Research Article http:sciencetechindonesia.com Science & Technology Indonesia p-ISSN: 2580-4405 e-ISSN: 2580-4391 Sci. Technol. Indonesia 1 (2018) 49-52 © 2018 The Authors. Production and hosting by ARTS Publishing in association with Indonesian Science and Technology Society. This is an open access article under the CC-BY-NC-SA license. Article History: Received 1 December 2017; revised 11 January 2018; accepted 11 January 2018 http://doi.org/10.26554/sti.2018.3.1.49-52 The Definite Positive Property of Characteristic Function from Compound Geometric Distribution as The Sum of Gamma Distribution Darvi Mailisa Putri1, Maiyastri2, and Dodi Devianto3* 1,2,3Department of Mathematics, Faculty of Mathematics and Natural Sciences, Andalas Univerrsity, Padang 25163, West Sumatra Province, Indonesia *Corresponding Author E-mail: ddevianto@fmipa.unand.ac.id ABSTRACT In this expository article it is given characterization of compound geometric distribution as the sum of gamma distribution. The char- acterization of this compound distribution is obtained by using the property of characteristic function as the Fourier-Stieltjes transform. The property of definite positive of characteristice function from compound geometric distribution as the sum of gamma distribution is exposed by analytical methods as the quadratic form of characteristic function. Keyword: compound geometric distribution, gamma distribution, characteristic function, definite positive. 1. INTRODUCTION Let us define S as the sum of independent and identically random variables, that is in the form S = x i i=1 N ∑ (1.1.) where the number of random variable Xi , that is N, itself a ran- dom variable. The property of this random variable has been paid great attention to many years. The most popular terminology for the distribution generated by such as a sum of random variables is called compound distribution. Compound distributions arise from many applied probabili- ty models and from insurance risk models in particular. One of the compound distributions that attracted the researcher was the compound geometric distribution. The compound geometric dis- tribution, as a special case of the compound negative binomial distribution plays a vital role in analysis of ruin probabilities and related problems in risk theory and insurance as in Willmot and Lin (2001). The most recently theoretical approach of compound geometric distribution has started by introducing on higher-order properties of compound geometric distributions (Willmot, 2002), this development is continued on the study of convolutions of compound geometric distributions (Psarrakos, 2009). Further- more, it has reviewed convolution on generated random variable from exponential distribution with stabilizer constant (Devianto et. al, 2015; Devianto, 2016) and some properties of hypoexpo- nential distribution with stabilizer constant (Devianto et. al, 2015). While in the last development of compound geometric distribu- tion was delivered by Koutras and Erylmaz (2016) when they stud- ied compound geometric distributions of order k . In the present paper, we study the property of compound ge- ometric distribution as the sum of gamma distribution. It will be discussed characterization of compound geometric distribution as the sum of gamma distribution in the form of definite positive property of characteristic function. In Section 2, we will explain about characterization of gamma distribution and compound geometric distribution. While the some properties of compound geometric distribution as the sum of gamma distribution will be given in Section 3. 2. THE CHARACTERIZATION OF GAMMA DISTRI- BUTION AND COMPOUND GEOMETRIC DISTRIBU- TION The gamma distribution is a family of continuous probability distri- bution with parameter α dan β. Let X be a random variable from gamma distribution with the form f(x)= 1 βαΓ(α) xα−1exp[− xβ ] (2.1.) for 0>x , α > 0, and β > 0. The expectation and variance of gamma distribution are given by E(x)=αβ (2.2) Putri et al. 2018 / Science & technology Indonesia 3 (1) 2018: 49-52 50 Var(X)=αβ2 (2.3) As for the moment generating function and characteristic function of the gamma distribution we can obtain by using defi- nition of Fourier-Stieltjes transform refers to Lukacs (2009) in the