Abstract
Empirical data on many types of variables across disciplines tend to exhibit unimodality and only a small amount of skewness. It is quite common to use a normal distribution as a model for such data. The normal distribution occupies the central place among all distributions in probability and statistics. When a new methodology is presented, it is usually first tested on the normal distribution. The most well-known procedures in the toolbox of a statistician have their exact inferential optimality properties when sample values come from a normal distribution. There is also the central limit theorem, which says that the sum of many small independent quantities approximately follows a normal distribution. Theoreticians sometimes think that empirical data are often approximately normal, while empiricists think that theory shows that many types of variables are approximately normally distributed. By a combination of reputation, convenience, mathematical justification, empirical experience, and habit, the normal distribution has become the most ubiquitous of all distributions. It is also the most studied; we know more theoretical properties of the normal distribution than of others. It satisfies intriguing and elegant characterizing properties not satisfied by any other distribution. Because of its clearly unique position and its continuing importance in every emerging problem, we discuss the normal distribution exclusively in this chapter.
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References
Bryc, W. (1995). The Normal Distribution, Springer, New York.
Feller, W. (1971). An Introduction to Probability Theory and Applications, Vol. II, Wiley, New York.
Freedman, D. (2005). Statistical Models, Cambridge University Press, New York.
Heyde, C. (1963). On a property of the lognormal distribution, J.R. Statist. Soc. Ser. B, 25(2), 392–393.
Johnson, N., Kotz, S., and Balakrishnan, N. (1994). Continuous Univariate Distributions, Vol. II, Wiley, New York.
Kendall, M. and Stuart, A. (1976). Advanced Theory of Statistics, Vol. I, Fourth ed., Macmillan, New York.
Patel, J. and Read, C. (1996). Handbook of the Normal Distribution, Marcel Dekker, New York.
Petrov, V. (1975). Limit Theorems of Probability Theory, Clarendon Press, London.
Rao, C.R. (1973). Linear Statistical Inference and Its Applications, Wiley, New York.
Stigler, S. (1975). Studies in the history of probability and statistics; Napoleonic statistics, the work of Laplace, Biometrika, 62, 503–517.
Stigler, S. (1986). The History of Statistics: The Measurement of Uncertainty Before 1900, Harvard University Press, Cambridge, MA.
Tong, Y. (1990). Multivariate Normal Distribution, Springer, New York.
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DasGupta, A. (2010). Normal Distribution. In: Fundamentals of Probability: A First Course. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5780-1_9
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DOI: https://doi.org/10.1007/978-1-4419-5780-1_9
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