incomplete beta functions

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1: 8.17 Incomplete Beta Functions

§8.17 Incomplete Beta Functions

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§8.17(ii) Hypergeometric Representations

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§8.17(iii) Integral Representation

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§8.17(iv) Recurrence Relations

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§8.17(vi) Sums

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2: 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). …

3: 8.24 Physical Applications

§8.24 Physical Applications

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§8.24(ii) Incomplete Beta Functions

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4: 8.1 Special Notation

… ►Unless otherwise indicated, primes denote derivatives with respect to the argument. ►The functions treated in this chapter are the incomplete gamma functions $\gamma (a,z)$, $\mathrm{\Gamma }(a,z)$, ${\gamma }^{\ast }(a,z)$, $P(a,z)$, and $Q(a,z)$; the incomplete beta functions ${\mathrm{B}}_x(a,b)$ and $I_x(a,b)$; the generalized exponential integral $E_p\left(z\right)$; the generalized sine and cosine integrals $\mathrm{si}(a,z)$, $\mathrm{ci}(a,z)$, $\mathrm{Si}(a,z)$, and $\mathrm{Ci}(a,z)$. ►Alternative notations include: Prym’s functions $P_z(a)=\gamma (a,z)$, $Q_z(a)=\mathrm{\Gamma }(a,z)$, Nielsen (1906a, pp. 25–26), Batchelder (1967, p. 63); $(a,z)!=\gamma (a+1,z)$, $[a,z]!=\mathrm{\Gamma }(a+1,z)$, Dingle (1973); $B(a,b,x)={\mathrm{B}}_x(a,b)$, $I(a,b,x)=I_x(a,b)$, Magnus et al. (1966); $\mathrm{Si}(a,x)\to \mathrm{Si}(1-a,x)$, $\mathrm{Ci}(a,x)\to \mathrm{Ci}(1-a,x)$, Luke (1975).

5: 8.26 Tables

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§8.26(iii) Incomplete Beta Functions

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6: 8.18 Asymptotic Expansions of $I_x(a,b)$

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8.18.1 $$I_x(a,b)=\mathrm{\Gamma }\left(a+b\right)x^a{(1-x)}^{b-1}\left(\sum _{k=0}^{n-1}\frac{1}{\mathrm{\Gamma }\left(a+k+1\right)\mathrm{\Gamma }\left(b-k\right)}{\left(\frac{x}{1-x}\right)}^k+O\left(\frac{1}{\mathrm{\Gamma }\left(a+n+1\right)}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $I_x(a,b)$: incomplete beta function, $k$: nonnegative integer, $n$: nonnegative integer, $a$: parameter, $b$: parameter and $x$: variable

Referenced by:

§8.18(i)

Permalink:

http://dlmf.nist.gov/8.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(i), §8.18 and Ch.8

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Symmetric Case

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General Case

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Inverse Function

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8.18.18 $$I_x(a,b)=p,$$ $0\le p\le 1$,

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Symbols:

$I_x(a,b)$: incomplete beta function, $p$: parameter, $a$: parameter, $b$: parameter and $x$: variable

Permalink:

http://dlmf.nist.gov/8.18.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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7: 8.28 Software

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§8.28(iv) Incomplete Beta Functions for Real Argument and Parameters

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§8.28(v) Incomplete Beta Functions for Complex Argument and Parameters

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8: Bibliography P

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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

Referenced by:

1st item.

Encodings:

BibTeX

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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).

Encodings:

BibTeX

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9: Bibliography N

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Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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External Links:

ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt

Referenced by:

§8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).

Encodings:

BibTeX

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10: Bibliography D

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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.

Encodings:

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).

Encodings:

BibTeX

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