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Statistics and Analysis of Scientific Data pp 35–54Cite as

Three Fundamental Distributions: Binomial, Gaussian, and Poisson

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Abstract

There are three distributions that play a fundamental role in statistics. The binomial distribution describes the number of positive outcomes in binary experiments, and it is the “mother” distribution from which the other two distributions can be obtained. The Gaussian distribution can be considered as a special case of the binomial, when the number of tries is sufficiently large. For this reason, the Gaussian distribution applies to a large number of variables, and it is referred to as the normal distribution. The Poisson distribution applies to counting experiments, and it can be obtained as the limit of the binomial distribution when the probability of success is small.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables. Dover, New York (1970)

    MATH  Google Scholar 

  2. Akritas, M.G., Bershady, M.A.: Linear regression for astronomical data with measurement errors and intrinsic scatter. Astrophys. J. 470, 706 (1996). doi:10.1086/177901

    Article  ADS  Google Scholar 

  3. Bayes, T., Price, R.: An essay towards solving a problem in the doctrine of chances. Philos. Trans. R. Soc. Lond. 53 (1763)

    Google Scholar 

  4. Bonamente, M., Swartz, D.A., Weisskopf, M.C., Murray, S.S.: Swift XRT observations of the possible dark galaxy VIRGOHI 21. Astrophys. J. Lett. 686, L71–L74 (2008). doi:10.1086/592819

    Article  ADS  Google Scholar 

  5. Bonamente, M., Hasler, N., Bulbul, E., Carlstrom, J.E., Culverhouse, T.L., Gralla, M., Greer, C., Hawkins, D., Hennessy, R., Joy, M., Kolodziejczak, J., Lamb, J.W., Landry, D., Leitch, E.M., Marrone, D.P., Miller, A., Mroczkowski, T., Muchovej, S., Plagge, T., Pryke, C., Sharp, M., Woody, D.: Comparison of pressure profiles of massive relaxed galaxy clusters using the Sunyaev–Zel’dovich and X-ray data. New. J. Phys. 14 (2), 025010 (2012). doi:10.1088/1367-2630/14/2/025010

    Article  ADS  Google Scholar 

  6. Brooks, S.P., Gelman, A.: General methods for monitoring convergence of iterative simulations. J. Comput. Graph. Stat. 7, 434–455 (1998)

    MathSciNet  Google Scholar 

  7. Bulmer, M.G.: Principles of Statistics. Dover, New York (1967)

    MATH  Google Scholar 

  8. Carlin, B., Gelfand, A., Smith, A.: Hierarchical Bayesian analysis for changepoint problems. Appl. Stat. 41, 389–405 (1992)

    Article  MATH  Google Scholar 

  9. Cash, W.: Parameter estimation in astronomy through application of the likelihood ratio. Astrophys. J. 228, 939–947 (1979)

    Article  ADS  Google Scholar 

  10. Cowan, G.: Statistical Data Analysis. Oxford University Press, Oxford (1998)

    Google Scholar 

  11. Cramer, H.: Mathematical Methods of Statistics. Princeton University Press, Princeton (1946)

    MATH  Google Scholar 

  12. Emslie, A.G., Massone, A.M.: Bayesian confidence limits of electron spectra obtained through regularized inversion of solar hard X-ray spectra. Astrophys. J. 759, 122 (2012)

    Article  ADS  Google Scholar 

  13. Fisher, R.A.: On a distribution yielding the error functions of several well known statistics. Proc. Int. Congr. Math. 2, 805–813 (1924)

    Google Scholar 

  14. Fisher, R.A.: The use of multiple measurements in taxonomic problems. Ann. Eugenics 7, 179–188 (1936)

    Article  Google Scholar 

  15. Gamerman, D.: Markov Chain Monte Carlo. Chapman and Hall/CRC, London/New York (1997)

    MATH  Google Scholar 

  16. Gehrels, N.: Confidence limits for small numbers of events in astrophysical data. Astrophys. J. 303, 336–346 (1986). doi:10.1086/164079

    Article  ADS  Google Scholar 

  17. Gelman, A., Rubin, D.: Inference from iterative simulation using multiple sequences. Stat. Sci. 7, 457–511 (1992)

    Article  Google Scholar 

  18. Gosset, W.S.: The probable error of a mean. Biometrika 6, 1–25 (1908)

    Article  Google Scholar 

  19. Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 (1), 97–109 (1970). doi:10.1093/biomet/57.1.97. http://biomet.oxfordjournals.org/content/57/1/97.abstract

    Article  MathSciNet  MATH  Google Scholar 

  20. Helmert, F.R.: Die genauigkeit der formel von peters zur berechnung des wahrscheinlichen fehlers director beobachtungen gleicher genauigkeit. Astron. Nachr. 88, 192–218 (1876)

