in terms of
the H-function, one has to split the coefficients bi in positive and negative groups such
that bi ≥ 0, for 1 ≤ bi ≤ N1 and bi ≤ 0, for N1 + 1 ≤ bi ≤ N . This way, by means of
the transformed complex integral and the definition of the H-function in (5), (13) can be
rewritten as:

fZ(x;σ, µ, α, β, b) =
1∏N

i=1 Γ(αi)Γ(βi)
× (14)



L. C. de S. M. Ozelim, P. N. Rathie / Eur. J. Pure Appl. Math, 12 (3) (2019), 722-733 728

×HN,N
N,N

ex− N∑
i=1

biµi
∣∣∣∣ (1− β1, σ1b1), ..., (1− βN1 , σN1bN1), (1− αN1+1, σN1+1|bN1+1|), ..., (1− αN , σN |bN |)

(α1, σ1b1), ..., (αN1 , σN1bN1), (βN1+1, σN1+1|bN1+1|), ..., (βN , σN |bN |)

 .
Equation (14) provides a closed form exact representation for the probability density

function of the linear combination of GLD random variables,valid for σi, αi, βi > 0, µi ∈ R
and bi ∈ R, i = 1, ..., N . The cumulative distribution function is given in the next
subsection.

3.4. The Cumulative Distribution Function of the Linear Combination of
GLD Random Variables

The cumulative distribution function of the random variable Z, FZ , whose p.d.f. is in
(14), is given by:

FZ(x;σ, µ, α, β, b) =
1∏N

i=1 Γ(αi)Γ(βi)
× (15)

×
∫ x
−∞H

N,N
N,N

ex− N∑
i=1

biµi
∣∣∣∣ (1− β1, σ1b1), ..., (1− βN1 , σN1bN1), (1− αN1+1, σN1+1|bN1+1|), ..., (1− αN , σN |bN |)

(α1, σ1b1), ..., (αN1 , σN1bN1), (βN1+1, σN1+1|bN1+1|), ..., (βN , σN |bN |)

 dx.
Equation (16) is nothing but nested real and contour integrals. By interchanging the

order of integration, (16) becomes [11]:

FZ(x;σ, µ, α, β, b) =
1∏N

i=1 Γ(αi)Γ(βi)
× (16)

×HN,N+1
N+1,N+1

ex− N∑
i=1

biµi
∣∣∣∣ (1− β1, σ1b1), ..., (1− βN1 , σN1bN1), (1− αN1+1, σN1+1|bN1+1|), ..., (1− αN , σN |bN |), (1, 1)

(α1, σ1b1), ..., (αN1 , σN1bN1), (βN1+1, σN1+1|bN1+1|), ..., (βN , σN |bN |), (0, 1)

.
Expression (17) is valid for the same values of parameters as that of (14).

4. Reliability P (X < Y )

The reliability measure R = P (X < Y ) = P (X − Y < 0) is of great interest to both
pure and applied scientists. In the next sub-section, the exact value of R is provided in
terms of the H-function by considering the difference of two generalized logistic random
variables.

4.1. Exact Expression

Let X ∼ GLD(µ1, σ1, α1, β1) and Y ∼ GLD(µ2, σ2, α2, β2). Then, by means of (17),
R = P (X < Y ) = P (X − Y < 0) can be exactly given as:

R =
1

Γ(α1)Γ(α2)Γ(β1)Γ(β2)
H2,3

3,3

[
eµ2−µ1

∣∣∣∣ (1− β1, σ1), (1− α2, σ2), (1, 1)
(α1, σ1), (β2, σ2), (0, 1)

]
. (17)



L. C. de S. M. Ozelim, P. N. Rathie / Eur. J. Pure Appl. Math, 12 (3) (2019), 722-733 729

Even though R is exactly expressed in (17), a mathematical software such as Math-
ematica is used to evaluate the H-function, as shown subsequently in the applications
section. On the other hand, when out-of-computer quick calculations are needed, a sim-
pler expression in terms of elementary functions is of great interest. In the next subsection,
the series expansion of (17) is presented.

4.2. Series Expansion Expression

The H-function can be evaluated by means of the residue theorem [11]. This way, the
contour integral can be calculated by summing the residues over the poles of the function.
The series representation of the H-function may become even simpler when the poles of
the Gamma functions in the numerator of fraction inside the contour integral are simple
[11]. In order to guarantee that, the following restrictions should be applied to (17):

• α1σ2 − σ1β2 + σ2r − σ1w 6= 0 ∀r, w ∈ N

and

• β1σ2 − α2σ1 + σ2u− σ1v 6= 0 ∀u, v ∈ N

The restrictions above arise from considering that none of the poles of Γ(α1 + sσ1)
coincide with the poles of Γ(β2 + sσ2) and that none of the poles of Γ(β1 − sσ1) coincide
with the poles of Γ(α2 − sσ2). Thus, if the conditions above are satisfied, the reliability
for two GLDs can be expressed as:

R =
∞∑
n=0

e
(n+α1)(µ2−µ1)

