gamma distribution

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1: 8.23 Statistical Applications

§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). …

2: 2.6 Distributional Methods

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2.6.38 $$t^{\mu -1}\ast {\delta }^{(s-1)}=\frac{\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\mu +1-s\right)}{t}^{\mu -s},$$ $t>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_x$: Dirac delta distribution, $\mu$: order and $\ast$: convolution

Referenced by:

§2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E38

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See also:

Annotations for §2.6(iii), §2.6 and Ch.2

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2.6.40 $$t^{\mu -1}\ast t^{-s-1}=\frac{{(-1)}^s}{\mu \cdot s!}{𝐷}^{s+1}\left(t^{\mu }\left(\ln t-\gamma -\psi \left(\mu +1\right)\right)\right),$$ $t>0$,

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Symbols:

$\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $\mu$: order, $\ast$: convolution and $𝐷$: derivative of distribution

Referenced by:

§2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E40

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See also:

Annotations for §2.6(iii), §2.6 and Ch.2

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2.6.44 $$t^{\mu -1}\ast f=\sum _{s=0}^{n-1}\frac{{(-1)}^s{a}_s}{\mu \cdot s!}{𝐷}^{s+1}\left(t^{\mu }\left(\ln t-\gamma -\psi \left(\mu +1\right)\right)\right)-\sum _{s=1}^n{d}_s\frac{\mathrm{\Gamma }\left(\mu \right)}{\mathrm{\Gamma }\left(\mu -s+1\right)}{t}^{\mu -s}+t^{\mu -1}\ast f_n$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\psi \left(z\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, $d_s$: coefficients, $\mu$: order, $\ast$: convolution, $𝐷$: derivative of distribution, $f(t)$: locally integrable function, $n$: nonnegative integer, $a_n$: coefficients and $f_n(t)$: remainder

Referenced by:

§2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E44

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Annotations for §2.6(iii), §2.6 and Ch.2

… ►Since the function $t^{\mu }\left(\ln t-\gamma -\psi \left(\mu +1\right)\right)$ and all its derivatives are locally absolutely continuous in $(0,\mathrm{\infty })$, the distributional derivatives in the first sum in (2.6.44) can be replaced by the corresponding ordinary derivatives. …

3: 8.13 Zeros

… ►For information on the distribution and computation of zeros of $\gamma (a,\lambda a)$ and $\mathrm{\Gamma }(a,\lambda a)$ in the complex $\lambda$-plane for large values of the positive real parameter $a$ see Temme (1995a). …

4: Bibliography K

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Tables of the Incomplete Gamma Function Ratio: The Chi-square Integral, the Poisson Distribution. Justus von Liebig Verlag, Darmstadt (German, English).

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Notes:

In cooperation with W. Rudert

External Links:

MathReview, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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5: Bibliography F

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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6: Philip J. Davis

… ►He also had a big influence on the development of the NBS Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (A&S), which became one of the most widely distributed and highly cited publications in NIST’s history. After being asked by Milton Abramowitz to work on the project, he chose to write the Chapter “Gamma Function and Related Functions. …

7: 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 …

8: Bibliography C

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Ellipsoidal distributions of charge or mass. J. Mathematical Phys. 2, pp. 441–450.

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External Links:

MathReview (H. Bremekamp), ZentralBlatt

Referenced by:

§19.33(iv), §19.37(iv).

Encodings:

BibTeX

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On Stieltjes’ continued fraction for the gamma function. Math. Comp. 34 (150), pp. 547–551.

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External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

§5.10.

Encodings:

BibTeX

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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.

Encodings:

BibTeX

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Calculation of angular distributions in complex angular momentum theories of elastic scattering. Molecular Physics 37 (6), pp. 1703–1712.

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External Links:

Document

Referenced by:

§14.31(iii).

Encodings:

BibTeX

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Some non-central distribution problems in multivariate analysis. Ann. Math. Statist. 34 (4), pp. 1270–1285.

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External Links:

FullText, MathReview (I. Olkin), ZentralBlatt

Referenced by:

§35.4(i), §35.4(ii).

Encodings:

BibTeX

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9: 31.15 Stieltjes Polynomials

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31.15.2 $$\sum _{j=1}^N\frac{{\gamma }_j/2}{z_k-a_j}+\sum _{\begin{array}{c}j=1\\ j\ne k\end{array}}^n\frac{1}{z_k-z_j}=0,$$ $k=1,2,\mathrm{\dots },n$.

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Referenced by:

§31.15(ii)

Permalink:

http://dlmf.nist.gov/31.15.E2

Encodings:

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See also:

Annotations for §31.15(ii), §31.15 and Ch.31

… ►The system (31.15.2) determines the $z_k$ as the points of equilibrium of $n$ movable (interacting) particles with unit charges in a field of $N$ particles with the charges ${\gamma }_j/2$ fixed at $a_j$. … ►

${\gamma }_j>0,$

… ►then there are exactly $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ polynomials $S(z)$, each of which corresponds to each of the $\left(\genfrac{}{}{0.0pt}{}{n+N-2}{N-2}\right)$ ways of distributing its $n$ zeros among $N-1$ intervals $(a_j,a_{j+1})$, $j=1,2,\mathrm{\dots },N-1$. … ►

31.15.7 $$q_j={\gamma }_j\sum _{k=1}^n\frac{1}{z_k-a_j},$$ $j=1,2,\mathrm{\dots },N$.

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Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E7

Encodings:

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See also:

Annotations for §31.15(ii), §31.15 and Ch.31

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10: Bibliography P

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Tablitsy nepolnoi gamma-funktsii. Vyčisl. Centr Akad. Nauk SSSR, Moscow (Russian).

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Notes:

In Russian. Translated title: Tables of the Incomplete Gamma Function.

External Links:

MathReview (John Todd)

Referenced by:

2nd item.

Encodings:

BibTeX

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An asymptotic formula for the incomplete gamma function. Ž. Vyčisl. Mat. i Mat. Fiz. 5, pp. 118–121 (Russian).

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Notes:

English translation in U.S.S.R. Computational Math. and Math. Phys. 5(1), pp. 162–166

External Links:

Document, MathReview (E. Lukacs), ZentralBlatt

Referenced by:

§8.11(iv).

Encodings:

BibTeX

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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).

Encodings:

BibTeX

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A Fortran routine for the computation of gamma percentiles. Adv. Eng. Software 10 (3), pp. 159–164.

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External Links:

Document, ZentralBlatt

Referenced by:

§8.28(ii).

Encodings:

BibTeX

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

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External Links:

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

Referenced by:

§35.9.

Encodings:

BibTeX

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