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PREFACE
First of all, we would like to tha k to all participa ts of The th “EAM“ UGM
5 I ter atio al
Co fere ce o Mathe atics a d Its Applicatio s which was held o August -21, 2015. The conference
is very important as a communication forum of the mathematicians, not only in Indonesia, but also in
Southeast Asia and surrounding areas. We also thank to the steering committee members and all of the
reviewers for all supports during the conference and the preparation of this proceedings.
As the scientific documentation of the conference, we provide two-types of proceedings. The first one is
the AIP Proceedings which contains the high quality paper selected by blind review process. The second
one is the regular proceedings, which contain the selected papers which are not published in the AIP
Proceedings and the paper of our invited speakers.
We would like to say thanks to all authors who have submitted the paper to our proceedings. During the
review process, we found that almost all papers has good quality. However due to the limitation number
of the paper which can be published in our proceedings, we should select the submitted paper based on
the reviewer recommendation and score. So, there are some papers which are not accepted to publish in
this proceedings, we apologize to the authors about this inconvenience.
Lastly, we would like to say thanks for all partners and all sponsors in supporting our conference.
Warm regards,
Dr. Fajar Adi Kusumo
Editor in Chief.
Proceedings of The 7th SEAMS UGM International Conference on Mathematics and Its Applications 2015
AIP Conf. Proc. 1707, 010001-1–010001-1; doi: 10.1063/1.4940800
© 2016 AIP Publishing LLC 978-0-7354-1354-2/$30.00
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List of Reviewers: The 7th SEAMS UGM International Conference on Mathematics and
its Applications 2015
Citation: AIP Conference Proceedings 1707, 010002 (2016); doi: 10.1063/1.4940801
View online: http://dx.doi.org/10.1063/1.4940801
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Photographs: Application of Mathematics in Technical and Natural Sciences 7th International Conference AMiTaNS’15
AIP Conf. Proc. 1684, 010004 (2015); 10.1063/1.4934281
List of Participants: Application of Mathematics in Technical and Natural Sciences 7th International Conference AMiTaNS’15
AIP Conf. Proc. 1684, 010003 (2015); 10.1063/1.4934280
Preface: Application of Mathematics in Technical and Natural Sciences 7th International Conference AMiTaNS’15
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LIST OF REVIEWERS
Abdurakhman
Agus Maman Abadi
Agus Suryanto
Andrei Sleptchenko
Ari Suparwanto
Atok Zulijanto
Budi Surodjo
Ch. Rini Indrati
Danardono
Dellavitha Nasution
Dhoriva Urwatul Wutsqa
Diari Indriati
Fajar Adi Kusumo
Fatmawati
Gunardi
Hasmawati
Hengki Tasman
Herni Utami
Herry Pribawanto Suryawan
I Nyoman Budiantara
I Wayan Sudarsana
Imam Solekhudin
Indah Emilia Wijayanti
Indarsih
Intan Muchtadi-Alamsyah
Irwan Endrayanto
Isnaini Rosyida
Khurul Wardati
L.H. Wiryanto
Lina Aryati
Manuharawati
Marcellinus Andy Rudhito
MHD. Reza M.I. Pulungan
Muhammad Wakhid Musthofa
Nor Haniza Sarmin
Nurdin
Nurhayadi
Nursanti Anggriani
Salmah
Sri Wahyuni
Sudi Mungkasi
Suhartono
Sumardi
Supama
Syafrizal Sy
Windarto
Yansen Marpaung
Yeni Susanti
Universitas Gadjah Mada
Universitas Negeri Yogyakarta
Universitas Brawijaya
Qatar University
Universitas Gadjah Mada
Universitas Gadjah Mada
Universitas Gadjah Mada
Universitas Gadjah Mada
Universitas Gadjah Mada
Institut Teknologi Bandung
Universitas Negeri Yogyakarta
Universitas Negeri Surakarta
Universitas Gadjah Mada
Universitas Airlangga
Universitas Gadjah Mada
Universitas Hasanuddin
Universitas Indonesia
Universitas Gadjah Mada
Universitas Sanata Dharma
Institut Teknologi Sepuluh Nopember
Universitas Tadulako
Universitas Gadjah Mada
Universitas Gadjah Mada
Universitas Gadjah Mada
Institut Teknologi Bandung
Universitas Gadjah Mada
Universitas Negeri Semarang
UIN Sunan Kalijaga
Institut Teknologi Bandung
Universitas Gadjah Mada
Universitas Negeri Surabaya
Universitas Sanata Dharma
Universitas Gadjah Mada
UIN Sunan Kalijaga
Universiti Teknologi Malaysia
Universitas Hasanuddin
Universitas Tadulako
Universitas Padjadjaran
Universitas Gadjah Mada
