伽马分布的性质

数理统计中的重要分布.

　　概率密度函数：

　　　　　　　　 $p(x ; \alpha, \lambda)=\frac{\lambda^{\alpha}}{\Gamma(\alpha)} x^{\alpha-1} e^{-\lambda x}, \quad x>0$

　　分布函数的性质：

$\begin{aligned} F(x ; \alpha, \lambda) &=\int_{0}^{x} \frac{\lambda^{\alpha} y^{\alpha-1}}{\Gamma(\alpha)} e^{-\lambda y} d y=\int_{0}^{\lambda x} \frac{t^{\alpha-1}}{\Gamma(\alpha)} e^{-t} d t \\ &=\frac{G(\lambda x ; \alpha)}{\Gamma(\alpha)}=F(\lambda x ; \alpha, 1) \end{aligned}\\$

　　伽马分布的K阶矩：

$\begin{aligned} \mathrm{E} X^{k} &=\int_{0}^{\infty} \frac{\lambda^{\alpha} x^{k+\alpha-1}}{\Gamma(\alpha)} e^{-\lambda x} d x \\ &=\frac{\Gamma(k+\alpha)}{\lambda^{k} \Gamma(\alpha)} \int_{0}^{\infty} \frac{\lambda^{k+\alpha} x^{k+\alpha-1}}{\Gamma(k+\alpha)} e^{-\lambda x} d x \\ &=\frac{\Gamma(k+\alpha)}{\lambda^{k} \Gamma(\alpha)} \end{aligned}\\$

　　期望和方差：

$E(\Gamma(\alpha, \lambda))=\frac{\alpha}{\lambda}, \quad \operatorname{Var}(\Gamma(\alpha, \lambda))=\frac{\alpha}{\lambda^{2}}\\$

　　矩母函数:

$\begin{aligned} \mathrm{E} e^{t X} &=\int_{0}^{\infty} \frac{\lambda^{\alpha} x^{\alpha-1}}{\Gamma(\alpha)} e^{t x-\lambda x} d x \\ &=\left(\frac{\lambda}{\lambda-t}\right)^{\alpha} \int_{0}^{\infty} \frac{(\lambda-t)^{\alpha} x^{\alpha-1}}{\Gamma(\alpha)} e^{-(\lambda-t) x} d x \\ &=\left(1-\frac{t}{\lambda}\right)^{-\alpha}, \quad t<\lambda \end{aligned}\\$