Una falacia en probabilidad ilustrada vía teoría de cópulas
A fallacy in probability illustrated via copula theory
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Abstract (en)
In basic probability courses, addressing the issue of random vectors shows that the marginal distributions of such vectors can be obtained in a unique way from the joint distribution. The reciprocal of this affirmation does not necessarily have. This article of informative type, tries, by means of a counterexample taken from Embrechts et al. (2002), and that makes use of the theory of copulas, to illustrate the fallacy: “ Marginal distributions and correlation determine the joint distribution ”.
Keywords:
correlation coefficient; joint distribution; marginal distribution; bivariate normal distribution; copulation theory; random vectors.
Abstract (es)
En cursos basicos de probabilidad, al abordar el tema de vectores aleatorios se demuestra que las distribuciones marginales de tales vectores pueden obtenerse de manera unica a partir de la distribucion conjunta. El recproco de esta armacion no necesariamente se tiene. Se construye aqu un contraejemplo haciendo uso de la teora de copulas que prentende ilustrar la falacia: "Distribuciones marginales y correla- cion determinan la distribucion conjunta".
Palabras clave:
coeficiente de correlación; distribución conjunta; distribución marginal; distribución normal bivariada; teoría de cópulas; vectores aleatorios.
References
Blanco, L., Arunachalam, V., and Dharmaraja, D. (2012). Introduction to Probability and Stochastic Processes with Applications. John Wiley & Sons, Inc., USA.
De la Peña, V., Ibragimov, R., and Sharakhmetov, S. (2006). Characterizations of joint distributions,copulas, information, dependence and decoupling, with applications to time series. Optimality. Institute of Mathematical Statistics, pages 183-209.
Embrechts, P., McNeil, A., and Straumann, D. (2002). Correlation and dependence in risk management: properties and pitfalls. Risk management: value at risk and beyond, pages 176-223.
Marimon, J. (2016). Una falacia sobre dependencia entre variables aleatorias ilustrada con teoría de copulas. Trabajo de grado Matemático, Bogota, Universidad Distrital Francisco Jose de Caldas.
McNeil, A., Frey, R., and Embrechts, P. (2005). Quantitative Risk Management: Concepts, techniques and tools. Princeton university press, New Jersey.
Nelsen, R. (2006). An introduction to copulas. Springer, USA.
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