Universitas Gadjah Mada
Universitas Sanata Dharma
Institut Teknologi Sepuluh Nopember
Universitas Gadjah Mada
Universitas Gadjah Mada
Universitas Andalas
Universitas Airlangga
Universitas Sanata Dharma
Universitas Gadjah Mada
Proceedings of The 7th SEAMS UGM International Conference on Mathematics and Its Applications 2015
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The uniform continuity of characteristic function from convoluted exponential
distribution with stabilizer constant
Dodi Devianto
Citation: AIP Conference Proceedings 1707, 080006 (2016); doi: 10.1063/1.4940863
View online: http://dx.doi.org/10.1063/1.4940863
View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1707?ver=pdfcov
Published by the AIP Publishing
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AIP Conf. Proc. 1479, 1212 (2012); 10.1063/1.4756369
Convolution Approximation for the n‐Particle Distribution Function
J. Math. Phys. 11, 1912 (1970); 10.1063/1.1665343
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1
2
3
4
5
6
7
8
9
The Uniform Continuity of Characteristic Function from
Convoluted Exponential Distribution with
Stabilizer Constant
Dodi Devianto
Department of Mathematics, Faculty of Mathematics and Natural Sciences,
Andalas University, Limau Manis Campus, Padang 25163, INDONESIA
Email: ddevianto@fmipa.unand.ac.id
Abstract. It is constructed convolution of generated random variable from independent and identically exponential
distribution with stabilizer constant. The characteristic function of this distribution is obtained by using Laplace-Stieltjes
transform. The uniform continuity property of characteristic function from this convolution is obtained by using
analytical methods as basic properties.
Keywords: convolution, exponential distribution with stabilizer constant, characteristic function, uniform continuity.
PACS: 02.50.Ng
INTRODUCTION
The characteristic function is a complex valued function which defines as Laplace-Stieltjes transform. This
transformation and its properties have been introduced in formal term [1] and [4]. Furthermore, the property of
characteristic function for infinite divisibility distribution has established in some well known works of Levy and
Khinchine in the form of canonical representation.
This property of Laplace-Stieltjes transform as characteristic function can be used to find the distribution of
convoluted random variables which are many applications in other area such as convoluted exponential random
variables as service time of the quequeing or reliability model by [5]. In other hand [2] has introduced properties of
the characteristic function from geometric distribution as discrete version of the exponential distribution. Furthermore,
the mathematical method to obtain the convolution of exponential distribution and its variations have been introduced
in some works of [6], [7], [8].
The research on characteristic functions of various kinds of exponential distribution are always interested to set
their properties such as in canonical representation or its uniform continuity. Devianto et al. [3] have introduced
convolution of generating random variables from exponential distribution with stabilizer constant. It is worth to give
some properties of characteristic function from this new distribution. Now let us adopt the process of constructing this
new introduced distribution. Let X be a random variable from exponential distribution with parameter λ. Probability
density function for this random variable is given by
f ( x ; O ) O e Ox
(1)
for positive λ and all real number x. It is denoted W as random variable related to xi which way generated from
exponential distribution with values spread in form
wi
xi
xm
(2)
Proceedings of The 7th SEAMS UGM International Conference on Mathematics and Its Applications 2015
AIP Conf. Proc. 1707, 080006-1–080006-5; doi: 10.1063/1.4940863
© 2016 AIP Publishing LLC 978-0-7354-1354-2/$30.00
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where it is defined x m max {x1 , x 2 , ..., x n } for x i [ 0, x m ] and n, m Z . Then random variable W has probability
density function as follows
f (w ; O ) T O e Ow
(3)
for positive λ and 0 w d 1 where stabilizer constant is to maintain the nature of probability density function defined
as T
(1 (e O 1)) .
