Linking the Negative Binomial and Logarithmic Series Distributions via their Associated Series

SADINLE, MAURICIO

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Revista Colombiana de Estadística

Print version ISSN 0120-1751

Rev.Colomb.Estad. vol.31 no.2 Bogotá July./Dec. 2008

Linking the Negative Binomial and Logarithmic Series Distributions via their Associated Series

Relacionando las distribuciones binomial negativa\\ y logarítmica vía sus series asociadas

MAURICIO SADINLE¹

¹Universidad Nacional de Colombia, Facultad de Ciencias, Departamento de Estadística, Bogotá, Colombia. Student. Email: msadinleg@unal.edu.co

Abstract

The negative binomial distribution is associated to the series obtained by taking derivatives of the logarithmic series. Conversely, the logarithmic series distribution is associated to the series found by integrating the series associated to the negative binomial distribution. The parameter of the number of failures of the negative binomial distribution is the number of derivatives needed to obtain the negative binomial series from the logarithmic series. The reasoning in this article could be used as an alternative method to prove that the probability mass function of the negative binomial distribution sums to one. Finally, an interpretation of the logarithmic series distribution is given by using the presented reasoning.

Key words: Convergent series, Logarithmic series distribution, Negative binomial distribution, Power series distributions.

Resumen

La distribución binomial negativa está asociada a la serie obtenida de derivar la serie logarítmica. Recíprocamente, la distribución logarítmica está asociada a la serie obtenida de integrar la serie asociada a la distribución binomial negativa. El parámetro del número de fallas de la distribución binomial negativa es el número de derivadas necesarias para obtener la serie binomial negativa de la serie logarítmica. El razonamiento presentado puede emplearse como un método alternativo para probar que la función de masa de probabilidad de la distribución binomial negativa suma uno. Finalmente, se presenta una interpretación de la distribución logarítmica usando el razonamiento planteado.

Palabras clave: distribución binomial negativa, distribución de series de potencias, distribución logarítmica, series convergentes.

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References

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[Recibido en marzo de 2008. Aceptado en octubre de 2008]

Este artículo se puede citar en LaTeX utilizando la siguiente referencia bibliográfica de BibTeX:

@ARTICLE{RCEv31n2a11,    

     AUTHOR  = {Sadinle, Mauricio},    

     TITLE   = {{Linking the Negative Binomial and Logarithmic Series Distributions via their Associated Series}},    

     JOURNAL = {Revista Colombiana de Estadística},    

    YEAR    = {2008},    

    volume  = {31},    

    number  = {2},    

    pages   = {311-319}    

}