following M X (t)= E[exp(tX)]= 1 1− βt ⎛ ⎝⎜ ⎞ ⎠⎟ α (2.4) φ X (t)= E[exp(itX)]= 1 1− βit ⎛ ⎝⎜ ⎞ ⎠⎟ α (2.5) The Figure 1 is the graph of parametric curves of characteristic function from gamma distribution with various parameter α. While the Figure 2 is the graph of parametric curves of characteristic func- tion from gamma distribution with various parameter β. The graph of these parametric curves of characteristic function are described in the complex plane that shows a smooth line, this is to confirm that its characteristic function is continuous and never vanish on the complex plane. The compound geometric distribution is the sum of inde- pendent and identically random variable, where the number of random variables has geometric distribution as follows P(N = n)= p(1− p)n (2.6) for ...,2,1,0=n and 10 << p , p is the probability of suc- cess. The geometric distribution has expectation and variance as follows E(N)= 1− p p (2.7) Var(N)= 1− p p2 (2.8) By using similar way with Equation (2.4) and (2.5), we have the moment generating function and characteristic function of the geometric distribution respectively as follows M N (t)= 1 1−(1− p)exp[t] (2.9) M N (t)= p 1−(1− p)exp[it] (2.10) The Figure 3 is the graph of parametric curves of character- istic function from geometric distribution with various parameter p. The graph of its characteristic function is described in the com- plex plane that shows a smooth line, this is also to confirm that this characteristic function is continuous and never vanish on the complex plane. 3. THE SOME PROPERTIES OF COMPOUND GE- OMETRIC DISTRIBUTION AS THE SUM OF GAMMA DISTRIBUTION The property of compound geometric distribution as the sum of gamma distribution will be stated on its definite positive of char- acteristic function. In order of the main result in this section, it is explained the property definite positive of characteristic function in Proposition 3.4, at the first we introduce the moment generat- ing function, characteristic function and continuity property at the following propositions. Proposition 3.1. Let the random variable S = xi i=1 N ∑ is defined as a com- pound geometric distribution as the sum of gamma distribution with parameter (p,α ,β ), then moment generating function is obtained as follows M S (t)= p 1−(1− p)(1− βt)−α (3.1) Proof. By using definition of conditional expectation and its line- arity property then it is obtained the moment generating function as follows M S (t)= E exp t X i i=1 N ∑ ⎛ ⎝⎜ ⎞ ⎠⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ = E E exp t Xi i=1 N ∑ ⎛ ⎝⎜ ⎞ ⎠⎟ N ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ (3.2) 001.0 005.0 009.0 5 15 30 1.0p 5.0p 9.0p 35.0p 65.0p 95.0p Figure 4. Parametric curves of characteristic function from compound geometric distribution as thesum of gamma distribution with various parameters 9.0,7.0,5.0 where 009.0 0.2 0.4 0.6 0.8 1.0 0.6 0.4 0.2 0.2 0.4 0.6 0.2 0.4 0.6 0.8 1.0 0.6 0.4 0.2 0.2 0.4 0.6 0.2 0.4 0.6 0.8 1.0 0.6 0.4 0.2 0.2 0.4 0.6 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 Figure 1. Parametric curves of characteristic function from gamma distribution with various parameters α = 0.5, 0.7, 0.9 001.0 005.0 009.0 5 15 30 1.0p 5.0p 9.0p 35.0p 65.0p 95.0p Figure 4. Parametric curves of characteristic function from compound geometric distribution as thesum of gamma distribution with various parameters 9.0,7.0,5.0 where 009.0 0.2 0.4 0.6 0.8 1.0 0.6 0.4 0.2 0.2 0.4 0.6 0.2 0.4 0.6 0.8 1.0 0.6 0.4 0.2 0.2 0.4 0.6 0.2 0.4 0.6 0.8 1.0 0.6 0.4 0.2 0.2 0.4 0.6 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 Figure 2. Parametric curves of characteristic function from gamma distribution with various parameters β = 0.5, 0.7, 0.9 001.0 005.0 009.0 5 15 30 1.0p 5.0p 9.0p 35.0p 65.0p 95.0p Figure 4. Parametric curves of characteristic function from compound geometric distribution as thesum of gamma distribution with various parameters 9.0,7.0,5.0 where 009.0 0.2 0.4 0.6 0.8 1.0 0.6 0.4 0.2 0.2 0.4 0.6 0.2 0.4 0.6 0.8 1.0 0.6 0.4 0.2 0.2 0.4 0.6 0.2 0.4 0.6 0.8 1.0 0.6 0.4 0.2 0.2 0.4 0.6 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 Figure 3. Parametric curves of characteristic function from geometric distribution with various parameters p = 0.1, 0.5, 0.9 Putri et al. 2018 / Science & technology Indonesia 3 (1) 2018: 49-52 51 Now, let us use the definition of moment generating function then we have M S (t)= E(M X (t)N )= M N (ln(M X (t))) (3.3) Base on the Equation (2.4) and (2.9) and we substitute to Equation (3.3), then we have ( )( ) . 