Devianto et al. [3] has shown the convolution of this new intoduced distribution by setting W1, W2, ..., Wn be n
independent and identically exponential distribution with stabilizer constant where probability density function is
defined as follows
f (wi ; O ) T O e Owi
(4)
for positive parameter λ, 0 wi d 1 and T 1 (exp (O ) 1) . Then the sum of random variables
S n W1 W2 ... Wn has probability density function
f (s n ; O ) T n
On
(n 1)!
sn
n 1
e O sn for 0 s n d n .
(5)
The properties of distribution from an exponential distribution with stabilizer constant and its convolution can be
given in the form of characteristic functions. This paper gives a brief explanation of these characteristic functions and
their properties of uniform continuity in Section 2 and Section 3.
THE PROPERTIES OF CHARACTERISTIC FUNCTION FROM GENERATING
RANDOM VARIABLE OF EXPONENTIAL DISTRIBUTION WITH STABILIZER
CONSTANT
The aim of this section is to present characteristic function of distribution from generating random variable of the
exponential distribution with stabilizer constant. Furthermore, it is given property of uniform continuity from this
characteristic function.
Theorem 1. Let W is generating random variable from exponential distribution with stabilier constant with probability
distribution function as in (3) that is
f (w ; O ) T O e Ow
for positive parameter λ and 0 w d 1 where stabilizer constant T
random variable W is
IW (t )
(1 (e O 1)) . Then characteristic function from
TO ( O it )
(e
1) for f t f .
O it
(6)
Proof. Characteristic function from random variable W is deriven by direct methods of Laplace-Stieltjes transform,
that is obtained as follows
IW (t ) E [e it W ]
³T O e
1
itw Ow
e
dw
(7)
0
TO ( O it )
(e
1) . ■
O it
Proposition 2. The Characteristic function of generating random variable from exponential distribution with stabilizer
constant is unifomly continuous.
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Proof. The uniform continuity of characteristic function from exponential distribution with stabilizer constant is
showing by define for every H ! 0 and G ! 0 such that
where G only depends on
| IW ( s ) IW (t ) | H for | s t | G
(8)
H . Now let us define a new function
e its e itw
.
st
\ ( w)
(9)
It is using Taylor expansion series to eisw and eitw then we have the new defined function can be rewritten as follows
·
§ ( s t ) w 2 ( s 3 s 2 t st 2 t 3 ) w 4
...¸¸ i
2
!
4
!
©
¹
\ ( w) ¨¨
·
§
( s 2 st t 2 ) w 3
¨w
...¸¸ .
¨
3
!
¹
©
(10)
We can establish uniform continuity of characteristic function by setting
| IW ( s) IW (t ) |
³T O e
1
e Ow dw ³ T O e itw e Ow dw
1
isw
0
(s t )
³T O e
1
0
0
Ow
(11)
\ ( w) dw ,
so that for every G ! H and | s t | o 0 then we have | IW ( s ) IW (t ) | o 0 . This confirms the characteristic function
from random variable W is uniform continuous. ■
0.5
0.35
0.25
0.30
0.4
0.20
0.25
0.3
0.20
0.15
0.15
0.2
0.10
0.10
0.1
0.05
0.05
0.15
O 0.2 and 0 d t d 50
0.10
0.05
0.05
0.10
0.05
O 1 and 0 d t d 50
0.05
0.10
O 4 and 0 d t d 50
0.1
0.2
0.3
0.4
FIGURE 1. Parametric curves of characteristic function from random variable W with various parameter λ and variable t
with 0 d T d 2S .
The parametric curves of characteristic function from random variable W from Fig. 1 show smooth line and never
vanish on the complex plane for all presented curves, this confirms the continuity of parametric curves by graphically.
THE PROPERTIES OF CHARACTERISTIC FUNCTION FROM CONVOLUTED
EXPONENTIAL DISTRIBUTION WITH STABILIZER CONSTANT
The aim of this section is to present characteristic function of distribution from convolution of generating random
variable of exponential distribution with stabilizer constant. Furthermore, it is given property of uniform continuity
from this characteristic function.
