Β分布

概率密度函数

Β分布的概率密度函数是：

${\displaystyle {\begin{aligned}f(x;\alpha ,\beta )&={\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\int _{0}^{1}u^{\alpha -1}(1-u)^{\beta -1}\,du}}\\[6pt]&={\frac {\Gamma (\alpha +\beta )}{\Gamma (\alpha )\Gamma (\beta )}}\,x^{\alpha -1}(1-x)^{\beta -1}\\[6pt]&={\frac {1}{\mathrm {B} (\alpha ,\beta )}}\,x^{\alpha -1}(1-x)^{\beta -1}\end{aligned}}}$ 

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

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

累积分布函数

Β分布的累积分布函数是：

${\displaystyle F(x;\alpha ,\beta )={\frac {\mathrm {B} _{x}(\alpha ,\beta )}{\mathrm {B} (\alpha ,\beta )}}=I_{x}(\alpha ,\beta )\!}$ 

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

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

${\displaystyle {\begin{aligned}{\frac {\alpha -1}{\alpha +\beta -2}}\\\end{aligned}}}$ ^[1]

期望值和方差分别是：

${\displaystyle \mu =\operatorname {E} (X)={\frac {\alpha }{\alpha +\beta }}}$ 

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

偏度是：

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

峰度是：

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

或：

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

${\displaystyle k}$ 阶矩是：

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

其中${\displaystyle (x)_{k}}$ 表示递进阶乘幂。${\displaystyle k}$ 阶矩还可以递归地表示为：

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

另外，

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

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：^[2]

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

其中${\displaystyle \psi }$ 表示双伽玛函数。

联合熵为：

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

其KL散度为：

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

• 概率论
• 概率分布
• Β函数

• Beta分布 （页面存档备份，存于互联网档案馆）
• LDA-math-认识Beta/Dirichlet分布 （页面存档备份，存于互联网档案馆）