베타분포 (Beta Distribution)

$$ X \sim Be(\alpha , \beta)$$

• $$ \alpha \in ( \ 0 \ , \ \infty \ ) $$
• $$ \beta \in ( \ 0 \ , \ \infty \ ) $$

$$ x \in [ \ 0 \ , \ 1 \ ] $$

$$f(x)= \left[ \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} \right] x^{a-1}(1-x)^{\beta-1} $$

(1) 그래프 : $\alpha \ \mathrm{or} \ \beta \ = \ 1$

(2) 그래프 : $\alpha \ = \ \beta \ < \ 1$

(3) 그래프 : $\alpha \ = \ \beta > 1$

(4) 그래프 : $1 \ < \ \alpha \ < \ \beta \ \mathrm{or} \ 1 \ < \ \beta \ < \ \alpha$

$$ F(x) = I( \ x \ ; \ \alpha \ , \ \beta \ ) $$

• 단, $I( \ x \ ; \ \alpha \ , \ \beta \ )$는 정칙 베타함수이다.

(1) 그래프 : $\alpha \ \mathrm{or} \ \beta \ = \ 1$

(2) 그래프 : $\alpha \ = \ \beta \ < \ 1$

(3) 그래프 : $\alpha \ = \ \beta > 1$

(4) 그래프 : $1 \ < \ \alpha \ < \ \beta \ \mathrm{or} \ 1 \ < \ \beta \ < \ \alpha$

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

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

$$ \gamma_{1} = \frac{2(\beta - \alpha) \sqrt{1 + \alpha + \beta}}{\sqrt{\alpha \beta} (2 + \alpha + \beta)} $$

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

$$ \phi \ (t) = \ _{1}F_{1} (\alpha \ ; \ \alpha + \beta \ ; \ i \cdot t) $$

$$ \mu'_{k} = \frac{\Gamma (\alpha + \beta) \cdot \Gamma (\alpha + k)}{\Gamma (\alpha + \beta + k) \cdot \Gamma (\alpha)} $$

$$ \mu_{k} = \left( - \frac{\alpha}{\alpha + \beta} \right)^{k} \ _{2}F_{1} \left( -k \ , \ \alpha \ ; \ \alpha + \beta \ ; \ \frac{\alpha + \beta}{\alpha} \right) $$

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