Theorem 3. Let Sn is generating random variable from exponential distribution with stabilier constant with probability
distribution function as in (5) that is
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f (s n ; O ) T n
On
(n 1)!
sn
n 1
for positive λ and 0 s n d n where stabilizer constant T
I S n (t ) E [e it S n ]
(1 (e O 1)) . Then characteristic function is
T n On
(n 1)! O it
e O sn for 0 s n d n .
*(n, n(O it )) for 0 t f .
n
(12)
Proof. Characteristic function is deriven by direct methods of Laplace-Stieltjes transform, that is obtained as follows
On
n 1
s n e O sn ds n
(n 1)!
its
n
³ e nT
I S n (t ) E [e itSn ]
n
0
s n (O it ) then we have
for f t f and 0 s n d n . Now, by using changing variable y
I S n (t ) T n
On
n O it
(n 1)! O it
1
n
³
(13)
y n 1 1e y dy
(14)
0
It is used the property of incomplete gamma function then characteristic function of random variable Sn can be written
as follows
T n On
*(n, n(O it )) for f t f ,
(n 1)!(O it ) n
I S n (t )
where it is defined
*(n, n(O it ))
³
(15)
n ( O it )
y n 1e y dy. ■
(16)
0
Proposition 4. The characteristic function of convoluted exponential distribution with stabilizer constant is unifomly
continuous.
Proof. The uniform continuity of characteristic function from exponential distribution with stabilizer constant is
showing by define for every H ! 0 and G ! 0 such that
| I Sn (s) M Sn (t ) | H for | s t | G
(17)
where G only depends on H . Now let us define a new function
\ (s n )
e its e itw
.
st
(18)
It is using Taylor expansion to eisw and eitw then we have the new defined function can be rewritten as follows
3
§ (s t ) s n 2 ( s 3 s 2 t st 2 t 3 ) s n 4
·
·
§
(s 2 st t 2 ) s n
...¸ i ¨ s n
...¸ .
\ (s n ) ¨
¨
©
2!
¸
¹
4!
¨
©
We can establish uniform continuity of characteristic function by setting
| I S n ( s ) I S n (t ) |
³e
n
0
issn
Tn
(s t )
On
(n 1)!
³T
n
0
n
sn
O
n 1
e O sn ds n ³ e itsn T n
n
0
n
(n 1)!
sn
n 1
e
O sn
¸
¹
3!
On
(n 1)!
\ ( s n ) ds n ,
sn
n 1
(19)
e O sn ds n
(20)
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so that for every G ! H and | s t | o 0 then we have |ISn (s) ISn (t ) | o 0 . This confirms the characteristic function
from random variable Sn is uniform continuous. ■
0.0006
40000
1.0
10 12
5.0
10 11
0.0004
20000
0.0002
0.0004
0.0002
0.0002
0.0004
0.0002
0.0006
40000
20000
20000
40000
1.5
10 12
1.0
10 12
20000
0.0004
5.0
10 11
5.0
5.0
10 11
1.0
10 12
10 11
1.0
10 12
1.5
10 12
40000
O
0.2 and 0 d t d 12
0.0006
O 1 and 0 d t d 6
O
4 and 0 d t d 4
1.5
10 12
FIGURE 2. Parametric curves of characteristic function from random variable Sn for 10-fold convolution with various
parameter λ and variabel t with 0 d T d 12S .
The parametric curves of characteristic function from random variable Sn from Fig. 2 show smooth line and never
vanish on the complex plane for all presented curves, this confirms the continuity of parametric curves by graphically.
CONCLUSION
The characteristic function of convolution of generated random variable from independent and identically
exponential distribution with stabilizer constant is contructed by using Laplace-Stieltjes transform. The uniform
continuity property of characteristic function from this convolution is obtained by using analytical methods in the
complex variable as basic properties. The characteristic function of this ditribution is governed by smooth line and
never vanish on the complex plane, this property confirms the uniform continuity of characteristic function by their
prametric curves.
REFERENCES
1. B. V. Gnedenko and A.N. Kolmogorov, Limit Distribution for Sums of Independent Random Variables, AddisonWiley, London, 1968.
2. D. Devianto and K. Takano, On necessary and sufficient conditions for convergence to the geometric distribution,
International Journal of Pure and Applied Mathematics 39, No. 2, 245-260 (2007).
