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# 我见过的最好的各统计分布之间关系的整理   [推广有奖]

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Diagram of distribution relationships

Probability distributions have a surprising number inter-connections. A dashed line in the chart below indicates an approximate (limit) relationship between two distribution families. A solid line indicates an exact relationship: special case, sum, or transformation.

Click on a distribution for the parameterization of that distribution. Click on an arrow for details on the relationship represented by the arrow. Other diagrams on this site:

The chart above is adapted from the chart originally published by Lawrence Leemis in 1986 (Relationships Among Common Univariate Distributions, American Statistician 40:143-146.) Leemis published a larger chart in 2008 which is available online.

Parameterizations

The precise relationships between distributions depend on parameterization. The relationships detailed below depend on the following parameterizations for the PDFs.

Let C(n, k) denote the binomial coefficient(n, k) and B(a, b) = Γ(a) Γ(b) / Γ(a + b).

Geometric: f(x) = p (1-p)x for non-negative integers x.

Discrete uniform: f(x) = 1/n for x = 1, 2, ..., n.

Negative binomial: f(x) = C(r + x - 1, x) pr(1-p)x for non-negative integers x. See notes on the negative binomial distribution.

Beta binomial: f(x) = C(n, x) B(α + x, n + β - x) / B(α, β) for x = 0, 1, ..., n.

Hypergeometric: f(x) = C(M, x) C(N-M, K - x) / C(N, K) for x = 0, 1, ..., N.

Poisson: f(x) = exp(-λ) λx/ x! for non-negative integers x. The parameter λ is both the mean and the variance.

Binomial: f(x) = C(n, x) px(1 - p)n-x for x = 0, 1, ..., n.

Bernoulli: f(x) = px(1 - p)1-x where x = 0 or 1.

Lognormal: f(x) = (2πσ2)-1/2 exp( -(log(x) - μ)2/ 2σ2) / x for positive x. Note that μ and σ2are not the mean and variance of the distribution.

Normal : f(x) = (2π σ2)-1/2 exp( - ½((x - μ)/σ)2 ) for all x.

Beta: f(x) = Γ(α + β) xα-1(1 - x)β-1 / (Γ(α) Γ(β)) for 0 ≤ x ≤ 1.

Standard normal: f(x) = (2π)-1/2 exp( -x2/2) for all x.

Chi-squared: f(x) = x-ν/2-1 exp(-x/2) / Γ(ν/2) 2ν/2 for positive x. The parameter ν is called the degrees of freedom.

Gamma: f(x) = β-α xα-1 exp(-x/β) / Γ(α) for positive x. The parameter α is called the shape and β is the scale.

Uniform: f(x) = 1 for 0 ≤ x ≤ 1.

Cauchy: f(x) = σ/(π( (x - μ)2 + σ2) ) for all x. Note that μ and σ are location and scale parameters. The Cauchy distribution has no mean or variance.

Snedecor F: f(x) is proportional to x(ν1 - 2)/2 / (1 + (ν1/ν2) x)(ν1 + ν2)/2 for positive x.

Exponential: f(x) = exp(-x/μ)/μ for positive x. The parameter μ is the mean.

Student t: f(x) is proportional to (1 + (x2/ν))-(ν + 1)/2 for positive x. The parameter ν is called the degrees of freedom.

Weibull: f(x) = (γ/β) xγ-1 exp(- xγ/β) for positive x. The parameter γ is the shape and β is the scale.

Double exponential : f(x) = exp(-|x-μ|/σ) / 2σ for all x. The parameter μ is the location and mean; σ is the scale.

For comparison, see distribution parameterizations in R/S-PLUS and Mathematica.

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### 本帖被以下文库推荐

 这么专业的东西，我只能说：不明觉厉啊！
hangtian8881   发表于 2014-3-10 13:33:33 |显示全部楼层 |坛友微信交流群
 必须赞一下，回来一定好好研究

 这是一篇文章的，一个是1986年的，另一个是2008年更新的 Univariate Distribution Relationships http://www.math.wm.edu/~leemis/2008amstat.pdf

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 蓝色 发表于 2014-3-10 14:00 这是一篇文章的 Univariate Distribution Relationships http://www.math.wm.edu/~leemis/2008amstat.pdf ...蓝版主果然厉害。pdf里的那个图更震撼……

 蓝色 发表于 2014-3-10 14:00 这是一篇文章的 Univariate Distribution Relationships http://www.math.wm.edu/~leemis/2008amstat.pdf ...是的，蓝色版主，在图下第一段已说明。我觉得这样帖子的形式，在文中加入超链接，学习起来感觉更方便，就分享给大家了。版主厉害。
Crsky7 发表于 2014-3-10 14:16:18 |显示全部楼层 |坛友微信交流群
 这个整理不错！我记得以前看到有本书专门介绍各种分布的，像概率分布百科全书一样。
Crsky7 发表于 2014-3-10 14:18:39 |显示全部楼层 |坛友微信交流群
 还看到过有本书专门讲统计检验的，各种各样的检验都有，像统计检验百科全书一样。

 Crsky7 发表于 2014-3-10 14:18 还看到过有本书专门讲统计检验的，各种各样的检验都有，像统计检验百科全书一样。Crsky7版主，你可以回忆一下是哪本书，给大家分享一下呀。
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