distribution functions

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1: 1.16 Distributions

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§1.16(i) Test Functions

… ► … ►A mapping $\mathrm{\Lambda }:𝒟(I)\to ℂ$ is a linear functional if …$\mathrm{\Lambda }:𝒟(I)\to ℂ$ is called a distribution, or generalized function, if it is a continuous linear functional on $𝒟(I)$, that is, it is a linear functional and for every ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►

§1.16(iv) Heaviside Function

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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: 32.14 Combinatorics

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

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Symbols:

$𝝅(m)$: permutations, ${\mathrm{\ell }}_N(𝝅)$: length and $F(s)$: distribution function

Permalink:

http://dlmf.nist.gov/32.14.E1

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Annotations for §32.14 and Ch.32

►where the distribution function $F(s)$ is defined here by ►

32.14.2 $$F(s)=\exp \left(-{\int }_s^{\mathrm{\infty }}(x-s)w^2(x)dx\right),$$

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $x$: real and $F(s)$: distribution function

Referenced by:

§32.14

Permalink:

http://dlmf.nist.gov/32.14.E2

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Annotations for §32.14 and Ch.32

… ►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). …

4: 7.20 Mathematical Applications

… ►The normal distribution function with mean $m$ and standard deviation $\sigma$ is given by …For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).

5: 1.1 Special Notation

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$x,y$	real variables.
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$⟨\mathrm{\Lambda },\varphi ⟩$	action of distribution $\mathrm{\Lambda }$ on test function $\varphi$.
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6: 13.22 Zeros

§13.22 Zeros

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7: 25.10 Zeros

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§25.10(i) Distribution

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8: 2.6 Distributional Methods

… ►To assign a distribution to the function $f_n(t)$, we first let $f_{n,n}(t)$ denote the $n$th repeated integral (§1.4(v)) of $f_n$: … ►We have now assigned a distribution to each function in (2.6.10). … ►On replacing the distributions by their corresponding functions, (2.6.43) and (2.6.44) give … ►

2.6.58 $${\int }_0^{\mathrm{\infty }}{t}^{\lambda }dt,$$ $\lambda \in ℝ$.

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Symbols:

$dx$: differential of $x$ , $\in$: element of, $\int$: integral and $ℝ$: real line

Permalink:

http://dlmf.nist.gov/2.6.E58

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

►However, in the theory of generalized functions (distributions), there is a method, known as “regularization”, by which these integrals can be interpreted in a meaningful manner. …

9: 13.9 Zeros

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§13.9(i) Zeros of $M(a,b,z)$

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10: 12.11 Zeros

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§12.11(i) Distribution of Real Zeros

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