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1: 8.23 Statistical Applications

§8.23 Statistical Applications

… ►The function ${\mathrm{B}}_x(a,b)$ and its normalization $I_x(a,b)$ play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). …

2: Bibliography C

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Algorithm AS 310. Computing the non-central beta distribution function. Appl. Statist. 46 (1), pp. 146–156.

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Notes:

Single-precision Fortran.

External Links:

FullText, Code, ZentralBlatt

Referenced by:

2nd item.

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BibTeX

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3: 19.33 Triaxial Ellipsoids

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§19.33(iv) Self-Energy of an Ellipsoidal Distribution

►Ellipsoidal distributions of charge or mass are used to model certain atomic nuclei and some elliptical galaxies. …where $\alpha ,\beta ,\gamma$ are dimensionless positive constants. The contours of constant density are a family of similar, rather than confocal, ellipsoids. In suitable units the self-energy of the distribution is given by …

4: Bibliography P

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Orthogonal polynomials and some $q$-beta integrals of Ramanujan. J. Math. Anal. Appl. 112 (2), pp. 517–540.

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External Links:

ISSN 0022-247X, Document, MathReview (Mourad E. H. Ismail), ZentralBlatt

Referenced by:

§18.33(iv), §18.33(v).

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Handbook of the Normal Distribution. Statistics: Textbooks and Monographs, Vol. 40, Marcel Dekker Inc., New York.

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Notes:

A revised and expanded 2nd edition was published in 1996.

External Links:

ISBN 0-8247-1541-1, MathReview (S. Kotz), ZentralBlatt

Referenced by:

§7.20(iii).

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BibTeX

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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

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1st item.

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A note on the computation of the incomplete beta function. Adv. Eng. Software 12 (1), pp. 39–44.

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Notes:

Includes Basic code.

External Links:

Document

Referenced by:

§8.28(iv).

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The exact distribution of the Wald statistic. Econometrica 54 (4), pp. 881–895.

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ISSN 0012-9682, FullText, MathReview (Sadanori Konishi), ZentralBlatt

Referenced by:

§35.9.

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5: 9.9 Zeros

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§9.9(i) Distribution and Notation

… ►They lie in the sectors and , and are denoted by ${\beta }_k$, ${\beta }_k^{\prime }$, respectively, in the former sector, and by $\overline{{\beta }_k}$, $\overline{{\beta }_k^{\prime }}$, in the conjugate sector, again arranged in ascending order of absolute value (modulus) for $k=1,2,\mathrm{\dots }.$ See §9.3(ii) for visualizations. ►For the distribution in $ℂ$ of the zeros of ${\mathrm{Ai}}^{\prime }\left(z\right)-\sigma \mathrm{Ai}\left(z\right)$, where $\sigma$ is an arbitrary complex constant, see Muraveĭ (1976) and Gil and Segura (2014). … ►

9.9.14 $${\beta }_k={\mathrm{e}}^{\pi \mathrm{i}/3}T\left(\frac{3}{8}\pi (4k-1)+\frac{3}{4}\mathrm{i}\ln 2\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, ${\beta }_k$: $k$th complex zero of Airy $\mathrm{Bi}$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $T$: expansion

Source:

Olver (1954, (A 22))

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http://dlmf.nist.gov/9.9.E14

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Annotations for §9.9(iv), §9.9 and Ch.9

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9.9.15 $${\mathrm{Bi}}^{\prime }\left({\beta }_k\right)={(-1)}^k\sqrt{2}{\mathrm{e}}^{-\pi \mathrm{i}/6}V\left(\frac{3}{8}\pi (4k-1)+\frac{3}{4}\mathrm{i}\ln 2\right),$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\beta }_k$: $k$th complex zero of Airy $\mathrm{Bi}$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $V$: expansion

Source:

Olver (1954, (A 22))

Permalink:

http://dlmf.nist.gov/9.9.E15

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Annotations for §9.9(iv), §9.9 and Ch.9

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6: 2.6 Distributional Methods

§2.6 Distributional Methods

… ►Motivated by the definition of distributional derivatives, we can assign them the distributions defined by … ►The Dirac delta distribution in (2.6.17) is given by … ►These equations again hold only in the sense of distributions. … ►

