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Generalized Probability Functions

Author

Listed:
  • Alexandre Souto Martinez
  • Rodrigo Silva González
  • César Augusto Sangaletti Terçariol

Abstract

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.

Suggested Citation

  • Alexandre Souto Martinez & Rodrigo Silva González & César Augusto Sangaletti Terçariol, 2009. "Generalized Probability Functions," Advances in Mathematical Physics, John Wiley & Sons, vol. 2009(1).
    • Handle: RePEc:wly:jnlamp:v:2009:y:2009:i:1:n:206176
    DOI: 10.1155/2009/206176
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    References listed on IDEAS

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    1. Martinez, Alexandre Souto & González, Rodrigo Silva & Terçariol, César Augusto Sangaletti, 2008. "Continuous growth models in terms of generalized logarithm and exponential functions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(23), pages 5679-5687.
    2. Takahashi, Taiki, 2007. "A comparison of intertemporal choices for oneself versus someone else based on Tsallis’ statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 385(2), pages 637-644.
    3. Cajueiro, Daniel O., 2006. "A note on the relevance of the q-exponential function in the context of intertemporal choices," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 364(C), pages 385-388.
    4. C. Anteneodo & C. Tsallis & A. S. Martinez, 2001. "Risk aversion in economic transactions," Papers cond-mat/0109203, arXiv.org, revised Jun 2002.
    5. Souto Martinez, Alexandre & Silva González, Rodrigo & Lauri Espíndola, Aquino, 2009. "Generalized exponential function and discrete growth models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 388(14), pages 2922-2930.
    6. Alexandre Souto Martinez & Rodrigo Silva Gonzalez & Cesar Augusto Sangaletti Tercariol, 2008. "Continuous growth models in terms of generalized logarithm and exponential functions," Papers 0803.2635, arXiv.org, revised May 2008.
    7. Naumis, G.G. & Cocho, G., 2008. "Tail universalities in rank distributions as an algebraic problem: The beta-like function," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(1), pages 84-96.
    8. Borges, Ernesto P., 2004. "A possible deformed algebra and calculus inspired in nonextensive thermostatistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 340(1), pages 95-101.
    9. Takahashi, Taiki & Oono, Hidemi & Radford, Mark H.B., 2008. "Psychophysics of time perception and intertemporal choice models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(8), pages 2066-2074.
    10. Takahashi, Taiki, 2007. "A probabilistic choice model based on Tsallis’ statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 386(1), pages 335-338.
    11. Kalogeropoulos, Nikos, 2005. "Algebra and calculus for Tsallis thermo-statistics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 356(2), pages 408-418.
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    1. Destefano, Natália & Martinez, Alexandre Souto, 2011. "The additive property of the inconsistency degree in intertemporal decision making through the generalization of psychophysical laws," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(10), pages 1763-1772.

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