Advertisement

Normal Approximations and the Central Limit Theorem

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

Abstract

Many of the special discrete and special continuous distributions that we have discussed can be well approximated by a normal distribution for suitable configurations of their underlying parameters. Typically, the normal approximation works well when the parameter values are such that the skewness of the distribution is small. For example, binomial distributions are well approximated by a normal distribution when n is large and p is not too small or too large. Gamma distributions are well approximated by a normal distribution when the shape parameter α is large. Whenever we see a certain phenomenon empirically all too often, we might expect that there is a unifyingmathematical result there, and in this case indeed there is. The unifyingmathematical result is one of the most important results in all of mathematics and is called the central limit theorem. The subject of central limit theorems is incredibly diverse. In this chapter, we present the basic or the canonical central limit theorem and its applications to certain problems with which we are already familiar. Among numerous excellent references on central limit theorems, we recommend Feller (1968, 1971) and Pitman (1992) for lucid expositions and examples. The subject of central limit theorems also has a really interesting history; we recommend Le Cam (1986) and Stigler (1986) in this area. Careful and comprehensive mathematical treatments are available in Hall (1992) and Bhattacharya and Rao (1986). For a diverse selection of examples, see DasGupta (2008).

Keywords

Central Limit Theorem Normal Approximation Continuity Correction Binomial Probability Fair Coin 
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. Abramowitz, M. and Stegun, I. (1970). Handbook of Mathematical Functions, Dover, New York.Google Scholar
  2. Bhattacharya, R. and Rao, R. (1986). Normal Approximation and Asymptotic Expansions, Wiley, New York.MATHGoogle Scholar
  3. Charlier, C. (1931). Applications de la théorie des probabilités á l’astronomie, Gauthier-Villars, Paris.MATHGoogle Scholar
  4. DasGupta, A. (2008). Asymptotic Theory of Statistics and Probability, Springer, New York.MATHGoogle Scholar
  5. Edgeworth, F. (1904). The law of error, Trans. Cambridge Philos. Soc., 20, 36–65, 113–141.Google Scholar
  6. Feller, W. (1968). Introduction to Probability Theory and Its Applications, Vol. I, Wiley, New York.Google Scholar
  7. Feller, W. (1971). Introduction to Probability Theory and Its Applications, Vol. II, Wiley, New York.MATHGoogle Scholar
  8. Hall, P. (1992). The Bootstrap and Edgeworth Expansion, Springer, New York.Google Scholar
  9. Le Cam, L. (1986). The central limit theorem around 1935, Statist. Sci., 1, 78–91.CrossRefMathSciNetGoogle Scholar
  10. Patel, J. and Read, C. (1996). Handbook of the Normal Distribution, Marcel Dekker, New York.MATHGoogle Scholar
  11. Pitman, J. (1992). Probability, Springer, New York.Google Scholar
  12. Stigler, S. (1986). History of Statistics: Measurement of Uncertainty before 1900, Harvard University press, Cambridge, MA.MATHGoogle Scholar

Copyright information

© Springer-Verlag New York 2010

Authors and Affiliations

  1. 1.Dept. Statistics & MathematicsPurdue UniversityWest LafayetteUSA

Personalised recommendations