双曲正割分布¶
与Logistic分布相关，并用于寿命分析。标准格式为(在所有情况下定义 \(x\) )
\BEGIN{eqnarray*}f\Left(x\Right)&=&\frac{1}{\pi}\mathm{sech}\Left(x\Right)\\
F\Left(x\Right)&=&\frac{2}{\pi}\arctan\Left(e^{x}\Right)\\
G\Left(Q\Right)&=&\log\left(\tan\left(\frac{\pi}{2}q\right)\right)\end{eqnarray*}
\[M\left(t\right)=\sec\left(\frac{\pi}{2}t\right)\]
\begin{eqnarray*} \mu_{n}^{\prime} & = & \frac{1+\left(-1\right)^{n}}{2\pi2^{2n}}n!\left[\zeta\left(n+1,\frac{1}{4}\right)-\zeta\left(n+1,\frac{3}{4}\right)\right]\\
& = & \left\{
\begin{array}{cc}
0 & n \text{ odd}\\
C_{n/2}\frac{\pi^{n}}{2^{n}} & n \text{ even}
\end{array}
\right.\end{eqnarray*}
哪里 \(C_{{m}}\) 是由给定的整数
\begin{eqnarray*} C_{m} & = & \frac{\left(2m\right)!\left[\zeta\left(2m+1,\frac{1}{4}\right)-\zeta\left(2m+1,\frac{3}{4}\right)\right]}{\pi^{2m+1}2^{2m}}\\
& = & 4\left(-1\right)^{m-1}\frac{16^{m}}{2m+1}B_{2m+1}\left(\frac{1}{4}\right)\end{eqnarray*}
哪里 \(B_{{2m+1}}\left(\frac{{1}}{{4}}\right)\) 是阶伯努利多项式吗？ \(2m+1\) 评估时间 \(1/4.\) 因此，
\[\begin{split}\MU_{n}^{\Prime}=\Left\{
\BEGIN{array}{cc}
0&n\文本{奇数}\\
文本{4\left(-1\right)^{n/2-1}\frac{\left(2\pi\right)^{n}}{n+1}B_{n+1}\left(\frac{1}{4}\right)}(&N\TEXT{EVEN})
\end{数组}
\对。\end{split}\]
\BEGIN{eqnarray*}m_{d}=m_{n}=\m&=&0\\
\MU_{2}&=&\frac{\pi^{2}}{4}\\
\Gamma_{1}&=&0\\
\Gamma_{2}&=&2\end{eqnarray*}
\[H\Left [X\right] =\log\Left(2\pi\Right)。\]