Β分布
概率密度函數 ![Probability density function for the Beta distribution](//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Beta_distribution_pdf.svg/325px-Beta_distribution_pdf.svg.png) |
累積分布函數 ![Cumulative distribution function for the Beta distribution](//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Beta_distribution_cdf.svg/325px-Beta_distribution_cdf.svg.png) |
参数 |
![{\displaystyle \beta >0}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/4a87dc52878418173659e6d0ff8e77ab2897eac9) |
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值域 |
![{\displaystyle x\in (0;1)\!}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/36d0ef7a9136c64d36b1b8e2bcee82c3f8ad7d5f) |
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概率密度函数 |
![{\displaystyle {\frac {x^{\alpha -1}(1-x)^{\beta -1}}{\mathrm {B} (\alpha ,\beta )}}\!}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/125fdaa41844a8703d1a8610ac00fbf3edacc8e7) |
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累積分布函數 |
![{\displaystyle I_{x}(\alpha ,\beta )\!}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/630767808887e1bd81c51a75934e8a196907bb93) |
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期望值 |
![{\displaystyle \operatorname {E} [x]={\frac {\alpha }{\alpha +\beta }}\!}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/0569ee58528ca526f9cdab57675a2d0d73bf4766)
![{\displaystyle \operatorname {E} [\ln x]=\psi (\alpha )-\psi (\alpha +\beta )\!}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/73a2d06fc2308f395e3dbaed6bb7d0b975d38eb1) (见双伽玛函数) |
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中位數 |
无解析表达 |
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眾數 |
for ![{\displaystyle \alpha >1,\beta >1}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/c2f1e017514157062ecc289c4042d17d99a1b77f) |
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方差 |
![{\displaystyle {\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}\!}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e5f3678db794b6d247e588b602bf565763dcb462) |
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偏度 |
![{\displaystyle {\frac {2\,(\beta -\alpha ){\sqrt {\alpha +\beta +1}}}{(\alpha +\beta +2){\sqrt {\alpha \beta }}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/43ec71817c032c8eb21b5feadd0ec9b91c747530) |
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峰度 |
见文字 |
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熵 |
见文字 |
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矩生成函数 |
![{\displaystyle 1+\sum _{k=1}^{\infty }\left(\prod _{r=0}^{k-1}{\frac {\alpha +r}{\alpha +\beta +r}}\right){\frac {t^{k}}{k!}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/b97b0e33f3134c2fc5c484016ab8e03e18d85481) |
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特徵函数 |
(见合流超几何函数) |
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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
区间的连续概率分布,有两个母数
。
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}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/835449e193daf41f7721dec385b81fb4a16375b2)
其中
是Γ函数。如果
為正整數,则有:
![{\displaystyle \Gamma (n)=(n-1)!}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/ef3f7eebd96f717c5f1fd154b3905af7fbcabf24)
随机变量X服从参数为
的Β分布通常写作
![{\displaystyle X\sim {\textrm {Be}}(\alpha ,\beta )}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/695ff6cf8cf9fb8433885c7116a0e4210fd457f8)
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 )\!}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e8cd1de9a12c65b8dd50df0f55568f9dd2312ac1)
其中
是不完全Β函数,
是正则不完全贝塔函数。
参数为
Β分布的众数是:
[1]
期望值和方差分别是:
![{\displaystyle \mu =\operatorname {E} (X)={\frac {\alpha }{\alpha +\beta }}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/dfdc0647128d7a401f049184fb71c5f9d3d27bbc)
![{\displaystyle \operatorname {Var} (X)=\operatorname {E} (X-\mu )^{2}={\frac {\alpha \beta }{(\alpha +\beta )^{2}(\alpha +\beta +1)}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/26a16cedc4b6bd697421d537ba5a85a198459dd3)
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 }}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/d12bd747cbf5cc3410db8103716da3f202eff5b7)
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)}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e70dc91216082cdf757ded4e3ab81c15418d8cb2)
或:
![{\displaystyle {\frac {6[(\alpha -\beta )^{2}(\alpha +\beta +1)-\alpha \beta (\alpha +\beta +2)]}{\alpha \beta (\alpha +\beta +2)(\alpha +\beta +3)}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/eea65a8d7c9e00ba6299b727eab679117776f41e)
阶矩是:
![{\displaystyle \operatorname {E} (X^{k})={\frac {\operatorname {B} (\alpha +k,\beta )}{\operatorname {B} (\alpha ,\beta )}}={\frac {(\alpha )_{k}}{(\alpha +\beta )_{k}}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/fbb4593a4b0a10930996c4233503bd2016cb3bbf)
其中
表示递进阶乘幂。
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})}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/924897ddc4941fe182cd240ab6023f1f0c46ae08)
另外,
![{\displaystyle \operatorname {E} (\log X)=\psi (\alpha )-\psi (\alpha +\beta )}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/c904f6828fe9f20c3e7561e3f379088e564e446a)
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}}}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/57c9a03b4e2fa6afdcc9812981979d7565ea8ab0)
其中
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 ).\,}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/574eafdd4451b313a4d7f5e56e6abf15a0ebf6e9)
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 ).}](https://backend.710302.xyz/https/wikimedia.org/api/rest_v1/media/math/render/svg/e92997734eccc39180e7a84414398c0a0b9aabd9)
- ^ 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.