probability theory

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1: 26.19 Mathematical Applications

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

2: 7.20 Mathematical Applications

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7.20.1 $$\frac{1}{\sigma \sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^x{\mathrm{e}}^{-{(t-m)}^2/(2{\sigma }^2)}dt=\frac{1}{2}\mathrm{erfc}\left(\frac{m-x}{\sigma \sqrt{2}}\right)=Q\left(\frac{m-x}{\sigma }\right)=P\left(\frac{x-m}{\sigma }\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: 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

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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).

3: Donald St. P. Richards

… ►Richards has published numerous papers on special functions of matrix argument, harmonic analysis, multivariate statistical analysis, probability inequalities, and applied probability. He is editor of the book Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, published by the American Mathematical Society in 1992, and coeditor of Representation Theory and Harmonic Analysis: A Conference in Honor of R. A. Kunze (with T. …

4: 8.23 Statistical Applications

… ►The functions $P(a,x)$ and $Q(a,x)$ are used extensively in statistics as the probability integrals of the gamma distribution; see Johnson et al. (1994, pp. 337–414). …In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of $Q(a,x)$; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319).

5: Bibliography L

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The Classical Theory of Fields. Pergamon Press, Oxford.

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Notes:

Revised second edition. Course of Theoretical Physics, Vol. 2. Translated from the Russian by Morton Hamermesh

External Links:

MathReview (E. L. Hill), ZentralBlatt

Referenced by:

§9.16.

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Essai sur la Théorie des Nombres. 2nd edition, Courcier, Paris.

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External Links:

ZentralBlatt

Referenced by:

§27.2(i).

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Zur Theorie der Gaußschen Summen. Math. Ann. 57 (4), pp. 554–567 (German).

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External Links:

ISSN 0025-5831, Document, MathReview Entry, ZentralBlatt

Referenced by:

§20.11(i).

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On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions. Constr. Approx. 10 (1), pp. 15–30.

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ISSN 0176-4276, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§8.13(i).

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Kinetic Theory: Classical, Quantum, and Relativistic Descriptions. third edition, Springer, New York.

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External Links:

ISBN 0387955518

Referenced by:

§18.39(iii).

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6: Bibliography P

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Statistical Field Theory. Addison-Wesley, Reading, MA.

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External Links:

ISBN 0-201-05985-1, MathReview (Tony C. Dorlas), ZentralBlatt

Referenced by:

§22.19(ii).

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La Théorie des Fonctions de Bessel Exposée en vue de ses Applications à la Physique Mathématique. Centre National de la Recherche Scientifique, Paris (French).

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.63(ii).

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Gauge Field Theories. Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-26537-1, MathReview (G. R. Allcock), ZentralBlatt

Referenced by:

§22.19(ii).

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Remarks on computing the probability integral in one and two dimensions. In Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1945, 1946, pp. 63–78.

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External Links:

MathReview (L. A. Aroian), ZentralBlatt

Referenced by:

(7.8.8), §7.8, Erratum (V1.0.17) for Equation (7.8.8).

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Catastrophe Theory and its Applications. Pitman, London.

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Notes:

Reprinted by Dover in 1996.

External Links:

ISBN 0-273-01029-8, MathReview (George K. Francis), ZentralBlatt

Referenced by:

§36.4(i), Ch.36.

Encodings:

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7: 18.33 Polynomials Orthogonal on the Unit Circle

… ►Simon (2005a, b) gives the general theory of these OP’s in terms of monic OP’s ${\mathrm{\Phi }}_n(x)$, see §18.33(vi). … ►See Baxter (1961) for general theory. … ►Let $\mu$ be a probability measure on the unit circle of which the support is an infinite set. … ►This states that for any sequence ${\{{\alpha }_n\}}_{n=0}^{\mathrm{\infty }}$ with ${\alpha }_n\in ℂ$ and the polynomials ${\mathrm{\Phi }}_n(z)$ generated by the recurrence relations (18.33.23), (18.33.24) with ${\mathrm{\Phi }}_0(z)=1$ satisfy the orthogonality relation (18.33.17) for a unique probability measure $\mu$ with infinite support on the unit circle. …

8: Bibliography M

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The Theory of Groups. Clarendon Press, Oxford.

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Notes:

Reprinted in 1988 by Krieger Publishing Co., Malabar, FL

External Links:

ISBN 0-19-853137-0;0-89464-287-1, MathReview (A. P. Street), ZentralBlatt

Referenced by:

§23.20(ii).

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Zur Theorie des zylindrisch-parabolischen Spiegels. Z. Physik 118, pp. 343–356 (German).

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§12.17.

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.

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ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

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Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.

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External Links:

ISSN 0047-259X, Document, MathReview (Peter A. Lachenbruch), ZentralBlatt

Referenced by:

§35.9.

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Algorithm 346: F-test probabilities [S14]. Comm. ACM 12 (3), pp. 184–185.

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Notes:

Includes Algol program, maximum accuracy 12S. For a remark see ACM Trans. Math. Software 14(1988), pp. 288-289

External Links:

ISSN 0001-0782, Document, Remark

Referenced by:

§8.28(iv).

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9: Bibliography B

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Moment equations for probability distributions in classical and quantum mechanics. Phys. Rev. A 58 (3), pp. 1799–1809.

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External Links:

Document

Referenced by:

§24.18.

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Random and Restricted Walks: Theory and Applications. Gordon and Breach, New York.

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External Links:

ZentralBlatt

Referenced by:

§16.24(i).

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Uniform approximation: A new concept in wave theory. Science Progress (Oxford) 57, pp. 43–64.

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Referenced by:

§36.14(i), §9.16.

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Efficiency and Security of Cryptosystems Based on Number Theory. Ph.D. Thesis, Swiss Federal Institute of Technology (ETH), Zurich.

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External Links:

FullText

Referenced by:

2nd item.

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The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. Dover Publications, New York.

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Notes:

Two volumes. A reproduction of the seventh edition [vol. 1, Longmans, Green, London 1912; vol. 2, Longmans, Green, London 1928].

External Links:

MathReview, ZentralBlatt

Referenced by:

§1.11(i), §1.11(ii), §1.11(iv).

Encodings:

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10: Bibliography H

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Computation of certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven 1968 (2), pp. 81–104. (errata insert) (Russian).

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External Links:

MathReview (H. A. Lauwerier)

Referenced by:

§7.14(iii).

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Certain integrals that contain a probability function and degenerate hypergeometric functions. Bul. Akad. S̆tiince RSS Moldoven 1969 (2), pp. 40–47 (Russian).

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External Links:

MathReview (Wazir Hasan Abdi)

Referenced by:

§7.14(iii).

Encodings:

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Some integrals that contain a probability function and hypergeometric functions. Bul. Akad. Štiince RSS Moldoven 1970 (1), pp. 49–62 (Russian).

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External Links:

MathReview (R. G. Langebartel)

Referenced by:

§7.14(iii).

Encodings:

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:

MathReview (L. Berg)

Referenced by:

§7.14(iii).

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Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).

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External Links:

MathReview (C. Tanaka)

Referenced by:

§7.14(iii).

Encodings:

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