    Article  Google Scholar 

  21. Hubble, E., Humason, M.: The velocity-distance relation among extra-galactic nebulae. Astrophys. J. 74, 43 (1931)

    Article  ADS  Google Scholar 

  22. Isobe, T., Feigelson, Isobe, T., Feigelson, E.D., Akritas, M.G., Babu, G.J.: Linear regression in astronomy. Astrophys. J. 364, 104 (1990)

    Google Scholar 

  23. Jeffreys, H.: Theory of Probability. Oxford University Press, London (1939)

    MATH  Google Scholar 

  24. Kelly, B.C.: Some aspects of measurement error in linear regression of astronomical data. Astrophys. J. 665, 1489–1506 (2007). doi: 10.1086/519947

    Article  ADS  Google Scholar 

  25. Kolmogorov, A.: Sulla determinazione empirica di una legge di distribuzione. Giornale dell’ Istituto Italiano degli Attuari 4, 1–11 (1933)

    Google Scholar 

  26. Kolmogorov, A.N.: Foundations of the Theory of Probability. Chelsea, New York (1950)

    MATH  Google Scholar 

  27. Lampton, M., Margon, B., Bowyer, S.: Parameter estimation in X-ray astronomy. Astrophys. J. 208, 177–190 (1976). doi:10.1086/154592

    Article  ADS  Google Scholar 

  28. Lewis, S.: gibbsit. http://lib.stat.cmu.edu/S/gibbsit

  29. Madsen, R.W., Moeschberger, M.L.: Introductory Statistics for Business and Economics. Prentice-Hall, Englewood Cliffs (1983)

    Google Scholar 

  30. Marsaglia, G., Tsang, W., Wang, J.: Evaluating Kolmogorov’s distribution. J. Stat. Softw. 8, 1–4 (2003)

    Google Scholar 

  31. Mendel, G.: Versuche über plflanzenhybriden (experiments in plant hybridization). Verhandlungen des naturforschenden Vereines in Brünn, pp. 3–47 (1865)

    Google Scholar 

  32. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953). doi: 10.1063/1.1699114

    Article  ADS  Google Scholar 

  33. Pearson, K., Lee, A.: On the laws on inheritance in men. Biometrika 2, 357–462 (1903)

    Article  Google Scholar 

  34. Plummer, M., Best, N., Cowles, K., Vines, K.: CODA: convergence diagnosis and output analysis for MCMC. R News 6 (1), 7–11 (2006). http://CRAN.R-project.org/doc/Rnews/

    Google Scholar 

  35. Press, W., Teukolski, S., Vetterling, W., Flannery, B.: Numerical Recipes 3rd Edition: The Art of Scientific Computing. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  36. Protassov, R., van Dyk, D.A., Connors, A., Kashyap, V.L., Siemiginowska, A.: Statistics, handle with care: detecting multiple model components with the likelihood ratio test. Astrophys. J. 571, 545–559 (2002). doi:1086/339856

    Google Scholar 

  37. Raftery, A., Lewis, S.: How many iterations in the gibbs sampler? Bayesian Stat. 4, 763–773 (1992)

    Google Scholar 

  38. Ross, S.M.: Introduction to Probability Models. Academic, San Diego (2003)

    MATH  Google Scholar 

  39. Thomson, J.J.: Cathode rays. Philos. Mag. 44, 293 (1897)

    Article  Google Scholar 

  40. Tremaine, S., Gebhardt, K., Bender, R., Bower, G., Dressler, A., Faber, S.M., Filippenko, A.V., Green, R., Grillmair, C., Ho, L.C., Kormendy, J., Lauer, T.R., Magorrian, J., Pinkney, J., Richstone, D.: The slope of the black hole mass versus velocity dispersion correlation. Astrophys. J. 574, 740–753 (2002). doi:10.1086/341002

    Article  ADS  Google Scholar 

  41. Von Eye, A., Schuster, C.: Regression Analysis for Social Sciences. Academic, New York (1998)

    Google Scholar 

  42. Wilks, S.S.: Mathematical Statistics. Princeton University Press, Princeton (1943)

    MATH  Google Scholar 

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Authors and Affiliations

  1. University of Alabama, Huntsville, Alabama, USA

    Massimiliano Bonamente

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  1. Massimiliano Bonamente
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Bonamente, M. (2017). Three Fundamental Distributions: Binomial, Gaussian, and Poisson. In: Statistics and Analysis of Scientific Data. Graduate Texts in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-6572-4_3

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  • DOI: https://doi.org/10.1007/978-1-4939-6572-4_3

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