Advertisement

Urn Models in Physics and Genetics

  • Anirban DasGupta
Chapter
Part of the Springer Texts in Statistics book series (STS)

Abstract

Urn models conceptualize general allocation problems in which we distribute, withdraw, and redistribute certain objects or units into a specified number of categories. We think of the categories as urns and the objects as balls. Depending on the specific urn model, the balls may be of different colors and distinguishable or indistinguishable. Urn models are special because they can be successfully used to model real phenomena in diverse areas such as physics, ecology, genetics, economics, clinical trials, modeling of networks, and many others. The aim is to understand the evolution of the content of the urns as distribution and redistribution according to some prespecified scheme progresses. There are many urn models in probability, and the allocation scheme depends on exactly which model one wishes to study. We introduce and provide basic information on some key urn models in this chapter. Classic references are Feller (1968) and Johnson and Kotz (1977). Bernoulli (1713) and Whitworth (1901) are two historically important monographs on urn models. More recent references include Gani (2004), Lange (2003), and Ivchenko and Medvedev (1997). Other specific references are given in the various sections of this chapter.

Keywords

Exact Distribution White Ball Factorial Moment Stirling Number Allele Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Balding, D., Bishop, M., and Cannings, C. (2007). Handbook of Statistical Genetics, third ed., Wiley, New York.MATHGoogle Scholar
  2. Barbour, A., Holst, L., and Janson, S. (1992). Poisson Approximation, Clarendon Press, Oxford.MATHGoogle Scholar
  3. Bernoulli, J. (1713). Ars Conjectandi, translated by Edith Sylla, 2005, Johns Hopkins University Press, Baltimore.Google Scholar
  4. DasGupta, A. (2008). Asymptotic Theory of Statistics and Probability, Springer, New York.MATHGoogle Scholar
  5. de Finetti, B. (1931). Funzione caratteristica di un fenomeno aleatorio, Atti R. Accad. Naz. Lincei, Ser. 6, Mem. Cl. Sci. Fis. Mat. Nat., 4, 251–299.Google Scholar
  6. Diaconis, P. (1988). Recent progress on de Finetti’s notions of exchangeability, in Bayesian Statistics, Vol. 3, J. Bernardo ed. 111–125, Oxford University Press, New York.Google Scholar
  7. Donnelly, P. and Tavaré, S. (1986). The ages of alleles and a coalescent, Adv. Appl. Prob., 18, 1–19.MATHCrossRefGoogle Scholar
  8. Ewens, W. (1972). The sampling theory of selectively neutral alleles, Theort. Pop. Biol., 3, 87–112.CrossRefMathSciNetGoogle Scholar
  9. Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Wiley, New York.MATHGoogle Scholar
  10. Gani, J. (2004). Random allocation and urn models, J. Appl. Prob., 41A, 313–320.MATHCrossRefMathSciNetGoogle Scholar
  11. Hoppe, F. (1984). Pólya-like urns and the Ewens sampling formula, J. Math. Biol., 20, 91–94.MATHCrossRefMathSciNetGoogle Scholar
  12. Ivchenko, G. and Medvedev, Y. (1997). The Contribution of the Russian Mathematicians to the Study of Urn Models, VSP, Utrecht.Google Scholar
  13. Johnson, N. and Kotz, S. (1977). Urn Models and Their Application, Wiley, New York.MATHGoogle Scholar
  14. Kolchin, V., Sevast’yanov, B., and Chistyakov, V. (1978). Random Allocations, V. H. Winston & Sons, Washington, DC.Google Scholar
  15. Lange, K. (2003). Applied Probability, Springer, New York.MATHGoogle Scholar
  16. Rao, C.R. and Shanbhag, D. (2001). Exchangeability, functional equations, and characterizations, in Stochastic Processes: Theory and Methods, Shanbhag, D. and Rao, C.R. eds. 733–763, North-Holland, Amsterdam.Google Scholar
  17. Regazzini, E. (1987). On the origins of the concept of exchangeability in probability and statistics, a reminiscence of Bruno de Finetti, Rend. Semin. Mat. Fis. Milano, 57, 261–273.MATHCrossRefMathSciNetGoogle Scholar
  18. Tomescu, I. (1985). Problems in Combinatorics and Graph Theory, Wiley Interscience, New York.MATHGoogle Scholar
  19. Whitworth, W. (1901). Choice and Chance, Hafner, New York.Google Scholar

Copyright information

© Springer-Verlag New York 2010

Authors and Affiliations

  1. 1.Dept. Statistics & MathematicsPurdue UniversityWest LafayetteUSA

Personalised recommendations