我见过的最好的各统计分布之间关系的整理-经管之家官网！

扫码加入统计交流群

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:

http://www.johndcook.com/distribution_chart.gif

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.

「经管之家」APP：经管人学习、答疑、交友，就上经管之家！
免流量费下载资料----在经管之家app可以下载论坛上的所有资源，并且不额外收取下载高峰期的论坛币。
涵盖所有经管领域的优秀内容----覆盖经济、管理、金融投资、计量统计、数据分析、国贸、财会等专业的学习宝库，各类资料应有尽有。
来自五湖四海的经管达人----已经有上千万的经管人来到这里，你可以找到任何学科方向、有共同话题的朋友。
经管之家（原人大经济论坛），跨越高校的围墙，带你走进经管知识的新世界。
扫描下方二维码下载并注册APP