伽马分布¶
伽马分布的标准形式是 \(\left(\alpha>0\right)\) 有效期为 \(x\geq0\) 。
\Begin{eqnarray*}f\Left(x；\Alpha\Right)&=&\frac{1}{\Gamma\left(\alpha\right)}x^{\alpha-1}e^{-x}\\
F\Left(x；\Alpha\Right)&=&\frac{\Gamma\Left(\Alpha，x\Right)}{\Gamma(\Alpha)}\\
g\Left(q；\alpha\right)&=&\Gamma^{-1}\Left(\alpha，Q\Gamma(\alpha)\Right)\end{等式*}
哪里 \(\gamma\) 是较低的不完全伽马函数， \(\gamma\left(s, x\right) = \int_0^x t^{{s-1}} e^{{-t}} dt\) 。
\[M\left(t\right)=\frac{1}{\left(1-t\right)^{\alpha}}\]
\BEGIN{eqnarray*}\mu&=&\alpha\\
\MU_{2}&=&\alpha\\
\Gamma_{1}&=&\frac{2}{\sqrt{\alpha}}\\
\Gamma_{2}&=&\frac{6}{\alpha}\\
M_{d}&=&\alpha-1\end{eqnarray*}
\[H\Left [X\right] =\PSI\Left(a\Right)\Left [1-a\right] +a+\log\Gamma\Left(a\Right)\]
哪里
\[\Psi\left(a\right)=\frac{\Gamma^{\prime}\left(a\right)}{\Gamma\left(a\right)}.\]