111 )( ab ---- = tp p tMS Proposition 3.2. Let the random variable is defined as a compound geometric distribution as the sum of gamma distribution with parameter , then ),,( bap characteristic function is obtained as follows ( )( ) . 111 )( ab f ---- = itp p tS (3.4) Proof. By using definition of conditional expectation and its lin- earity property then we can write the characteristic function as follows ÷ ÷ ø ö ç ç è æ ÷ ÷ ø ö ç ç è æ ÷÷ ø ö çç è æ =÷ ÷ ø ö ç ç è æ ÷÷ ø ö çç è æ = åå == NXitEEXitEt N i i N i iS 11 expexp)(f (3.5) Now, let us use the definition of moment generating function and definition of characteristic function to write )))((ln())(()( tMtEt XN N XS fff == (3.6) Base on the Equation (2.4) and (2.5) then we have ( )( ) . 111 )( ab f ---- = itp p tS Proposition 3.3. Let the random variable å = = N i iXS 1 is defined as a com- pound geometric distribution as the sum of gamma distribution with parameter ),,( bap and characteristic function ( )( ) ,)( ab f ---- = itp p tS 111 then )(tSf is continuous. Proof. The continuity property is explained by using definition of uniform continuity, that is for every 0>e there exists 0>d such that eff <- )()( 21 tt SS for d<- 21 tt where d depends only on e . Then the uniform continuity of character- istic function is obtained by the following way. First we write the following equation ( )( ) ( )( ) ( ) ( ) . )(11 1 )(11 1 111111 )()( 21 21 21 tptp p itp p itp p tt XX SS ff bb ff aa -- - -- = --- - --- =- -- (3.7) Now let us define 21 tth -= , so that for 0®h we have the following limit .0 )()1(1 1 )()1(1 1 220 = -- - +--® tpthp Lim XXh ff (3.8) This hold for ed < where eff <-+ )()( 22 tth SS for d<- 21 tt . Then )(tSf is uniformly continuous. Proposition 3.4. Let the random variable S = x i i=1 N ∑ is defined as a compound geometric distribution as the sum of gamma distribution with pa- rameter (p, α , β ) and characteristic function ( )( ) ,)( ab f ---- = itp p tS 111 then fs(t) is positively defined function with quadratic form 0)( 1 1 ³-å å ££ ££nj ljXlj nl ttcc f (3.9) for any complex number nccc ...,,, 21 and real number nttt ...,,, 21 . Proof. We will show the characteristic function )(tSf is positive function that is to satisfy the quadratic form 0)( 1 1 ³-å å ££ ££nj ljXlj nl ttcc f for any complex numbers nccc ...,,, 21 and real numbers nttt ...,,, 21 . It is used definition of characteristic function of compound geometric distribution as the sum of gamma distribu- tion and a geometric series with ratio (1 - p)E(exp(i(tj-tl)X) where p ∈(0.1) and 1))(((exp £- XttiE lj , hence we have the follow- ing equation ( ) å åå å å å åå å ££ ¥ =££ ££ ££ ££ - ££££ ££ ÷÷ ø ö çç è æ --= --- = ---- =- nj m m ljlj nl nj lj lj nl nj lj lj nlnlj ljSlj nl XttiEpccp XttiEp p cc ttip p ccttcc 1 01 1 1 1 1,1 1 ))(((exp)1( )))(((exp)1(1 )(1)1(1 )( ab f (3.10) 001.0 005.0 009.0 5 15 30 1.0p 5.0p 9.0p 35.0p 65.0p 95.0p Figure 4. Parametric curves of characteristic function from compound geometric distribution as thesum of gamma distribution with various parameters 9.0,7.0,5.0 where 009.0 0.2 0.4 0.6 0.8 1.0 0.6 0.4 0.2 0.2 0.4 0.6 0.2 0.4 0.6 0.8 1.0 0.6 0.4 0.2 0.2 0.4 0.6 0.2 0.4 0.6 0.8 1.0 0.6 0.4 0.2 0.2 0.4 0.6 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 1.0 0.5 0.5 1.0 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 Figure 4. Parametric curves of characteristic function from com- pound geometric distribution as thesum of gamma distribution with various parameters α = 0.5, 0.7, 0.9 where β = 0.009 Putri et al. 2018 / Science & technology Indonesia 3 (1) 2018: 49-52 52 We rewrite the Equation (3.10) by using its complex conjugate property and exponential function property as the follows ÷ ÷ ø ö ç ç è æ ÷÷ ø ö çç è æ -´ ÷ ÷ ø ö ç ç è æ ÷÷ ø ö çç è æ -= - ååå å å å ¥ =££££ ¥ = ££ ££ m m ll nlnj m m jj nlj ljSlj nl XtiEpcXtiEpcp ttcc 011 0 ,1 1 )((exp)1()((exp)1( )(f (3.11) Next, base on the quadratic form property of modulus complex number, then we obtain ( ) 01)1( )((exp)1()( 2 1 0 2 1 0,1 