Β分布，亦称贝它分布、Beta 分布（Beta distribution），在概率论中，是指一组定义在 ( 0 , 1 ) {\displaystyle (0,1)} 区间的连续概率分布，有两个母数 α , β > 0 {\displaystyle \alpha ,\beta >0} 。 事实速览 参数, 值域 ...Β分布 概率密度函数 累积分布函数参数 α > 0 {\displaystyle \alpha >0} β > 0 {\displaystyle \beta >0} 值域 x ∈ ( 0 ; 1 ) {\displaystyle x\in (0;1)\!} 概率密度函数 x α − 1 ( 1 − x ) β − 1 B ( α , β ) {\displaystyle {\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\!} 累积分布函数 I x ( α , β ) {\displaystyle I_{x}(\alpha ,\beta )\!} 期望 E [ x ] = α α + β {\displaystyle \operatorname {E} [x]={\frac {\alpha }{\alpha +\beta }}\!} E [ ln x ] = ψ ( α ) − ψ ( α + β ) {\displaystyle \operatorname {E} [\ln x]=\psi (\alpha )-\psi (\alpha +\beta )\!} (见双伽玛函数)中位数 I 0.5 − 1 ( α , β ) {\displaystyle I_{0.5}^{-1}(\alpha ,\beta )} 无解析表达众数 α − 1 α + β − 2 {\displaystyle {\frac {\alpha -1}{\alpha +\beta -2}}\!} for α > 1 , β > 1 {\displaystyle \alpha >1,\beta >1} 方差 α β ( α + β ) 2 ( α + β + 1 ) {\displaystyle {\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!} 偏度 2 ( β − α ) α + β + 1 ( α + β + 2 ) α β {\displaystyle {\frac {2\,(\beta -\alpha ){\sqrt {\alpha +\beta +1}}}{(\alpha +\beta +2){\sqrt {\alpha \beta }}}}} 峰度 见文字熵 见文字矩生成函数 1 + ∑ k = 1 ∞ ( ∏ r = 0 k − 1 α + r α + β + r ) t k k ! {\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{k!}}} 特征函数 1 F 1 ( α ; α + β ; i t ) {\displaystyle {}_{1}F_{1}(\alpha ;\alpha +\beta ;i\,t)\!} (见合流超几何函数)关闭 定义 概率密度函数 Β分布的概率密度函数是： f ( x ; α , β ) = x α − 1 ( 1 − x ) β − 1 ∫ 0 1 u α − 1 ( 1 − u ) β − 1 d u = Γ ( α + β ) Γ ( α ) Γ ( β ) x α − 1 ( 1 − x ) β − 1 = 1 B ( α , β ) x α − 1 ( 1 − x ) β − 1 {\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 Γ ( z ) {\displaystyle \Gamma (z)} 是Γ函数。如果 n {\displaystyle n} 为正整数，则有： Γ ( n ) = ( n − 1 ) ! {\displaystyle \Gamma (n)=(n-1)!} 随机变量X服从参数为 α , β {\displaystyle \alpha ,\beta } 的Β分布通常写作 X ∼ Be ( α , β ) {\displaystyle 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 ; α , β ) = B x ( α , β ) B ( α , β ) = I x ( α , β ) {\displaystyle F(x;\alpha ,\beta )={\frac {\mathrm {B} _{x}(\alpha ,\beta )}{\mathrm {B} (\alpha ,\beta )}}=I_{x}(\alpha ,\beta )\!} 其中 B x ( α , β ) {\displaystyle \mathrm {B} _{x}(\alpha ,\beta )} 是不完全Β函数， I x ( α , β ) {\displaystyle I_{x}(\alpha ,\beta )} 是正则不完全贝塔函数。 性质 参数为 α , β {\displaystyle \alpha ,\beta } Β分布的众数是： α − 1 α + β − 2 {\displaystyle {\begin{aligned}{\frac {\alpha -1}{\alpha +\beta -2}}\\\end{aligned}}} [1] 期望和方差分别是： μ = E ( X ) = α α + β {\displaystyle \mu =\operatorname {E} (X)={\frac {\alpha }{\alpha +\beta }}} Var ( X ) = E ( X − μ ) 2 = α β ( α + β ) 2 ( α + β + 1 ) {\displaystyle \operatorname {Var} (X)=\operatorname {E} (X-\mu )^{2}={\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}} 偏度是： E ( X − μ ) 3 [ E ( X − μ ) 2 ] 3 / 2 = 2 ( β − α ) α + β + 1 ( α + β + 2 ) α β {\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： E ( X − μ ) 4 [ E ( X − μ ) 2 ] 2 − 3 = 6 [ α 3 − α 2 ( 2 β − 1 ) + β 2 ( β + 1 ) − 2 α β ( β + 2 ) ] α β ( α + β + 2 ) ( α + β + 3 ) {\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： 6 [ ( α − β ) 2 ( α + β + 1 ) − α β ( α + β + 2 ) ] α β ( α + β + 2 ) ( α + β + 3 ) {\displaystyle {\frac {6[(\alpha -\beta )^{2}(\alpha +\beta +1)-\alpha \beta (\alpha +\beta +2)]}{\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)}}} k {\displaystyle k} 阶矩是： E ( X k ) = B ( α + k , β ) B ( α , β ) = ( α ) k ( α + β ) k {\displaystyle \operatorname {E} (X^{k})={\frac {\operatorname {B} (\alpha +k,\beta )}{\operatorname {B} (\alpha ,\beta )}}={\frac {(\alpha )_{k}}{(\alpha +\beta )_{k}}}} 其中 ( x ) k {\displaystyle (x)_{k}} 表示递进阶乘幂。 k {\displaystyle k} 阶矩还可以递归地表示为： E ( X k ) = α + k − 1 α + β + k − 1 E ( X k − 1 ) {\displaystyle \operatorname {E} (X^{k})={\frac {\alpha +k-1}{\alpha +\beta +k-1}}\operatorname {E} (X^{k-1})} 另外， E ( log X ) = ψ ( α ) − ψ ( α + β ) {\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] h ( X ) = ln B ( α , β ) − ( α − 1 ) ψ ( α ) − ( β − 1 ) ψ ( β ) + ( α + β − 2 ) ψ ( α + β ) {\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 ψ {\displaystyle \psi } 表示双伽玛函数。 联合熵为： H ( X , Y ) = ln B ( α ′ , β ′ ) − ( α ′ − 1 ) ψ ( α ) − ( β ′ − 1 ) ψ ( β ) + ( α ′ + β ′ − 2 ) ψ ( α + β ) . {\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： D K L ( X , Y ) = ln B ( α ′ , β ′ ) B ( α , β ) − ( α ′ − α ) ψ ( α ) − ( β ′ − β ) ψ ( β ) + ( α ′ − α + β ′ − β ) ψ ( α + β ) . {\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 ).} 2 sources参见 概率论 概率分布 Β函数 外部链接 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） 参考文献 [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. Wikiwand in your browser!Seamless Wikipedia browsing. On steroids.Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.Wikiwand for ChromeWikiwand for EdgeWikiwand for Firefox
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![{\displaystyle \operatorname {E} [x]={\frac {\alpha }{\alpha +\beta }}\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/0569ee58528ca526f9cdab57675a2d0d73bf4766)
![{\displaystyle \operatorname {E} [\ln x]=\psi (\alpha )-\psi (\alpha +\beta )\!}](//wikimedia.org/api/rest_v1/media/math/render/svg/73a2d06fc2308f395e3dbaed6bb7d0b975d38eb1)
无解析表达
for 
(见合流超几何函数)
![{\displaystyle \operatorname {E} [x]={\frac {\alpha }{\alpha +\beta }}\!}](http://wikimedia.org/api/rest_v1/media/math/render/svg/0569ee58528ca526f9cdab57675a2d0d73bf4766)
![{\displaystyle \operatorname {E} [\ln x]=\psi (\alpha )-\psi (\alpha +\beta )\!}](http://wikimedia.org/api/rest_v1/media/math/render/svg/73a2d06fc2308f395e3dbaed6bb7d0b975d38eb1)
![{\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}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/835449e193daf41f7721dec385b81fb4a16375b2)
![{\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 }}}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/d12bd747cbf5cc3410db8103716da3f202eff5b7)
![{\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)}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/e70dc91216082cdf757ded4e3ab81c15418d8cb2)
![{\displaystyle {\frac {6[(\alpha -\beta )^{2}(\alpha +\beta +1)-\alpha \beta (\alpha +\beta +2)]}{\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/eea65a8d7c9e00ba6299b727eab679117776f41e)
