diff --git a/docs/chapter2/chapter2.md b/docs/chapter2/chapter2.md index edd2fda..513a3a2 100644 --- a/docs/chapter2/chapter2.md +++ b/docs/chapter2/chapter2.md @@ -112,8 +112,10 @@ $$\begin{aligned} &=\sum_{i=C+1}^{m}\left(\begin{array}{c}{m} \\ {i}\end{array}\right) p_0^{i} (1-p_0)^{m-i} \end{aligned}$$ 对于此方程,通常不一定正好解得一个整数$C$使得方程成立,较常见的情况是存在这样一个$\overline{C}$使得 -$$\sum_{i=\overline{C}+1}^{m}\left(\begin{array}{c}{m} \\ {i}\end{array}\right) p_0^{i} (1-p_0)^{m-i}<\alpha \\ -\sum_{i=\overline{C}}^{m}\left(\begin{array}{c}{m} \\ {i}\end{array}\right) p_0^{i} (1-p_0)^{m-i}>\alpha$$ +$$\begin{aligned} +\sum_{i=\overline{C}+1}^{m}\left(\begin{array}{c}{m} \\ {i}\end{array}\right) p_0^{i} (1-p_0)^{m-i}<\alpha \\ +\sum_{i=\overline{C}}^{m}\left(\begin{array}{c}{m} \\ {i}\end{array}\right) p_0^{i} (1-p_0)^{m-i}>\alpha +\end{aligned}$$ 此时,$C$只能取$\overline{C}$或者$\overline{C}+1$,若$C$取$\overline{C}$,则相当于升高了检验水平$\alpha$,若$C$取$\overline{C}+1$则相当于降低了检验水平$\alpha$,具体如何取舍需要结合实际情况,但是通常为了减小犯第一类错误的概率,会倾向于令$C$取$\overline{C}+1$。下面考虑如何求解$\overline{C}$:易证$\beta_{\varphi}(p_0)$是关于$C$的减函数,所以再结合上述关于$\overline{C}$的两个不等式易推得 $$\overline{C}=\min C\quad\text { s.t. } \sum_{i=C+1}^{m}\left(\begin{array}{c}{m} \\ {i}\end{array}\right) p_0^{i}(1-p_0)^{m-i}<\alpha$$