# §19.31 Probability Distributions

$\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ and $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ 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: $n$: nonnegative integer Permalink: http://dlmf.nist.gov/19.31.E1 Encodings: TeX, pMML, png See also: Annotations for 19.31

and

 19.31.2 $\int_{{\mathbb{R}^{n}}}(\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x})^{\mu}% \mathop{\exp\/}\nolimits\!\left(-\mathbf{x}^{\mathrm{T}}\mathbf{B}\mathbf{x}% \right)\mathrm{d}x_{1}\cdots\mathrm{d}x_{n}=\frac{\pi^{n/2}\mathop{\Gamma\/}% \nolimits\!\left(\mu+\tfrac{1}{2}n\right)}{\sqrt{\det\mathbf{B}}\mathop{\Gamma% \/}\nolimits\!\left(\tfrac{1}{2}n\right)}\mathop{R_{\mu}\/}\nolimits\!\left(% \tfrac{1}{2},\dots,\tfrac{1}{2};\lambda_{1},\dots,\lambda_{n}\right),$ $\mu>-\tfrac{1}{2}n$.

§19.16(iii) shows that for $n=3$ the incomplete cases of $\mathop{R_{F}\/}\nolimits$ and $\mathop{R_{G}\/}\nolimits$ 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).