Advances in Mathematical Physics
Volume 2009 (2009), Article ID 206176, 13 pages
Research Article

Generalized Probability Functions

1Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto, Universidade de São Paulo, Avenida Bandeirantes, 3900, 14040-901 Ribeirão Preto, SP, Brazil
2National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil
3Centro Universitário Barão de Mauá, Rua Ramos de Azevedo, 423, 14090-180 Ribeirão Preto, SP, Brazil

Received 12 June 2009; Revised 17 September 2009; Accepted 8 October 2009

Academic Editor: Giorgio Kaniadakis

Copyright © 2009 Alexandre Souto Martinez et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


From the integration of nonsymmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. Motivated by the mathematical curiosity, we show that these generalized functions are suitable to generalize some probability density functions (pdfs). A very reliable rank distribution can be conveniently described by the generalized exponential function. Finally, we turn the attention to the generalization of one- and two-tail stretched exponential functions. We obtain, as particular cases, the generalized error function, the Zipf-Mandelbrot pdf, the generalized Gaussian and Laplace pdf. Their cumulative functions and moments were also obtained analytically.