Β分布

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Β分布
Probability density function for the Beta distribution
概率密度函數
Cumulative distribution function for the Beta distribution
累積分佈函數
參數 \alpha > 0
\beta > 0
支撑集 x \in (0; 1)\!
概率密度函數 \frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)}\!
累積分佈函數 I_x(\alpha,\beta)\!
期望值 \operatorname{E}[x] = \frac{\alpha}{\alpha+\beta}\!
\operatorname{E}[\ln x] = \psi(\alpha) - \psi(\alpha + \beta)\!
(见双伽玛函数)
中位數 I_{0.5}^{-1}(\alpha,\beta) 无解析表达
眾數 \frac{\alpha-1}{\alpha+\beta-2}\! for \alpha>1, \beta>1
方差 \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\!
偏度 \frac{2\,(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}
峰度 见文字
信息熵 见文字
動差生成函數 1  +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} \right) \frac{t^k}{k!}
特性函数 {}_1F_1(\alpha; \alpha+\beta; i\,t)\! (见合流超几何函数)

概率论中,Β分布也称贝塔分布,是指一组定义在(0,1)区间的连续概率分布,有两个参数\alpha, \beta>0

定义[编辑]

概率密度函数[编辑]

Β分布的概率密度函数是:


\begin{align}
f(x;\alpha,\beta) & = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{\int_0^1 u^{\alpha-1} (1-u)^{\beta-1}\, du} \\[6pt]
& = \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\, x^{\alpha-1}(1-x)^{\beta-1} \\[6pt]
& = \frac{1}{\mathrm{B}(\alpha,\beta)}\, x
^{\alpha-1}(1-x)^{\beta-1}
\end{align}

其中\Gamma(z)Γ函数。随机变量X服从参数为\alpha, \beta的Β分布通常写作

X \sim \textrm{Be}(\alpha, \beta)

累积分布函数[编辑]

Β分布的累积分布函数是:

F(x;\alpha,\beta) = \frac{\mathrm{B}_x(\alpha,\beta)}{\mathrm{B}(\alpha,\beta)} = I_x(\alpha,\beta) \!

其中\mathrm{B}_x(\alpha,\beta)不完全Β函数I_x(\alpha,\beta)正则不完全贝塔函数

性质[编辑]

参数为\alpha, \betaΒ分布的众数是:

\begin{align}
 \frac{\alpha - 1}{\alpha + \beta - 2} \\
\end{align}[1]

期望值方差分别是:

 \mu = \operatorname{E}(X) = \frac{\alpha}{\alpha + \beta}
 \operatorname{Var}(X) = \operatorname{E}(X - \mu)^2 = \frac{\alpha \beta}{(\alpha + \beta)^2(\alpha + \beta + 1)}

偏度是:

\frac{\operatorname{E}(X - \mu)^3}{[\operatorname{E}(X - \mu)^2]^{3/2}} = \frac{2 (\beta - \alpha) \sqrt{\alpha + \beta + 1} }
        {(\alpha + \beta + 2) \sqrt{\alpha \beta}}

峰度是:

\frac{\operatorname{E}(X - \mu)^4}{[\operatorname{E}(X - \mu)^2]^{2}}-3 = \frac{6[\alpha^3-\alpha^2(2\beta - 1) + \beta^2(\beta + 1) - 2\alpha\beta(\beta + 2)]}
{\alpha \beta (\alpha + \beta + 2) (\alpha + \beta + 3)}

或:

\frac{6[(\alpha - \beta)^2 (\alpha +\beta + 1) - \alpha \beta (\alpha + \beta + 2)]}
{\alpha \beta (\alpha + \beta + 2) (\alpha + \beta + 3)}

k是:

\operatorname{E}(X^k) = \frac{\operatorname{B}(\alpha + k, \beta)}{\operatorname{B}(\alpha,\beta)} = \frac{(\alpha)_{k}}{(\alpha + \beta)_{k}}

其中(x)_{k}表示下降阶乘幂k还可以递归地表示为:

\operatorname{E}(X^k) = \frac{\alpha + k - 1}{\alpha + \beta + k - 1}\operatorname{E}(X^{k - 1})

另外,

\operatorname{E}(\log X) = \psi(\alpha) - \psi(\alpha + \beta)

给定两个Β分布随机变量, X ~ Beta(α, β) and Y ~ Beta(α', β'), X微分熵为:[2]

\begin{align}
  h(X) &= \ln\mathrm{B}(\alpha,\beta)-(\alpha-1)\psi(\alpha)-(\beta-1)\psi(\beta)+(\alpha+\beta-2)\psi(\alpha+\beta)
\end{align}

其中\psi表示双伽玛函数

联合熵为:

H(X,Y) = \ln\mathrm{B}(\alpha',\beta')-(\alpha'-1)\psi(\alpha)-(\beta'-1)\psi(\beta)+(\alpha'+\beta'-2)\psi(\alpha+\beta).\,

KL散度为:


 D_{\mathrm{KL}}(X,Y) = \ln\frac{\mathrm{B}(\alpha',\beta')}
                                {\mathrm{B}(\alpha,\beta)} -
                        (\alpha'-\alpha)\psi(\alpha) - (\beta'-\beta)\psi(\beta) +
                        (\alpha'-\alpha+\beta'-\beta)\psi(\alpha+\beta).

參見[编辑]

外部連接[编辑]

参考文献[编辑]

  1. ^ Johnson, Norman L., Samuel Kotz, and N. Balakrishnan (1995). "Continuous Univariate Distributions, Vol. 2", Wiley, ISBN 978-0-471-58494-0.
  2. ^ A. C. G. Verdugo Lazo and P. N. Rathie. "On the entropy of continuous probability distributions," IEEE Trans. Inf. Theory, IT-24:120–122,1978.