3. D. Devianto, Maiyastri, L. Oktasari and M. Anas, Convolution of generated random variable from exponential
distribution with stabilizer constant, Applied Mathematical Sciences 9, No. 96, 4781-4789 (2015).
4. E. Lukacs, Characteristic Function, Hafner Publishing Company, London, 1970.
5. H. Jasiulewicz and W. Kordecki, Convolution of erlang and of pascal distributions with application to reliability,
Demonstratio Mathematica 36, 231-238 (2003).
6. K. Smaili, T. Kadri and S. Kadry, Hypoexponential distribution with different parameters, Applied mathematics
4, 624-631 (2013).
7. M. Akkouchi, On the convolution of exponential distributions, Journal of the Chungcheong Mathematical Society,
21, No. 4, 501-510 (2008).
8. N.-Y. Ma and F. Liu, A novel analytical scheme to compute the n-fold convolution of exponential-sum distribution
functions, Applied Mathematics and Computation 158, 225-235 (2004).
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Applications 2015
Citation: AIP Conference Proceedings 1707, 010001 (2016); doi: 10.1063/1.4940800
View online: http://dx.doi.org/10.1063/1.4940800
View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1707?ver=pdfcov
Published by the AIP Publishing
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List of Reviewers: The 7th SEAMS UGM International Conference on Mathematics and its Applications 2015
AIP Conf. Proc. 1707, 010002 (2016); 10.1063/1.4940801
Preface: Application of Mathematics in Technical and Natural Sciences 7th International Conference AMiTaNS’15
AIP Conf. Proc. 1684, 010001 (2015); 10.1063/1.4934278
Preface: International Conference on Advancements in Mathematical Sciences (AMS 2015)
AIP Conf. Proc. 1676, 010001 (2015); 10.1063/1.4930426
Preface: 39th International Conference "Applications of Mathematics in Engineering and Economics"
AIP Conf. Proc. 1570, 1 (2013); 10.1063/1.4854735
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PREFACE
First of all, we would like to tha k to all participa ts of The th “EAM“ UGM
5 I ter atio al
Co fere ce o Mathe atics a d Its Applicatio s which was held o August -21, 2015. The conference
is very important as a communication forum of the mathematicians, not only in Indonesia, but also in
Southeast Asia and surrounding areas. We also thank to the steering committee members and all of the
reviewers for all supports during the conference and the preparation of this proceedings.
As the scientific documentation of the conference, we provide two-types of proceedings. The first one is
the AIP Proceedings which contains the high quality paper selected by blind review process. The second
one is the regular proceedings, which contain the selected papers which are not published in the AIP
Proceedings and the paper of our invited speakers.
We would like to say thanks to all authors who have submitted the paper to our proceedings. During the
review process, we found that almost all papers has good quality. However due to the limitation number
of the paper which can be published in our proceedings, we should select the submitted paper based on
the reviewer recommendation and score. So, there are some papers which are not accepted to publish in
this proceedings, we apologize to the authors about this inconvenience.
Lastly, we would like to say thanks for all partners and all sponsors in supporting our conference.
Warm regards,
Dr. Fajar Adi Kusumo
Editor in Chief.
