### The Characterization of Infinite Divisibility of Compound Negative Binomial Distribution as the Sum of Exponential Distribution

Anis Nur Afifah, Maiyastri Maiyastri, Dodi Devianto

#### 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.

#### Keywords

Compound distribution; Characteristic function; Infinitely divisible distributions; Negative binomial distribution; Exponential distribution.

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DOI: http://dx.doi.org/10.31958/js.v13i1.2529

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