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11: Karl Dilcher

… … ► 1954 in Wabern-Harle, Germany) is Professor in the Department of Mathematics and Statistics at Dalhousie University in Halifax, Nova Scotia, Canada. …

12: T. Mark Dunster

… … ►is a Professor in the Department of Mathematics and Statistics, San Diego State University, California. …

13: Javier Segura

… … ► 1965 in Novelda, Spain) is professor of mathematical analysis at the Department of Mathematics, Statistics and Computation, University of Cantabria, Spain. …

14: 17.17 Physical Applications

§17.17 Physical Applications

►In exactly solved models in statistical mechanics (Baxter (1981, 1982)) the methods and identities of §17.12 play a substantial role. …

15: 35 Functions of Matrix Argument

…

16: Peter A. Clarkson

… … ►is Professor of Mathematics in the School of Mathematics, Statistics, and Actuarial Science at the University of Kent, Canterbury, U. …

17: 7.20 Mathematical Applications

… ►

§7.20(iii) Statistics

… ►

7.20.1

\frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{x}e^{-(t-m)^{2}/(2\sigma^{2})}\,% \mathrm{d}t=\frac{1}{2}\operatorname{erfc}\left(\frac{m-x}{\sigma\sqrt{2}}% \right)=Q\left(\frac{m-x}{\sigma}\right)=P\left(\frac{x-m}{\sigma}\right).

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $x$ : real variable, $m$ : mean and $\sigma$ : standard deviation
A&S Ref:: 7.1.22 (slightly modified)
Permalink:: http://dlmf.nist.gov/7.20.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §7.20(iii), §7.20 and Ch.7

►For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).

18: 8.24 Physical Applications

… ►

§8.24(ii) Incomplete Beta Functions

►The function

I_{x}\left(a,b\right)

appears in: Monte Carlo sampling in statistical mechanics (Kofke (2004)); analysis of packings of soft or granular objects (Prellberg and Owczarek (1995)); growth formulas in cosmology (Hamilton (2001)). …

19: 26.19 Mathematical Applications

… ►These have applications in operations research, probability theory, and statistics. …

20: 20.12 Mathematical Applications

… ►This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)).