Proceedings of The 7th SEAMS UGM International Conference on Mathematics and Its Applications 2015
AIP Conf. Proc. 1707, 010001-1–010001-1; doi: 10.1063/1.4940800
© 2016 AIP Publishing LLC 978-0-7354-1354-2/$30.00
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is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
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List of Reviewers: The 7th SEAMS UGM International Conference on Mathematics and
its Applications 2015
Citation: AIP Conference Proceedings 1707, 010002 (2016); doi: 10.1063/1.4940801
View online: http://dx.doi.org/10.1063/1.4940801
View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1707?ver=pdfcov
Published by the AIP Publishing
Articles you may be interested in
Preface: The 7th SEAMS UGM International Conference on Mathematics and its Applications 2015
AIP Conf. Proc. 1707, 010001 (2016); 10.1063/1.4940800
Photographs: Application of Mathematics in Technical and Natural Sciences 7th International Conference AMiTaNS’15
AIP Conf. Proc. 1684, 010004 (2015); 10.1063/1.4934281
List of Participants: Application of Mathematics in Technical and Natural Sciences 7th International Conference AMiTaNS’15
AIP Conf. Proc. 1684, 010003 (2015); 10.1063/1.4934280
Preface: Application of Mathematics in Technical and Natural Sciences 7th International Conference AMiTaNS’15
AIP Conf. Proc. 1684, 010001 (2015); 10.1063/1.4934278
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180.250.36.4 On: Thu, 10 Mar 2016 02:53:27
LIST OF REVIEWERS
Abdurakhman
Agus Maman Abadi
Agus Suryanto
Andrei Sleptchenko
Ari Suparwanto
Atok Zulijanto
Budi Surodjo
Ch. Rini Indrati
Danardono
Dellavitha Nasution
Dhoriva Urwatul Wutsqa
Diari Indriati
Fajar Adi Kusumo
Fatmawati
Gunardi
Hasmawati
Hengki Tasman
Herni Utami
Herry Pribawanto Suryawan
I Nyoman Budiantara
I Wayan Sudarsana
Imam Solekhudin
Indah Emilia Wijayanti
Indarsih
Intan Muchtadi-Alamsyah
Irwan Endrayanto
Isnaini Rosyida
Khurul Wardati
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Proceedings of The 7th SEAMS UGM International Conference on Mathematics and Its Applications 2015
AIP Conf. Proc. 1707, 010002-1–010002-1; doi: 10.1063/1.4940801
© 2016 AIP Publishing LLC 978-0-7354-1354-2/$30.00
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The uniform continuity of characteristic function from convoluted exponential
distribution with stabilizer constant
Dodi Devianto
Citation: AIP Conference Proceedings 1707, 080006 (2016); doi: 10.1063/1.4940863
View online: http://dx.doi.org/10.1063/1.4940863
View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1707?ver=pdfcov
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The Uniform Continuity of Characteristic Function from
Convoluted Exponential Distribution with
Stabilizer Constant
Dodi Devianto
Department of Mathematics, Faculty of Mathematics and Natural Sciences,
Andalas University, Limau Manis Campus, Padang 25163, INDONESIA
Email: ddevianto@fmipa.unand.ac.id
Abstract. It is constructed convolution of generated random variable from independent and identically exponential
distribution with stabilizer constant. The characteristic function of this distribution is obtained by using Laplace-Stieltjes
transform. The uniform continuity property of characteristic function from this convolution is obtained by using
analytical methods as basic properties.
Keywords: convolution, exponential distribution with stabilizer constant, characteristic function, uniform continuity.
PACS: 02.50.Ng
INTRODUCTION
The characteristic function is a complex valued function which defines as Laplace-Stieltjes transform. This
transformation and its properties have been introduced in formal term [1] and [4]. Furthermore, the property of
characteristic function for infinite divisibility distribution has established in some well known works of Levy and
Khinchine in the form of canonical representation.
This property of Laplace-Stieltjes transform as characteristic function can be used to find the distribution of
convoluted random variables which are many applications in other area such as convoluted exponential random
variables as service time of the quequeing or reliability model by [5]. In other hand [2] has introduced properties of
the characteristic function from geometric distribution as discrete version of the exponential distribution. Furthermore,
the mathematical method to obtain the convolution of exponential distribution and its variations have been introduced
in some works of [6], [7], [8].
The research on characteristic functions of various kinds of exponential distribution are always interested to set
their properties such as in canonical representation or its uniform continuity. Devianto et al. [3] have introduced
convolution of generating random variables from exponential distribution with stabilizer constant. It is worth to give
some properties of characteristic function from this new distribution. Now let us adopt the process of constructing this
new introduced distribution. Let X be a random variable from exponential distribution with parameter λ. Probability
density function for this random variable is given by
f ( x ; O ) O e Ox
(1)
for positive λ and all real number x. It is denoted W as random variable related to xi which way generated from
exponential distribution with values spread in form
wi
xi
xm
(2)
Proceedings of The 7th SEAMS UGM International Conference on Mathematics and Its Applications 2015
AIP Conf. Proc. 1707, 080006-1–080006-5; doi: 10.1063/1.4940863
© 2016 AIP Publishing LLC 978-0-7354-1354-2/$30.00
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where it is defined x m max {x1 , x 2 , ..., x n } for x i [ 0, x m ] and n, m Z . Then random variable W has probability
density function as follows
f (w ; O ) T O e Ow
(3)
for positive λ and 0 w d 1 where stabilizer constant is to maintain the nature of probability density function defined
as T
(1 (e O 1)) .
