| 分布类型 | 概率密度函数 | 基尼系数 | | Lognormal | $\begin{aligned}f(x ; \mu, \sigma) & =\frac{1}{\sqrt{2 \pi} \sigma x} \exp \left[-\frac{1}{2 \sigma^2}(\ln x-\mu)^2\right], x>0 \\& =0 \text {, 否则 }\end{aligned} $, | $ 2 \Phi\left(\frac{\sigma}{\sqrt{2}}\right)-1$ | | Gamma | $f(x ; \alpha, \beta)=\frac{x^{\alpha-1} \exp \left(-\frac{x}{\beta}\right)}{\beta^\alpha \Gamma(\alpha)}, x>0, \alpha>0, \beta>0 $ | $\frac{\Gamma(\alpha+0.5)}{\Gamma(\alpha+1) \sqrt{\pi}} $ | | Weibull | $ f(x ; \alpha, \beta)=\frac{\alpha}{\beta}\left(\frac{x}{\beta}\right)^{\alpha-1} \exp \left(\left(-\frac{x}{\beta}\right)^\alpha\right)$ | $1-\left(\frac{1}{2}\right)^\alpha $ | | SM | $\begin{aligned}f(x ; a, b, q) & =\frac{a x^{a-1}}{b^a B(1, q)\left(1+\left(\frac{x}{b}\right)\right)^{1+q}}, x \geq 0 \\& =0, \text { 否则 }\end{aligned} $ | $1-\frac{\Gamma(q) \Gamma\left(2 q-\frac{1}{a}\right)}{\Gamma\left(q-\frac{1}{a}\right) \Gamma(2 q)} $ | | Beta2 | $\begin{aligned}f(x ; b, p, q) & =\frac{x^{p-1}}{b^p B(p, q)\left(1+\left(\frac{x}{b}\right)\right)^{p+q}}, x \geq 0 \\& =0, \text { 否则 }\end{aligned} $ | $\frac{2 B(2 p, 2 q-1)}{p B^2(p, q)} $ | | Dagum | $\begin{aligned}f(x ; a, b, p) & =\frac{a x^{a p-1}}{b^{a p} B(p, 1)\left(1+\left(\frac{x}{b}\right)^a\right)^{p+1}}, x \geq 0 \\& =0, \text { 否则 }\end{aligned} $ | $-1+\frac{\Gamma(p) \Gamma\left(2 p+\frac{1}{a}\right)}{\Gamma\left(p+\frac{1}{a}\right) \Gamma(2 p)} $ | | GB2 | $\begin{aligned}f(x ; a, b, p, q) & =\frac{a x^{a p-1}}{b^{a p} B(p, q)\left(1+\left(\frac{x}{b}\right)^a\right)^{p+q}}, x \geq 0 \\& =0, \text { 否则 }\end{aligned} $ | $\begin{aligned}& \frac{B\left(2 p-\frac{1}{a}, 2 q+\frac{1}{a}\right)}{B(p, q) B\left(p+\frac{1}{a}, q-\frac{1}{a}\right)}\left\{(\frac { 1 } { p }) _ { 3 } F _ { 2 } \left[1, p+q, 2 q+\frac{1}{a} ;\right.\right. \\& p+1, 2(p+q) ; 1]-\left(\frac{1}{p+\frac{1}{a}}\right)_3 F_2\left[1, p+q, 2 p+\frac{1}{a} ;\right. \\& \left.\left.p+\frac{1}{a}+1, 2(p+q) ; 1\right]\right\}\end{aligned} $ | | Pareto | $\begin{aligned}f\left(x ; x_{\min }, k\right) & =k \frac{x_{\min }^k}{x^{k+1}}, x \geq x_{\min } \\& =0, \text { 否则 }\end{aligned} $ | $\frac{k}{2 k-1} $ |
|