Logistic(二次方)分布¶
广义Logistic分布的一个特例 \(c=1.\) 定义为 \(x\geq0\)
\Begin{eqnarray*}f\Left(x\Right)&=&\frac{\exp\left(-x\right)}{\left(1+\exp\left(-x\right)\right)^{2}}\\
F\Left(x\Right)&=&\frac{1}{1+\exp\Left(-x\Right)}\\
G\LEFT(Q\RIGHT)&=&-\LOG\LEFT(1/Q-1\RIGHT)\end{eqnarray*}
\BEGIN{eqnarray*}\mu&=&\Gamma+\psi_{0}\Left(1\Right)=0\\
\MU_{2}&=&\frac{\pi^{2}}{6}+\psi_{1}\left(1\right)=\frac{\pi^{2}}{3}\\
\Gamma_{1}&=&\frac{\psi_{2}\left(1\right)+2\zeta\left(3\right)}{\mu_{2}^{3/2}}=0\\
\Gamma_{2}&=&\frac{\left(\frac{\pi^{4}}{15}+\psi_{3}\left(1\right)\right)}{\mu_{2}^{2}}=\frac{6}{5}\\
M_{d}&=&\log1=0\\
m_{n}&=&-\log\Left(2-1\Right)=0\end{eqnarray*}
哪里 \(\psi_m\) 是多伽马函数 \(\psi_m(z) = \frac{{d^{{m+1}}}}{{dz^{{m+1}}}} \log(\Gamma(z))\) 。
\[H\Left [X\right] =1。\]