19 Elliptic Integrals Applications 19.30 Lengths of Plane Curves 19.32 Conformal Map onto a Rectangle

§19.31 Probability Distributions

ⓘ

Keywords:: applications, probability distribution, statistical, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.31
See also:: Annotations for Ch.19

$R_{G}\left(x,y,z\right)$ and $R_{F}\left(x,y,z\right)$ occur as the expectation values, relative to a normal probability distribution in ${\mathbb{R}}^{2}$ or ${\mathbb{R}}^{3}$ , of the square root or reciprocal square root of a quadratic form. More generally, let $\mathbf{A}$ ( $=[a_{r,s}]$ ) and $\mathbf{B}$ ( $=[b_{r,s}]$ ) be real positive-definite matrices with $n$ rows and $n$ columns, and let $\lambda_{1},\dots,\lambda_{n}$ be the eigenvalues of $\mathbf{A}\mathbf{B}^{-1}$ . If $\mathbf{x}$ is a column vector with elements $x_{1},x_{2},\dots,x_{n}$ and transpose $\mathbf{x}^{\mathrm{T}}$ , then

19.31.1		$\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x}=\sum_{r=1}^{n}\sum_{s=1}^{n}a_{r,s% }x_{r}x_{s},$
ⓘ Symbols: $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix and $n$ : nonnegative integer Permalink: http://dlmf.nist.gov/19.31.E1 Encodings: TeX, pMML, png See also: Annotations for §19.31 and Ch.19

and

19.31.2		$\int_{{\mathbb{R}}^{n}}(\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x})^{\mu}\exp% \left(-\mathbf{x}^{\mathrm{T}}\mathbf{B}\mathbf{x}\right)\,\mathrm{d}x_{1}% \cdots\,\mathrm{d}x_{n}=\frac{\pi^{n/2}\Gamma\left(\mu+\tfrac{1}{2}n\right)}{% \sqrt{\det\mathbf{B}}\Gamma\left(\tfrac{1}{2}n\right)}R_{\mu}\left(\tfrac{1}{2% },\dots,\tfrac{1}{2};\lambda_{1},\dots,\lambda_{n}\right),$
$\mu>-\tfrac{1}{2}n$ .
ⓘ 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: TeX, pMML, png See also: Annotations for §19.31 and Ch.19

§19.16(iii) shows that for $n=3$ the incomplete cases of $R_{F}$ and $R_{G}$ occur when $\mu=-1/2$ and $\mu=1/2$ , respectively, while their complete cases occur when $n=2$ .

For (19.31.2) and generalizations see Carlson (1972b).

19.30 Lengths of Plane Curves 19.32 Conformal Map onto a Rectangle

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