# distributions

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##### 1: 1.16 Distributions

###### §1.16 Distributions

… ► $\mathrm{\Lambda}:\mathcal{D}(I)\to \u2102$ is called a*distribution*if it is a continuous linear functional on $\mathcal{D}(I)$, that is, it is a linear functional and for every ${\varphi}_{n}\to \varphi $ in $\mathcal{D}(I)$, … ► … ► … ►

##### 2: 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). Particular forms are the chi-square distribution functions; see Johnson et al. (1994, pp. 415–493). 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: 1.1 Special Notation

##### 4: 24.18 Physical Applications

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►Bernoulli polynomials appear in statistical physics (Ordóñez and Driebe (1996)), in discussions of Casimir forces (Li et al. (1991)), and in a study of quark-gluon plasma (Meisinger et al. (2002)).
►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).

##### 5: 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

…##### 6: 32.14 Combinatorics

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►

32.14.1
$$\underset{N\to \mathrm{\infty}}{lim}\mathrm{Prob}\left(\frac{{\mathrm{\ell}}_{N}(\mathit{\pi})-2\sqrt{N}}{{N}^{1/6}}\le s\right)=F(s),$$

►where the *distribution function*$F(s)$ is defined here by ►
32.14.2
$$F(s)=\mathrm{exp}\left(-{\int}_{s}^{\mathrm{\infty}}(x-s){w}^{2}(x)dx\right),$$

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►The distribution function $F(s)$ given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of $n\times n$ Hermitian matrices; see Tracy and Widom (1994).
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##### 7: Sidebar 7.SB1: Diffraction from a Straightedge

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►The intensity distribution follows ${|\mathcal{F}\left(x\right)|}^{2}$, where $\mathcal{F}$ is the Fresnel integral (See 7.3.4).
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##### 8: Funding

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►The NIST DLMF project has been funded, in part, by the Knowledge & Distributed Intelligence Program of the National Science Foundation.
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##### 9: 35.9 Applications

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►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument ${}_{p}F_{q}$, with $p\le 2$ and $q\le 1$.
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►For other statistical applications of ${}_{p}F_{q}$ functions of matrix argument see Perlman and Olkin (1980), Groeneboom and Truax (2000), Bhaumik and Sarkar (2002), Richards (2004) (monotonicity of power functions of multivariate statistical test criteria), Bingham et al. (1992) (Procrustes analysis), and Phillips (1986) (exact distributions of statistical test criteria).
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►In the nascent area of applications of zonal polynomials to the limiting probability distributions of symmetric random matrices, one of the most comprehensive accounts is Rains (1998).

##### 10: Ingram Olkin

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►Olkin’s research covered a broad range of areas, including multivariate analysis, reliability theory, matrix theory, statistical models in the social and behavioral sciences, life distributions, and meta-analysis.
… Hedges), published by Academic Press in 1985, and

*Life Distributions: Non-Parametric, Semi-Parametric, and Parametric Families*(with A. …