§2.6(iv) Regularization

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7: 18.16 Zeros

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§18.16(i) Distribution

… ►Then ${\theta }_{n,m}$ is strictly increasing in $\alpha$ and strictly decreasing in $\beta$; furthermore, if $\alpha =\beta$, then ${\theta }_{n,m}$ is strictly increasing in $\alpha$. … ►except when ${\alpha }^2={\beta }^2=\frac{1}{4}$. … ►Then as $n\to \mathrm{\infty }$, with $\alpha$ ($>-\frac{1}{2}$) and $\beta$ ($\ge -1-\alpha$) fixed, … ►For ultraspherical and Legendre polynomials, set $\alpha =\beta$ and $\alpha =\beta =0$, respectively, in the results given in §18.16(ii). …

8: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►of the Dirac delta distribution. … ►

1.18.21 $$T(\alpha v+\beta w)=\alpha Tv+\beta Tw,$$ $v,w\in V$, $\alpha ,\beta \in ℂ$.

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Symbols:

$ℂ$: complex plane, $\in$: element of and $w$: variable

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http://dlmf.nist.gov/1.18.E21

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Annotations for §1.18(iii), §1.18(iii), §1.18 and Ch.1

… ►For a formally self-adjoint second order differential operator $ℒ$, such as that of (1.18.28), the space $𝒟({ℒ}^{\ast })$ can be seen to consist of all $f\in L^2\left(X\right)$ such that the distribution $ℒf$ can be identified with a function in $L^2\left(X\right)$, which is the function ${ℒ}^{\ast }f$. … ►

1.18.71 $$ℬ(f)=\underset{x\to a+}{lim}\left(\alpha (x)f(x)+\beta (x)f^{\prime }(x)\right),$$ $f\in 𝒟({ℒ}^{\ast })$,

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Symbols:

${𝐀}^{\ast }$: adjoint of matrix and $\in$: element of

Referenced by:

Erratum (V1.2.0) Section 1.18

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http://dlmf.nist.gov/1.18.E71

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Annotations for §1.18(ix), §1.18(ix), §1.18 and Ch.1

►where $\alpha$ and $\beta$ are given functions on $X$, and where the limit has to exist for all $f$. …

9: Bibliography D

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The computation of lattice frequency distribution functions by means of continued fractions. Physica 39 (3), pp. 413–423.

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External Links:

ISSN 0031-8914, Document, Link

Referenced by:

§18.40(ii).

Encodings:

BibTeX

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Algorithm 708: Significant digit computation of the incomplete beta function ratios. ACM Trans. Math. Software 18 (3), pp. 360–373.

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Notes:

Single-precision Fortran, maximum accuracy 14D.

External Links:

ISSN 0098-3500, Document, GAMS (module 708; package TOMS), ZentralBlatt

Referenced by:

3rd item.

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BibTeX

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Algorithm 322. F-distribution. Comm. ACM 11 (2), pp. 116–117.

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Notes:

Includes Algol program, maximum accuracy 10S. For certification see same journal 12(1969), p. 39

External Links:

ISSN 0001-0782, Document

Referenced by:

§8.28(iv).

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BibTeX

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The incomplete beta function—a historical profile. Arch. Hist. Exact Sci. 24 (1), pp. 11–29.

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External Links:

ISSN 0003-9519, MathReview (Willard Parker), ZentralBlatt

Referenced by:

§8.17(i).

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BibTeX

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10: 1.2 Elementary Algebra

… ►distributive if $𝐁$ and $𝐂$ have the same dimensions … ►for $\alpha ,\beta \in ℂ$ ►

1.2.42 $$⟨\alpha 𝐮,\beta 𝐯⟩=\alpha \overline{\beta }⟨𝐮,𝐯⟩,$$

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Symbols:

$\overline{z}$: complex conjugate, $⟨𝐮,𝐯⟩$: inner product over vectors, $𝐯$: column vector and $𝐮$: column vector

Referenced by:

§1.2(v)

Permalink:

http://dlmf.nist.gov/1.2.E42

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Annotations for §1.2(v), §1.2(v), §1.2 and Ch.1

… ►If $𝐮$, $𝐯$, $\alpha$ and $\beta$ are real the complex conjugate bars can be omitted in (1.2.40)–(1.2.42). …