Devianto et al. [3] has shown the convolution of this new intoduced distribution by setting W1, W2, ..., Wn be n
independent and identically exponential distribution with stabilizer constant where probability density function is
defined as follows
f (wi ; O ) T O e Owi
(4)
for positive parameter λ, 0 wi d 1 and T 1 (exp (O ) 1) . Then the sum of random variables
S n W1 W2 ... Wn has probability density function
f (s n ; O ) T n
On
(n 1)!
sn
n 1
e O sn for 0 s n d n .
(5)
The properties of distribution from an exponential distribution with stabilizer constant and its convolution can be
given in the form of characteristic functions. This paper gives a brief explanation of these characteristic functions and
their properties of uniform continuity in Section 2 and Section 3.
THE PROPERTIES OF CHARACTERISTIC FUNCTION FROM GENERATING
RANDOM VARIABLE OF EXPONENTIAL DISTRIBUTION WITH STABILIZER
CONSTANT
The aim of this section is to present characteristic function of distribution from generating random variable of the
exponential distribution with stabilizer constant. Furthermore, it is given property of uniform continuity from this
characteristic function.
Theorem 1. Let W is generating random variable from exponential distribution with stabilier constant with probability
distribution function as in (3) that is
f (w ; O ) T O e Ow
for positive parameter λ and 0 w d 1 where stabilizer constant T
random variable W is
IW (t )
(1 (e O 1)) . Then characteristic function from
TO ( O it )
(e
1) for f t f .
O it
(6)
Proof. Characteristic function from random variable W is deriven by direct methods of Laplace-Stieltjes transform,
that is obtained as follows
IW (t ) E [e it W ]
³T O e
1
itw Ow
e
dw
(7)
0
TO ( O it )
(e
1) . ■
O it
Proposition 2. The Characteristic function of generating random variable from exponential distribution with stabilizer
constant is unifomly continuous.
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Proof. The uniform continuity of characteristic function from exponential distribution with stabilizer constant is
showing by define for every H ! 0 and G ! 0 such that
where G only depends on
| IW ( s ) IW (t ) | H for | s t | G
(8)
H . Now let us define a new function
e its e itw
.
st
\ ( w)
(9)
It is using Taylor expansion series to eisw and eitw then we have the new defined function can be rewritten as follows
·
§ ( s t ) w 2 ( s 3 s 2 t st 2 t 3 ) w 4
...¸¸ i
2
!
4
!
©
¹
\ ( w) ¨¨
·
§
( s 2 st t 2 ) w 3
¨w
...¸¸ .
¨
3
!
¹
©
(10)
We can establish uniform continuity of characteristic function by setting
| IW ( s) IW (t ) |
³T O e
1
e Ow dw ³ T O e itw e Ow dw
1
isw
0
(s t )
³T O e
1
0
0
Ow
(11)
\ ( w) dw ,
so that for every G ! H and | s t | o 0 then we have | IW ( s ) IW (t ) | o 0 . This confirms the characteristic function
from random variable W is uniform continuous. ■
0.5
0.35
0.25
0.30
0.4
0.20
0.25
0.3
0.20
0.15
0.15
0.2
0.10
0.10
0.1
0.05
0.05
0.15
O 0.2 and 0 d t d 50
0.10
0.05
0.05
0.10
0.05
O 1 and 0 d t d 50
0.05
0.10
O 4 and 0 d t d 50
0.1
0.2
0.3
0.4
FIGURE 1. Parametric curves of characteristic function from random variable W with various parameter λ and variable t
with 0 d T d 2S .
The parametric curves of characteristic function from random variable W from Fig. 1 show smooth line and never
vanish on the complex plane for all presented curves, this confirms the continuity of parametric curves by graphically.
THE PROPERTIES OF CHARACTERISTIC FUNCTION FROM CONVOLUTED
EXPONENTIAL DISTRIBUTION WITH STABILIZER CONSTANT
The aim of this section is to present characteristic function of distribution from convolution of generating random
variable of exponential distribution with stabilizer constant. Furthermore, it is given property of uniform continuity
from this characteristic function.
