SAINSTEK: JURNAL SAINS DAN TEKNOLOGI Publisher: AMSET IAIN Batusangkar and IAIN Batusangkar Press Website: http://ecampus.iainbatusangkar.ac.id/ojs/index.php/sainstek E-mail: sainstek@iainbatusangkar.ac.id June 2021 Vol 13 No 1 ISSN: 2085-8019 (p) ISSN: 2580-278X (e) pp : 58-65 Sainstek: Jurnal Sains dan Teknologi 58 Vol 13 No 1, June 2021 ISSN: 2085-8019 (p), ISSN: 2580-278x (e) The Characterization of Infinite Divisibility of Compound Negative Binomial Distribution as the Sum of Exponential Distribution Anis Nur Afifah 1 , Maiyastri 1 , Dodi Devianto 1* 1 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Andalas University, Padang 25163, West Sumatra Province, INDONESIA *Email: ddevianto@sci.unand.ac.id Article History Received: 8 January 2021 Revised: 9 February 2021 Accepted: 6 June 2021 Published: 30 June 2021 Key Words Compound distribution; Characteristic function; Infinitely divisible distributions; Negative binomial distribution; Exponential distribution. Abstract The sum of random variables that are identical and independent from an exponential distribution creates the compound distribution. It is called compound negative binomial distribution as the sum of exponential distribution when the number of random variables added follows the negative binomial distribution. This compound distribution’s characteristic function is established by using mathematical analysis methods, included its uniform continuity property. The characteristic function's parametric curves never disappear from the complex plane, which means it is a positively defined function. Another characteristic function's property shows that this compound distribution is one of infinitely divisible distribution. INTRODUCTION The sum of random variables that are identical and independent where the number of random variables added follows a negative binomial distribution forms as the compound negative binomial distribution. These identical and independent random variables added follow an exponential distribution, it is called compound negative binomial distribution with the sum of exponential distribution. For an instant, suppose the random variable N has negative binomial distribution and X1, X2,…, XN are identical and independent random variables from an exponential distribution. We denote the sum of these random variables by N N X X X S     ... 2 1 (1) as the compound distribution. (Furman, 2007), who delivered the convolution of some independent random variables as the sum of negative binomial distribution and its properties, introduced the previous research on compound distribution. The negative binomial-exponential model, as introduced by (Panjer & Willmot, 1981) is one of compound negative binomial distribution where its convolution is evaluated as finite sums. Furthermore, (Z. Wang, 2011) applied one mixed negative binomial distribution in the insurance. We have found another application of compound negative binomial distribution in many fields, such as it has done by (Rono et al., 2020) in modeling natural disaster in Kenya. (Girondot, 2017) used the convolution of negative binomial distribution to optimize the turtle research sampling design. (Omair et al., 2018) carried another application out by utilizing this distribution to develop a new bivariate model that is suitable for accident data. This compound distribution has better theoretical development from statistical mathematics views, as explained by previous authors and its applications. (Chen & Guisong, 2017) derived the exact distribution using the generating function and the convolution, while (X. Wang et al., 2019) studied the characters of