# Β分布

參數 概率密度函數 累積分佈函數 $\alpha > 0$ $\beta > 0$ $x \in (0; 1)\!$ $\frac{x^{\alpha-1}(1-x)^{\beta-1}} {\mathrm{B}(\alpha,\beta)}\!$ $I_x(\alpha,\beta)\!$ $\operatorname{E}[x] = \frac{\alpha}{\alpha+\beta}\!$ $\operatorname{E}[\ln x] = \psi(\alpha) - \psi(\alpha + \beta)\!$ (见双伽玛函数) $I_{0.5}^{-1}(\alpha,\beta)$ 无解析表达 $\frac{\alpha-1}{\alpha+\beta-2}\!$ for $\alpha>1, \beta>1$ $\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}\!$ $\frac{2\,(\beta-\alpha)\sqrt{\alpha+\beta+1}}{(\alpha+\beta+2)\sqrt{\alpha\beta}}$ 见文字 见文字 $1 +\sum_{k=1}^{\infty} \left( \prod_{r=0}^{k-1} \frac{\alpha+r}{\alpha+\beta+r} \right) \frac{t^k}{k!}$ ${}_1F_1(\alpha; \alpha+\beta; i\,t)\!$ (见合流超几何函数)

## 定义

### 概率密度函数

Β分布的概率密度函数是：

\begin{align} 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{align}

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

### 累积分布函数

Β分布的累积分布函数是：

$F(x;\alpha,\beta) = \frac{\mathrm{B}_x(\alpha,\beta)}{\mathrm{B}(\alpha,\beta)} = I_x(\alpha,\beta) \!$

## 性质

\begin{align} \frac{\alpha - 1}{\alpha + \beta - 2} \\ \end{align}[1]

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

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

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

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

$k$是：

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

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

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

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

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

KL散度为：

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

## 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.