Β分布

Β分布，亦称貝它分布、Beta 分布（Beta distribution），在概率论中，是指一组定义在 $(0,1)$ 区间的连续概率分布，有两个母数 $\alpha ,\beta >0$ 。

定义

{\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 $\Gamma (z)$ 是Γ函数。如果 $n$ 為正整數，则有：

\Gamma (n)=(n-1)!

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 )

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

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

参数为 $\alpha ,\beta$ Β分布的众数是：

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

\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)}}

{\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 )^{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)}}

$k$ 阶矩是：

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

其中 $(x)_{k}$ 表示递进阶乘幂。 $k$ 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})

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的微分熵为：[2]

{\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 $\psi$ 表示双伽玛函数。

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 ).

Johnson, Norman L., Samuel Kotz, and N. Balakrishnan (1995). "Continuous Univariate Distributions, Vol. 2", Wiley, ISBN 978-0-471-58494-0.
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.

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密度函數
累積分布函數
	$\alpha >0$ $\beta >0$
值域	$x\in (0;1)\!$
	${\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\!$
累積分布函數	$I_{x}(\alpha ,\beta )\!$
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 $\alpha >1,\beta >1$
	${\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!$
偏度	${\frac {2\,(\beta -\alpha ){\sqrt {\alpha +\beta +1}}}{(\alpha +\beta +2){\sqrt {\alpha \beta }}}}$
峰度	见文字
熵	见文字
	$1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{k!}}$
特徵函数	${\displaystyle {}_{1}F_{1}(\alpha$ (见合流超几何函数)