Β分布

Β分佈，亦稱貝它分佈、Beta 分佈（Beta distribution），在機率論中，是指一組定義在${\displaystyle (0,1)}$區間的連續機率分佈，有兩個母數${\displaystyle \alpha ,\beta >0}$。

Β分佈

機率密度函數
累積分佈函數
參數	${\displaystyle \alpha >0}$ ${\displaystyle \beta >0}$
值域	${\displaystyle x\in (0;1)\!}$
機率密度函數	${\displaystyle {\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\!}$
累積分佈函數	${\displaystyle I_{x}(\alpha ,\beta )\!}$
期望值	${\displaystyle \operatorname {E} [x]={\frac {\alpha }{\alpha +\beta }}\!}$ ${\displaystyle \operatorname {E} [\ln x]=\psi (\alpha )-\psi (\alpha +\beta )\!}$ (見雙伽瑪函數)
中位數	${\displaystyle I_{0.5}^{-1}(\alpha ,\beta )}$ 無解析表達
眾數	${\displaystyle {\frac {\alpha -1}{\alpha +\beta -2}}\!}$ for ${\displaystyle \alpha >1,\beta >1}$
變異數	${\displaystyle {\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!}$
偏度	${\displaystyle {\frac {2\,(\beta -\alpha ){\sqrt {\alpha +\beta +1}}}{(\alpha +\beta +2){\sqrt {\alpha \beta }}}}}$
峰度	見文字
熵	見文字
動差母函數	${\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{k!}}}$
特徵函數	${\displaystyle {}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!}$ (見合流超幾何函數)

定義

機率密度函數

Β分佈的機率密度函數是：

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

其中${\displaystyle \Gamma (z)}$是Γ函數。如果${\displaystyle n}$為正整數，則有：

${\displaystyle \Gamma (n)=(n-1)!}$

隨機變量X服從參數為${\displaystyle \alpha ,\beta }$的Β分佈通常寫作

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

累積分佈函數

Β分佈的累積分佈函數是：

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

其中${\displaystyle \mathrm {B} _{x}(\alpha ,\beta )}$是不完全Β函數，${\displaystyle I_{x}(\alpha ,\beta )}$是正則不完全貝塔函數。

性質

參數為${\displaystyle \alpha ,\beta }$Β分佈的眾數是：

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

期望值和方差分別是：

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

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

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：

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

${\displaystyle k}$階矩是：

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

其中${\displaystyle (x)_{k}}$表示遞進階乘冪。${\displaystyle k}$階矩還可以遞歸地表示為：

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

另外，

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

給定兩個Β分佈隨機變量, X ~ Beta(α, β) and Y ~ Beta(α', β'), X的微分熵為：^[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${\displaystyle \psi }$表示雙伽瑪函數。

聯合熵為：

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

其KL散度為：

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

參見

• 機率論
• 機率分佈
• Β函數

外部連結

• Beta分佈 （頁面存檔備份，存於互聯網檔案館）
• LDA-math-認識Beta/Dirichlet分佈 （頁面存檔備份，存於互聯網檔案館）

參考文獻

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.