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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
7.20.1 1 σ 2 π x e ( t m ) 2 / ( 2 σ 2 ) d t = 1 2 erfc ( m x σ 2 ) = Q ( m x σ ) = P ( x m σ ) .
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
  • L. D. Landau and E. M. Lifshitz (1962) The Classical Theory of Fields. Pergamon Press, Oxford.
  • A. M. Legendre (1808) Essai sur la Théorie des Nombres. 2nd edition, Courcier, Paris.
  • M. Lerch (1903) Zur Theorie der Gaußschen Summen. Math. Ann. 57 (4), pp. 554–567 (German).
  • J. S. Lew (1994) On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions. Constr. Approx. 10 (1), pp. 15–30.
  • R. L. Liboff (2003) Kinetic Theory: Classical, Quantum, and Relativistic Descriptions. third edition, Springer, New York.
  • 6: Bibliography P
  • G. Parisi (1988) Statistical Field Theory. Addison-Wesley, Reading, MA.
  • G. Petiau (1955) La Théorie des Fonctions de Bessel Exposée en vue de ses Applications à la Physique Mathématique. Centre National de la Recherche Scientifique, Paris (French).
  • S. Pokorski (1987) Gauge Field Theories. Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge.
  • G. Pólya (1949) 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.
  • T. Poston and I. Stewart (1978) Catastrophe Theory and its Applications. Pitman, London.
  • 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 Φ n ( x ) , see §18.33(vi). … See Baxter (1961) for general theory. … Let μ be a probability measure on the unit circle of which the support is an infinite set. … This states that for any sequence { α n } n = 0 with α n and | α n | < 1 the polynomials Φ n ( z ) generated by the recurrence relations (18.33.23), (18.33.24) with Φ 0 ( z ) = 1 satisfy the orthogonality relation (18.33.17) for a unique probability measure μ with infinite support on the unit circle. …
    8: Bibliography M
  • I. D. Macdonald (1968) The Theory of Groups. Clarendon Press, Oxford.
  • W. Magnus (1941) Zur Theorie des zylindrisch-parabolischen Spiegels. Z. Physik 118, pp. 343–356 (German).
  • H. R. McFarland and D. St. P. Richards (2001) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. I. The equal-means case. J. Multivariate Anal. 77 (1), pp. 21–53.
  • H. R. McFarland and D. St. P. Richards (2002) Exact misclassification probabilities for plug-in normal quadratic discriminant functions. II. The heterogeneous case. J. Multivariate Anal. 82 (2), pp. 299–330.
  • J. Morris (1969) Algorithm 346: F-test probabilities [S14]. Comm. ACM 12 (3), pp. 184–185.
  • 9: Bibliography B
  • L. E. Ballentine and S. M. McRae (1998) Moment equations for probability distributions in classical and quantum mechanics. Phys. Rev. A 58 (3), pp. 1799–1809.
  • M. N. Barber and B. W. Ninham (1970) Random and Restricted Walks: Theory and Applications. Gordon and Breach, New York.
  • M. V. Berry (1969) Uniform approximation: A new concept in wave theory. Science Progress (Oxford) 57, pp. 43–64.
  • D. Bleichenbacher (1996) Efficiency and Security of Cryptosystems Based on Number Theory. Ph.D. Thesis, Swiss Federal Institute of Technology (ETH), Zurich.
  • W. S. Burnside and A. W. Panton (1960) The Theory of Equations: With an Introduction to the Theory of Binary Algebraic Forms. Dover Publications, New York.
  • 10: Bibliography H
  • P. I. Hadži (1968) Computation of certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven 1968 (2), pp. 81–104. (errata insert) (Russian).
  • P. I. Hadži (1969) Certain integrals that contain a probability function and degenerate hypergeometric functions. Bul. Akad. S̆tiince RSS Moldoven 1969 (2), pp. 40–47 (Russian).
  • P. I. Hadži (1970) Some integrals that contain a probability function and hypergeometric functions. Bul. Akad. Štiince RSS Moldoven 1970 (1), pp. 49–62 (Russian).
  • P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
  • P. I. Hadži (1975a) Certain integrals that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).