Theorem 3. Let Sn is generating random variable from exponential distribution with stabilier constant with probability
distribution function as in (5) that is
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f (s n ; O ) T n
On
(n 1)!
sn
n 1
for positive λ and 0 s n d n where stabilizer constant T
I S n (t ) E [e it S n ]
(1 (e O 1)) . Then characteristic function is
T n On
(n 1)! O it
e O sn for 0 s n d n .
*(n, n(O it )) for 0 t f .
n
(12)
Proof. Characteristic function is deriven by direct methods of Laplace-Stieltjes transform, that is obtained as follows
On
n 1
s n e O sn ds n
(n 1)!
its
n
³ e nT
I S n (t ) E [e itSn ]
n
0
s n (O it ) then we have
for f t f and 0 s n d n . Now, by using changing variable y
I S n (t ) T n
On
n O it
(n 1)! O it
1
n
³
(13)
y n 1 1e y dy
(14)
0
It is used the property of incomplete gamma function then characteristic function of random variable Sn can be written
as follows
T n On
*(n, n(O it )) for f t f ,
(n 1)!(O it ) n
I S n (t )
where it is defined
*(n, n(O it ))
³
(15)
n ( O it )
y n 1e y dy. ■
(16)
0
Proposition 4. The characteristic function of convoluted exponential distribution with stabilizer constant is unifomly
continuous.
Proof. The uniform continuity of characteristic function from exponential distribution with stabilizer constant is
showing by define for every H ! 0 and G ! 0 such that
| I Sn (s) M Sn (t ) | H for | s t | G
(17)
where G only depends on H . Now let us define a new function
\ (s n )
e its e itw
.
st
(18)
It is using Taylor expansion to eisw and eitw then we have the new defined function can be rewritten as follows
3
§ (s t ) s n 2 ( s 3 s 2 t st 2 t 3 ) s n 4
·
·
§
(s 2 st t 2 ) s n
...¸ i ¨ s n
...¸ .
\ (s n ) ¨
¨
©
2!
¸
¹
4!
¨
©
We can establish uniform continuity of characteristic function by setting
| I S n ( s ) I S n (t ) |
³e
n
0
issn
Tn
(s t )
On
(n 1)!
³T
n
0
n
sn
O
n 1
e O sn ds n ³ e itsn T n
n
0
n
(n 1)!
sn
n 1
e
O sn
¸
¹
3!
On
(n 1)!
\ ( s n ) ds n ,
sn
n 1
(19)
e O sn ds n
(20)
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so that for every G ! H and | s t | o 0 then we have |ISn (s) ISn (t ) | o 0 . This confirms the characteristic function
from random variable Sn is uniform continuous. ■
0.0006
40000
1.0
10 12
5.0
10 11
0.0004
20000
0.0002
0.0004
0.0002
0.0002
0.0004
0.0002
0.0006
40000
20000
20000
40000
1.5
10 12
1.0
10 12
20000
0.0004
5.0
10 11
5.0
5.0
10 11
1.0
10 12
10 11
1.0
10 12
1.5
10 12
40000
O
0.2 and 0 d t d 12
0.0006
O 1 and 0 d t d 6
O
4 and 0 d t d 4
1.5
10 12
FIGURE 2. Parametric curves of characteristic function from random variable Sn for 10-fold convolution with various
parameter λ and variabel t with 0 d T d 12S .
The parametric curves of characteristic function from random variable Sn from Fig. 2 show smooth line and never
vanish on the complex plane for all presented curves, this confirms the continuity of parametric curves by graphically.
CONCLUSION
The characteristic function of convolution of generated random variable from independent and identically
exponential distribution with stabilizer constant is contructed by using Laplace-Stieltjes transform. The uniform
continuity property of characteristic function from this convolution is obtained by using analytical methods in the
complex variable as basic properties. The characteristic function of this ditribution is governed by smooth line and
never vanish on the complex plane, this property confirms the uniform continuity of characteristic function by their
prametric curves.
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distribution with stabilizer constant, Applied Mathematical Sciences 9, No. 96, 4781-4789 (2015).
4. E. Lukacs, Characteristic Function, Hafner Publishing Company, London, 1970.
5. H. Jasiulewicz and W. Kordecki, Convolution of erlang and of pascal distributions with application to reliability,
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