Β分布

Β分布

概率密度函數
累積分佈函數
參數	$\alpha >0$$\alpha >0$ $\beta >0$$\beta >0$
支撑集	$x\in (0;1)\!$$x\in (0;1)\!$
概率密度函数	${\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\!$${\frac {x^{{\alpha -1}}(1-x)^{{\beta -1}}}{{\mathrm {B}}(\alpha ,\beta )}}\!$
累積分佈函數	$I_{x}(\alpha ,\beta )\!$$I_{x}(\alpha ,\beta )\!$
期望值	$\operatorname {E} [x]={\frac {\alpha }{\alpha +\beta }}\!$$\operatorname {E}[x]={\frac {\alpha }{\alpha +\beta }}\!$ $\operatorname {E} [\ln x]=\psi (\alpha )-\psi (\alpha +\beta )\!$$\operatorname {E}[\ln x]=\psi (\alpha )-\psi (\alpha +\beta )\!$ (见双伽玛函数)
中位數	$I_{0.5}^{-1}(\alpha ,\beta )$$I_{{0.5}}^{{-1}}(\alpha ,\beta )$ 无解析表达
眾數	${\frac {\alpha -1}{\alpha +\beta -2}}\!$${\frac {\alpha -1}{\alpha +\beta -2}}\!$ for $\alpha >1,\beta >1$$\alpha >1,\beta >1$
方差	${\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!$${\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!$
偏度	${\frac {2\,(\beta -\alpha ){\sqrt {\alpha +\beta +1}}}{(\alpha +\beta +2){\sqrt {\alpha \beta }}}}$${\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!}}$$1+\sum _{{k=1}}^{{\infty }}\left(\prod _{{r=0}}^{{k-1}}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{k!}}$
特性函数	${}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!$${}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!$ (见合流超几何函数)

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

定义

概率密度函数

Β分布的概率密度函数是：

${\begin{aligned}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{aligned}}$${\begin{aligned}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{aligned}}$

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

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

累积分布函数

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 )\!$$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 )$${\mathrm {B}}_{x}(\alpha ,\beta )$是不完全Β函数，$I_{x}(\alpha ,\beta )$$I_{x}(\alpha ,\beta )$是正则不完全贝塔函数。

性质

参数为$\alpha ,\beta$$\alpha ,\beta$Β分布的众数是：

${\begin{aligned}{\frac {\alpha -1}{\alpha +\beta -2}}\\\end{aligned}}$${\begin{aligned}{\frac {\alpha -1}{\alpha +\beta -2}}\\\end{aligned}}$

期望值和方差分别是：

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

$k$$k$阶矩是：

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

其中$(x)_{k}$$(x)_{{k}}$表示下降阶乘幂。$k$$k$阶矩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})$$\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 )$$\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的微分熵为：

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

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

其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 ).$$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

• 概率论
• 機率分佈
• Β函数

外部連結

• Beta分佈
• LDA-math-认识Beta/Dirichlet分布