σ1 (−1)nΓ(n+ α1 + β1)Γ(− (n+α1)σ2
σ1

+ β2)Γ( (n+α1)σ2
σ1

+ α2)

n!(n+ α1)Γ(α1)Γ(α2)Γ(β1)Γ(β2)
(18)

+
∞∑
n=0

e
(n+β2)(µ2−µ1)

σ2 (−1)nΓ(n+ α2 + β2)Γ(− (n+β2)σ1
σ2

+ α1)Γ( (n+β2)σ1σ2
+ β1)

n!(n+ β2)Γ(α1)Γ(α2)Γ(β1)Γ(β2)
,

when µ2 < µ1, and

R = 1−
∞∑
n=0

e
(n+β1)(µ1−µ2)

σ1 (−1)nΓ(n+ α1 + β1)Γ(− (n+β1)σ2
σ1

+ α2)Γ( (n+β1)σ2σ1
+ β2)

n!(n+ β1)Γ(α1)Γ(α2)Γ(β1)Γ(β2)
(19)

−
∞∑
n=0

e
(n+α2)(µ1−µ2)

σ2 (−1)nΓ(n+ α2 + β2)Γ( (n+α2)σ1
σ2

+ α1)Γ(− (n+α2)σ1
σ2

+ β1)

n!(n+ α2)Γ(α1)Γ(α2)Γ(β1)Γ(β2)
,

otherwise.



L. C. de S. M. Ozelim, P. N. Rathie / Eur. J. Pure Appl. Math, 12 (3) (2019), 722-733 730

5. Numerical Applications of the Results: Reliability of Generalized
Logistic Distributions

The formulas developed in the present paper are numerically evaluated in order to
show their applicability.

5.1. Reliability of the type R = P (X < Y )

The present paper provides both the exact and approximated formulas for evaluating
the reliability measure R when two generalized logistic distributions are considered. A set
of four generalized logistic random variables is considered to show the applicability of (17),
(18)and (19). Such variables are shown in Table 1 and graphically in Figure 1. Using the
generalized logistic variables considered in Table 1, the reliability measures are obtained
numerically by means of a computational code in Mathematica and given in Table 2.

Table 1: Logistic Random Variables Considered

Random Variable µ σ α β

X1 1.7 1.4 2.3 1.1

X2 0.3 0.8 3.1 1.4

X3 -1.3 2.1 0.9 2.3

X4 -0.5 1.7 2.5 0.7

-10 -5 0 5 10

0.0

0.1

0.2

0.3

0.4

x

D
e
n
s
it
y
fu
n
c
ti
o
n

Figure 1: Probability Density Functions of the Random Variables from Table 1 (X1 full, X2 dotted, X3

dashed and X4 dot-dashed).

The values of R estimated from random data have been obtained by the procedure
below:

• For the random variables X and Y generate random samples with 105 elements each,
xi and yi, i = 1, ..., 105.



L. C. de S. M. Ozelim, P. N. Rathie / Eur. J. Pure Appl. Math, 12 (3) (2019), 722-733 731

Table 2: Reliability Measures

R Random (17) (18) or (19) w/ (18) or (19) w/ (18) or (19) w/
Data 20 term 40 terms 60 terms

P (X1 < X2) 0.16764 0.16748 0.16795 0.16748 0.16748

P (X2 < X3) 0.04099 0.04080 0.04529 0.04080 0.04080

P (X3 < X4) 0.95795 0.95807 -43.1999 0.83806 0.95799

• Consider the indicator function I(x, y) = 1 − u(x − y), where u(x) = 0, x < 0 and

u(x) = 1, otherwise. The value of R is estimated by Re =
105∑
i=1

I(xi, yi);

• Repeat the above process 1000 times and then take the mean value of the Res
generated. These values are shown in Table 2.

It is worth noticing that the variance of the estimator used above was of order 10−7 for
all the cases. As can be seen from the analysis of Table 2, both the exact and approximate
expressions obtained in the present paper correctly model the random data generated.
Depending on the parameters of the distribution, the series presented may be slowly
convergent. On the other hand, as the series themselves are quite simple to implement,
taking more than 60 terms is not a hard task for any commercial computational software.

6. Conclusions

Skewness is present in most of the random data collected from nature. Even though
mathematically simple, generalized logistic models have shown to be useful tools to model
skewed data. The probability density and the cummulative distribution functions of the
linear combination of N independent and not identically distributed generalized logistic
random variables have been obtained in terms of the H-function.

The c.d.f. of the linear combination can be used to build reliability measures of the

type P (
N1∑
i=1

Xi <
N2∑
j=1

Yj) when Xi, i = 1, ..., N1 and Yj , j = 1, ..., N2 are generalized logistic

random variables of type IV with different location, scale and shape parameters. Besides,
a series expansion alternative expression has been derived for the case N1 = N2 = 1.

The applicability of the expressions developed has been illustrated by numerical exper-
iments, indicating a very good accordance between the exact and the estimated reliability
measures.

Acknowledgements

The authors acknowledge the support provided by the following institutions: The
Brazilian National Council for Scientific and Technological Development (CNPq Grant
151778/2018-3) and the University of Brasilia (UnB).



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