¹Universidad del Valle, Facultad de Ciencias, Departamento de Matemáticas, Cali, Colombia. Lecturer. Email: paula.bran@gmail.com
²Universidad de Antioquía, Facultad de Ciencias Naturales y Exactas, Departamento de Matemáticas, Medellín, Colombia. Lecturer. Email: jmoc03@gmail.com
³Universidad de Antioquía, Facultad de Ciencias Naturales y Exactas, Departamento de Matemáticas, Medellín, Colombia. Professor. Email: dayaknagar@yahoo.com

In this article, we study several properties such as marginal and conditional distributions, joint moments, and mixture representation of the bivariate generalization of the Kummer-Beta distribution. To show the behavior of the density function, we give some graphs of the density for different values of the parameters. Finally, we derive the exact and approximate distribution of the product of two random variables which are distributed jointly as bivariate Kummer-Beta. The exact distribution of the product is derived as an infinite series involving Gauss hypergeometric function, whereas the beta distribution has been used as an approximate distribution. Further, to show the closeness of the approximation, we have compared the exact distribution and the approximate distribution by using several graphs. An application of the results derived in this article is provided to visibility data from Colombia.

Key words: Beta distribution, Bivariate distribution, Dirichlet distribution, Hypergeometric function, Moments, Transformation.

En este artículo, definimos la función de densidad de la generalización bivariada de la distribución Kummer-Beta. Estudiamos algunas de sus propiedades y casos particulares, así como las distribuciones marginales y condicionales. Para ilustrar el comportamiento de la función de densidad, mostramos algunos gráficos para diferentes valores de los parámetros. Finalmente, encontramos la distribución del producto de dos variables cuya distribución conjunta es Kummer-Beta bivariada y utilizamos la distribución beta como una aproximación. Además, con el fin de comparar la distribución exacta y la aproximada de este producto, mostramos algunos gráficos. Se presenta una aplicación a datos climáticos sobre niebla y neblina de Colombia.

Palabras clave: distribución Beta, distribución bivariada, distribución Dirichlet, función hipergeométrica, momentos, transformación.

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