1 ³÷÷ ø ö çç è æ --= ÷÷ ø ö çç è æ -=- å å å åå å ££ ¥ = - ££ ¥ =££ ££ nj m m jj nj m m jj nlj ljSlj nl itpcp XtiEpcpttcc ab f (3.12) Then it is proved that fs (t) as positively defined function where the quadratic form has nonnegative values. The graph of parametric curves of characteristic function from compound geometric distribution as the sum of gamma dis- tribution will be presented in Figure 4, Figure 5, and Figure 6 as follows Based on the graph presented in Figures 4, 5, and 6, it can be concluded that the characteristic function of compound geomet- ric distribution as the sum of gamma distribution is continuous, as well as it has discussed on Proposition 3.3 . In Figure 4 and 6, the shape of parametric curves form three smooth lines. While in Fig- ure 5 the shape of parametric curves only form the same smooth line for various parameter β, and tends to be one for all various parameter α dan β when fixed on parameter p. 4. CONCLUSION The compound geometric distribution as the sum of gamma dis- tribution is the sum of independent and identically random var- iable from gamma distribution, where the number of these ran- dom variables has geometric distribution. The characterization of compound geometric distribution as the sum of gamma distribu- tion is obtained by using the property of characteristic function, where the characteristic function is defined as the Forier-Stieltjes transform. The property of definite positive of characteristice function from compound geometric distribution as the sum of gamma distribution is exposed by showing its quadratic form of characteristic function. The addition property of this compound geometric distribution as the sum of gamma distribution is con- inuity peroperty of characteristic function, where the graph of the parametric curves of characteristic function is in the smooth line and never vanish on the complex plane. REFERENCES Devianto, D., (2016). The uniform continuity of characteristic function from convoluted exponential distribution with stabi- lizer constant. AIP Conference Proceedings, American Insti- tute of Physics. Devianto, D., Maiyastri, Oktasari, L. & Anas, M. (2015). Convolu- tion on generated random variable from exponential distribu- tion with stabilizer constant. Applied Mathematical Sciences, 9(96), 4781-4789. Devianto, D., Oktasari, L., & Maiyastri. (2015). Some properties of hypoexponential distribution with stabilizer constant. Ap- plied Mathematical Sciences, 9(142), 7063-7070. Koutras, M.V., & Eryilmaz, S. (2016). Compound geometric dis- tribution of order k. Methodol. Comput. Appl. Proba, 19(2), 377-393. Lukacs, E. (1970). Characteristic function (2nd edition). London: Griffin. Psarrakos, G. (2009). A note on convolutions of compound ge- ometric distributions. Statistics & Probability letters, 79(9), 1231-1237. Willmot, G.E. (2002). On higher-order properties of compound geometric distributions. Applied Probability, 39(2), 324-340. Willmot, G.E., & Lin, X.S. (2001). Lunberg approximations for compound distributions with insurance applications. New York, NY: Springer. 5.0 7.0 9.0 Figure 5. Parametric curves of characteristic function from compound geometric distribution as thesum of gamma distribution with various parameters 09.0,07.0,05.0 where 5.0p 5.0 7.0 9.0 Figure 6. Parametric curves of characteristic function from compound geometric distribution as thesum of gamma distribution with various parameters 95.0,65.0,35.0p where 7.0 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 Figure 5. Parametric curves of characteristic function from compound geometric distribution as thesum of gamma distribu- tion with various parameters β = 0.05, 0.07, 0.09 where p = 0.5 5.0 7.0 9.0 Figure 5. Parametric curves of characteristic function from compound geometric distribution as thesum of gamma distribution with various parameters 09.0,07.0,05.0 where 5.0p 5.0 7.0 9.0 Figure 6. Parametric curves of characteristic function from compound geometric distribution as thesum of gamma distribution with various parameters 95.0,65.0,35.0p where 7.0 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 0.2 0.4 0.6 0.8 1.0 0.4 0.2 0.2 0.4 Figure 6. Parametric curves of characteristic function from compound geometric distribution as thesum of gamma distribu- tion