probability functions

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11: Bibliography L

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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.

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External Links:: ISSN 0176-4276, Document, MathReview (P. Anandani), ZentralBlatt
Referenced by:: §8.13(i).
Encodings:: BibTeX

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12: 2.3 Integrals of a Real Variable

… ►This result is probably the most frequently used method for deriving asymptotic expansions of special functions. …

13: 18.39 Applications in the Physical Sciences

… ►These eigenfunctions are quantum wave-functions whose absolute values squared give the probability density of finding the single particle at hand at position

x

in the

n

th eigenstate, namely that probability is

P(x\to x+\Delta x)

|\psi_{n}(x)|^{2}\Delta(x)

\Delta(x)

being a localized interval on the

x

-axis. …

14: 5.20 Physical Applications

§5.20 Physical Applications

… ►The probability density of the positions when the gas is in thermodynamic equilibrium is: … ►and the partition function is given by … ►

Elementary Particles

… ►

15: 26.19 Mathematical Applications

… ►Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)). … ►These have applications in operations research, probability theory, and statistics. …

16: Bibliography B

… ►

P. L. Butzer, S. Flocke, and M. Hauss (1994) Euler functions

{E}_{\alpha}(z)

with complex

\alpha

and applications. In Approximation, probability, and related fields (Santa Barbara, CA, 1993), G. Anastassiou and S. T. Rachev (Eds.), pp. 127–150.

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External Links:: MathReview (R. N. Kalia), ZentralBlatt
Referenced by:: §24.16(iii).
Encodings:: BibTeX

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17: 3.5 Quadrature

… ►which depends on function values computed previously. … ►

Gauss Formula for a Logarithmic Weight Function

… ►

Example

… ► … ►In more advanced methods points are sampled from a probability distribution, so that they are concentrated in regions that make the largest contribution to the integral. …

18: Bibliography P

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E. Pairman (1919) Tables of Digamma and Trigamma Functions. In Tracts for Computers, No. 1, K. Pearson (Ed.),

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External Links:: ZentralBlatt
Referenced by:: §5.1, Notations F.
Encodings:: BibTeX

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R. B. Paris (2002c) Exponential asymptotics of the Mittag-Leffler function. Proc. Roy. Soc. London Ser. A 458, pp. 3041–3052.

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External Links:: ISSN 1364-5021, Document, MathReview (Roderick Wong), ZentralBlatt
Referenced by:: §10.46.
Encodings:: BibTeX

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K. Pearson (Ed.) (1968) Tables of the Incomplete Beta-function. 2nd edition, Published for the Biometrika Trustees at the Cambridge University Press, Cambridge.

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External Links:: MathReview, ZentralBlatt
Referenced by:: 1st item.
Encodings:: BibTeX

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E. Petropoulou (2000) Bounds for ratios of modified Bessel functions. Integral Transform. Spec. Funct. 9 (4), pp. 293–298.

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External Links:: ISSN 1065-2469, Document, MathReview, ZentralBlatt
Referenced by:: §10.37.
Encodings:: BibTeX

… ►

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.

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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).
Encodings:: BibTeX

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19: 18.33 Polynomials Orthogonal on the Unit Circle

… ►

Szegő–Askey

… ►For the hypergeometric function

{{}_{2}F_{1}}

see §§15.1 and 15.2(i). … ►See Al-Salam and Ismail (1994) for special biorthogonal rational functions on the unit circle. … ►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}^{\infty}

with

\alpha_{n}\in\mathbb{C}

and

|\alpha_{n}|<1

the polynomials

\Phi_{n}(z)

generated by the recurrence relations (18.33.23), (18.33.24) with

\Phi_{0}(z)=1

satisfy the orthogonality relation (18.33.17) for a unique probability measure

\mu

with infinite support on the unit circle. …

20: 19.31 Probability Distributions

§19.31 Probability Distributions

►

R_{G}\left(x,y,z\right)

and

R_{F}\left(x,y,z\right)

occur as the expectation values, relative to a normal probability distribution in

{\mathbb{R}}^{2}

{\mathbb{R}}^{3}

, of the square root or reciprocal square root of a quadratic form. … ►

19.31.2

\int_{{\mathbb{R}}^{n}}(\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x})^{\mu}\exp% \left(-\mathbf{x}^{\mathrm{T}}\mathbf{B}\mathbf{x}\right)\,\mathrm{d}x_{1}% \cdots\,\mathrm{d}x_{n}=\frac{\pi^{n/2}\Gamma\left(\mu+\tfrac{1}{2}n\right)}{% \sqrt{\det\mathbf{B}}\Gamma\left(\tfrac{1}{2}n\right)}R_{\mu}\left(\tfrac{1}{2% },\dots,\tfrac{1}{2};\lambda_{1},\dots,\lambda_{n}\right),

\mu>-\tfrac{1}{2}n

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Symbols:: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\det$ : determinant, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\int$ : integral, $\mathbb{R}$ : real line, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $n$ : nonnegative integer and $\lambda_{n}$ : eigenvalues
Referenced by:: §19.31
Permalink:: http://dlmf.nist.gov/19.31.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §19.31 and Ch.19

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