probability functions
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1: 7.1 Special Notation
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►The notations , , and are used in mathematical statistics, where these functions are called the normal or Gaussian probability functions.
2: 12.7 Relations to Other Functions
3: 7.18 Repeated Integrals of the Complementary Error Function
4: Bibliography H
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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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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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Certain integrals that contain a probability function.
Bul. Akad. Štiince RSS Moldoven. 1975 (2), pp. 86–88, 95 (Russian).
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Expansions for the probability function in series of Čebyšev polynomials and Bessel functions.
Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).
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Sums with cylindrical functions that reduce to the probability function and to related functions.
Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).
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5: 35.9 Applications
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►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument , with and .
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►In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions.
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6: 7.20 Mathematical Applications
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7.20.1
►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).
7: Donald St. P. Richards
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►Richards has published numerous papers on special functions of matrix argument, harmonic analysis, multivariate statistical analysis, probability inequalities, and applied probability.
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8: Bibliography C
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9: 8.23 Statistical Applications
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►The functions
and are used extensively in statistics as the probability integrals of the gamma distribution; see Johnson et al. (1994, pp. 337–414).
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10: Bibliography M
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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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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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